diff --git a/.cirrus.yml b/.cirrus.yml index 9399fbd..506a860 100644 --- a/.cirrus.yml +++ b/.cirrus.yml @@ -1,6 +1,5 @@ env: WIDEMUL: auto - BIGNUM: auto STATICPRECOMPUTATION: yes ECMULTGENPRECISION: auto ASM: no @@ -59,9 +58,8 @@ task: - env: {WIDEMUL: int128, RECOVERY: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes} - env: {WIDEMUL: int128, ECDH: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes} - env: {WIDEMUL: int128, ASM: x86_64} - - env: {BIGNUM: no} - - env: {BIGNUM: no, RECOVERY: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes} - - env: {BIGNUM: no, STATICPRECOMPUTATION: no} + - env: { RECOVERY: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes} + - env: { STATICPRECOMPUTATION: no} - env: {BUILD: distcheck, WITH_VALGRIND: no, CTIMETEST: no, BENCH: no} - env: {CPPFLAGS: -DDETERMINISTIC} - env: {CFLAGS: -O0, CTIMETEST: no} @@ -69,7 +67,6 @@ task: CFLAGS: "-fsanitize=undefined -fno-omit-frame-pointer" LDFLAGS: "-fsanitize=undefined -fno-omit-frame-pointer" UBSAN_OPTIONS: "print_stacktrace=1:halt_on_error=1" - BIGNUM: no ASM: x86_64 ECDH: yes RECOVERY: yes @@ -80,7 +77,6 @@ task: - env: { ECMULTGENPRECISION: 8 } - env: RUN_VALGRIND: yes - BIGNUM: no ASM: x86_64 ECDH: yes RECOVERY: yes @@ -115,12 +111,6 @@ task: CC: i686-linux-gnu-gcc - env: CC: clang --target=i686-pc-linux-gnu -isystem /usr/i686-linux-gnu/include - matrix: - - env: - BIGNUM: gmp - - env: - BIGNUM: no - << : *MERGE_BASE test_script: - ./ci/cirrus.sh << : *CAT_LOGS @@ -178,7 +168,7 @@ task: # If we haven't restored from cached (and just run brew install), this is a no-op. - brew link valgrind brew_script: - - brew install automake libtool gmp gcc@9 + - brew install automake libtool gcc@9 << : *MERGE_BASE test_script: - ./ci/cirrus.sh @@ -195,7 +185,6 @@ task: HOST: s390x-linux-gnu BUILD: WITH_VALGRIND: no - BIGNUM: no ECDH: yes RECOVERY: yes EXPERIMENTAL: yes diff --git a/Makefile.am b/Makefile.am index 023fa60..58c9635 100644 --- a/Makefile.am +++ b/Makefile.am @@ -14,8 +14,6 @@ noinst_HEADERS += src/scalar_8x32_impl.h noinst_HEADERS += src/scalar_low_impl.h noinst_HEADERS += src/group.h noinst_HEADERS += src/group_impl.h -noinst_HEADERS += src/num_gmp.h -noinst_HEADERS += src/num_gmp_impl.h noinst_HEADERS += src/ecdsa.h noinst_HEADERS += src/ecdsa_impl.h noinst_HEADERS += src/eckey.h @@ -26,14 +24,16 @@ noinst_HEADERS += src/ecmult_const.h noinst_HEADERS += src/ecmult_const_impl.h noinst_HEADERS += src/ecmult_gen.h noinst_HEADERS += src/ecmult_gen_impl.h -noinst_HEADERS += src/num.h -noinst_HEADERS += src/num_impl.h noinst_HEADERS += src/field_10x26.h noinst_HEADERS += src/field_10x26_impl.h noinst_HEADERS += src/field_5x52.h noinst_HEADERS += src/field_5x52_impl.h noinst_HEADERS += src/field_5x52_int128_impl.h noinst_HEADERS += src/field_5x52_asm_impl.h +noinst_HEADERS += src/modinv32.h +noinst_HEADERS += src/modinv32_impl.h +noinst_HEADERS += src/modinv64.h +noinst_HEADERS += src/modinv64_impl.h noinst_HEADERS += src/assumptions.h noinst_HEADERS += src/util.h noinst_HEADERS += src/scratch.h diff --git a/README.md b/README.md index 9918678..197a56f 100644 --- a/README.md +++ b/README.md @@ -34,11 +34,11 @@ Implementation details * Optimized implementation of arithmetic modulo the curve's field size (2^256 - 0x1000003D1). * Using 5 52-bit limbs (including hand-optimized assembly for x86_64, by Diederik Huys). * Using 10 26-bit limbs (including hand-optimized assembly for 32-bit ARM, by Wladimir J. van der Laan). - * Field inverses and square roots using a sliding window over blocks of 1s (by Peter Dettman). * Scalar operations * Optimized implementation without data-dependent branches of arithmetic modulo the curve's order. * Using 4 64-bit limbs (relying on __int128 support in the compiler). * Using 8 32-bit limbs. +* Modular inverses (both field elements and scalars) based on [safegcd](https://gcd.cr.yp.to/index.html) with some modifications, and a variable-time variant (by Peter Dettman). * Group operations * Point addition formula specifically simplified for the curve equation (y^2 = x^3 + 7). * Use addition between points in Jacobian and affine coordinates where possible. diff --git a/build-aux/m4/bitcoin_secp.m4 b/build-aux/m4/bitcoin_secp.m4 index 7b48a5e..e57888c 100644 --- a/build-aux/m4/bitcoin_secp.m4 +++ b/build-aux/m4/bitcoin_secp.m4 @@ -75,19 +75,6 @@ if test x"$has_libcrypto" = x"yes" && test x"$has_openssl_ec" = x; then fi ]) -dnl -AC_DEFUN([SECP_GMP_CHECK],[ -if test x"$has_gmp" != x"yes"; then - CPPFLAGS_TEMP="$CPPFLAGS" - CPPFLAGS="$GMP_CPPFLAGS $CPPFLAGS" - LIBS_TEMP="$LIBS" - LIBS="$GMP_LIBS $LIBS" - AC_CHECK_HEADER(gmp.h,[AC_CHECK_LIB(gmp, __gmpz_init,[has_gmp=yes; GMP_LIBS="$GMP_LIBS -lgmp"; AC_DEFINE(HAVE_LIBGMP,1,[Define this symbol if libgmp is installed])])]) - CPPFLAGS="$CPPFLAGS_TEMP" - LIBS="$LIBS_TEMP" -fi -]) - AC_DEFUN([SECP_VALGRIND_CHECK],[ if test x"$has_valgrind" != x"yes"; then CPPFLAGS_TEMP="$CPPFLAGS" diff --git a/ci/cirrus.sh b/ci/cirrus.sh index f223a91..f26ca98 100755 --- a/ci/cirrus.sh +++ b/ci/cirrus.sh @@ -14,7 +14,7 @@ valgrind --version || true ./configure \ --enable-experimental="$EXPERIMENTAL" \ - --with-test-override-wide-multiply="$WIDEMUL" --with-bignum="$BIGNUM" --with-asm="$ASM" \ + --with-test-override-wide-multiply="$WIDEMUL" --with-asm="$ASM" \ --enable-ecmult-static-precomputation="$STATICPRECOMPUTATION" --with-ecmult-gen-precision="$ECMULTGENPRECISION" \ --enable-module-ecdh="$ECDH" --enable-module-recovery="$RECOVERY" \ --enable-module-schnorrsig="$SCHNORRSIG" \ diff --git a/ci/linux-debian.Dockerfile b/ci/linux-debian.Dockerfile index 201ace4..5967cf8 100644 --- a/ci/linux-debian.Dockerfile +++ b/ci/linux-debian.Dockerfile @@ -8,6 +8,6 @@ RUN apt-get update RUN apt-get install --no-install-recommends --no-upgrade -y \ git ca-certificates \ make automake libtool pkg-config dpkg-dev valgrind qemu-user \ - gcc clang libc6-dbg libgmp-dev \ - gcc-i686-linux-gnu libc6-dev-i386-cross libc6-dbg:i386 libgmp-dev:i386 \ + gcc clang libc6-dbg \ + gcc-i686-linux-gnu libc6-dev-i386-cross libc6-dbg:i386 \ gcc-s390x-linux-gnu libc6-dev-s390x-cross libc6-dbg:s390x diff --git a/configure.ac b/configure.ac index fd15d34..e84005e 100644 --- a/configure.ac +++ b/configure.ac @@ -48,17 +48,12 @@ case $host_os in # in expected paths because they may conflict with system files. Ask # Homebrew where each one is located, then adjust paths accordingly. openssl_prefix=`$BREW --prefix openssl 2>/dev/null` - gmp_prefix=`$BREW --prefix gmp 2>/dev/null` valgrind_prefix=`$BREW --prefix valgrind 2>/dev/null` if test x$openssl_prefix != x; then PKG_CONFIG_PATH="$openssl_prefix/lib/pkgconfig:$PKG_CONFIG_PATH" export PKG_CONFIG_PATH CRYPTO_CPPFLAGS="-I$openssl_prefix/include" fi - if test x$gmp_prefix != x; then - GMP_CPPFLAGS="-I$gmp_prefix/include" - GMP_LIBS="-L$gmp_prefix/lib" - fi if test x$valgrind_prefix != x; then VALGRIND_CPPFLAGS="-I$valgrind_prefix/include" fi @@ -164,9 +159,6 @@ AC_ARG_ENABLE(external_default_callbacks, # Legal values are int64 (for [u]int64_t), int128 (for [unsigned] __int128), and auto (the default). AC_ARG_WITH([test-override-wide-multiply], [] ,[set_widemul=$withval], [set_widemul=auto]) -AC_ARG_WITH([bignum], [AS_HELP_STRING([--with-bignum=gmp|no|auto], -[bignum implementation to use [default=auto]])],[req_bignum=$withval], [req_bignum=auto]) - AC_ARG_WITH([asm], [AS_HELP_STRING([--with-asm=x86_64|arm|no|auto], [assembly optimizations to use (experimental: arm) [default=auto]])],[req_asm=$withval], [req_asm=auto]) @@ -245,32 +237,6 @@ else esac fi -if test x"$req_bignum" = x"auto"; then - SECP_GMP_CHECK - if test x"$has_gmp" = x"yes"; then - set_bignum=gmp - fi - - if test x"$set_bignum" = x; then - set_bignum=no - fi -else - set_bignum=$req_bignum - case $set_bignum in - gmp) - SECP_GMP_CHECK - if test x"$has_gmp" != x"yes"; then - AC_MSG_ERROR([gmp bignum explicitly requested but libgmp not available]) - fi - ;; - no) - ;; - *) - AC_MSG_ERROR([invalid bignum implementation selection]) - ;; - esac -fi - # Select assembly optimization use_external_asm=no @@ -308,24 +274,6 @@ auto) ;; esac -# Select bignum implementation -case $set_bignum in -gmp) - AC_DEFINE(HAVE_LIBGMP, 1, [Define this symbol if libgmp is installed]) - AC_DEFINE(USE_NUM_GMP, 1, [Define this symbol to use the gmp implementation for num]) - AC_DEFINE(USE_FIELD_INV_NUM, 1, [Define this symbol to use the num-based field inverse implementation]) - AC_DEFINE(USE_SCALAR_INV_NUM, 1, [Define this symbol to use the num-based scalar inverse implementation]) - ;; -no) - AC_DEFINE(USE_NUM_NONE, 1, [Define this symbol to use no num implementation]) - AC_DEFINE(USE_FIELD_INV_BUILTIN, 1, [Define this symbol to use the native field inverse implementation]) - AC_DEFINE(USE_SCALAR_INV_BUILTIN, 1, [Define this symbol to use the native scalar inverse implementation]) - ;; -*) - AC_MSG_ERROR([invalid bignum implementation]) - ;; -esac - # Set ecmult window size if test x"$req_ecmult_window" = x"auto"; then set_ecmult_window=15 @@ -390,11 +338,6 @@ else enable_openssl_tests=no fi -if test x"$set_bignum" = x"gmp"; then - SECP_LIBS="$SECP_LIBS $GMP_LIBS" - SECP_INCLUDES="$SECP_INCLUDES $GMP_CPPFLAGS" -fi - if test x"$enable_valgrind" = x"yes"; then SECP_INCLUDES="$SECP_INCLUDES $VALGRIND_CPPFLAGS" fi @@ -571,7 +514,6 @@ echo " module extrakeys = $enable_module_extrakeys" echo " module schnorrsig = $enable_module_schnorrsig" echo echo " asm = $set_asm" -echo " bignum = $set_bignum" echo " ecmult window size = $set_ecmult_window" echo " ecmult gen prec. bits = $set_ecmult_gen_precision" # Hide test-only options unless they're used. diff --git a/doc/safegcd_implementation.md b/doc/safegcd_implementation.md new file mode 100644 index 0000000..8346d22 --- /dev/null +++ b/doc/safegcd_implementation.md @@ -0,0 +1,750 @@ +# The safegcd implementation in libsecp256k1 explained + +This document explains the modular inverse implementation in the `src/modinv*.h` files. It is based +on the paper +["Fast constant-time gcd computation and modular inversion"](https://gcd.cr.yp.to/papers.html#safegcd) +by Daniel J. Bernstein and Bo-Yin Yang. The references below are for the Date: 2019.04.13 version. + +The actual implementation is in C of course, but for demonstration purposes Python3 is used here. +Most implementation aspects and optimizations are explained, except those that depend on the specific +number representation used in the C code. + +## 1. Computing the Greatest Common Divisor (GCD) using divsteps + +The algorithm from the paper (section 11), at a very high level, is this: + +```python +def gcd(f, g): + """Compute the GCD of an odd integer f and another integer g.""" + assert f & 1 # require f to be odd + delta = 1 # additional state variable + while g != 0: + assert f & 1 # f will be odd in every iteration + if delta > 0 and g & 1: + delta, f, g = 1 - delta, g, (g - f) // 2 + elif g & 1: + delta, f, g = 1 + delta, f, (g + f) // 2 + else: + delta, f, g = 1 + delta, f, (g ) // 2 + return abs(f) +``` + +It computes the greatest common divisor of an odd integer *f* and any integer *g*. Its inner loop +keeps rewriting the variables *f* and *g* alongside a state variable *δ* that starts at *1*, until +*g=0* is reached. At that point, *|f|* gives the GCD. Each of the transitions in the loop is called a +"division step" (referred to as divstep in what follows). + +For example, *gcd(21, 14)* would be computed as: +- Start with *δ=1 f=21 g=14* +- Take the third branch: *δ=2 f=21 g=7* +- Take the first branch: *δ=-1 f=7 g=-7* +- Take the second branch: *δ=0 f=7 g=0* +- The answer *|f| = 7*. + +Why it works: +- Divsteps can be decomposed into two steps (see paragraph 8.2 in the paper): + - (a) If *g* is odd, replace *(f,g)* with *(g,g-f)* or (f,g+f), resulting in an even *g*. + - (b) Replace *(f,g)* with *(f,g/2)* (where *g* is guaranteed to be even). +- Neither of those two operations change the GCD: + - For (a), assume *gcd(f,g)=c*, then it must be the case that *f=a c* and *g=b c* for some integers *a* + and *b*. As *(g,g-f)=(b c,(b-a)c)* and *(f,f+g)=(a c,(a+b)c)*, the result clearly still has + common factor *c*. Reasoning in the other direction shows that no common factor can be added by + doing so either. + - For (b), we know that *f* is odd, so *gcd(f,g)* clearly has no factor *2*, and we can remove + it from *g*. +- The algorithm will eventually converge to *g=0*. This is proven in the paper (see theorem G.3). +- It follows that eventually we find a final value *f'* for which *gcd(f,g) = gcd(f',0)*. As the + gcd of *f'* and *0* is *|f'|* by definition, that is our answer. + +Compared to more [traditional GCD algorithms](https://en.wikipedia.org/wiki/Euclidean_algorithm), this one has the property of only ever looking at +the low-order bits of the variables to decide the next steps, and being easy to make +constant-time (in more low-level languages than Python). The *δ* parameter is necessary to +guide the algorithm towards shrinking the numbers' magnitudes without explicitly needing to look +at high order bits. + +Properties that will become important later: +- Performing more divsteps than needed is not a problem, as *f* does not change anymore after *g=0*. +- Only even numbers are divided by *2*. This means that when reasoning about it algebraically we + do not need to worry about rounding. +- At every point during the algorithm's execution the next *N* steps only depend on the bottom *N* + bits of *f* and *g*, and on *δ*. + + +## 2. From GCDs to modular inverses + +We want an algorithm to compute the inverse *a* of *x* modulo *M*, i.e. the number a such that *a x=1 +mod M*. This inverse only exists if the GCD of *x* and *M* is *1*, but that is always the case if *M* is +prime and *0 < x < M*. In what follows, assume that the modular inverse exists. +It turns out this inverse can be computed as a side effect of computing the GCD by keeping track +of how the internal variables can be written as linear combinations of the inputs at every step +(see the [extended Euclidean algorithm](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm)). +Since the GCD is *1*, such an algorithm will compute numbers *a* and *b* such that a x + b M = 1*. +Taking that expression *mod M* gives *a x mod M = 1*, and we see that *a* is the modular inverse of *x +mod M*. + +A similar approach can be used to calculate modular inverses using the divsteps-based GCD +algorithm shown above, if the modulus *M* is odd. To do so, compute *gcd(f=M,g=x)*, while keeping +track of extra variables *d* and *e*, for which at every step *d = f/x (mod M)* and *e = g/x (mod M)*. +*f/x* here means the number which multiplied with *x* gives *f mod M*. As *f* and *g* are initialized to *M* +and *x* respectively, *d* and *e* just start off being *0* (*M/x mod M = 0/x mod M = 0*) and *1* (*x/x mod M += 1*). + +```python +def div2(M, x): + """Helper routine to compute x/2 mod M (where M is odd).""" + assert M & 1 + if x & 1: # If x is odd, make it even by adding M. + x += M + # x must be even now, so a clean division by 2 is possible. + return x // 2 + +def modinv(M, x): + """Compute the inverse of x mod M (given that it exists, and M is odd).""" + assert M & 1 + delta, f, g, d, e = 1, M, x, 0, 1 + while g != 0: + # Note that while division by two for f and g is only ever done on even inputs, this is + # not true for d and e, so we need the div2 helper function. + if delta > 0 and g & 1: + delta, f, g, d, e = 1 - delta, g, (g - f) // 2, e, div2(M, e - d) + elif g & 1: + delta, f, g, d, e = 1 + delta, f, (g + f) // 2, d, div2(M, e + d) + else: + delta, f, g, d, e = 1 + delta, f, (g ) // 2, d, div2(M, e ) + # Verify that the invariants d=f/x mod M, e=g/x mod M are maintained. + assert f % M == (d * x) % M + assert g % M == (e * x) % M + assert f == 1 or f == -1 # |f| is the GCD, it must be 1 + # Because of invariant d = f/x (mod M), 1/x = d/f (mod M). As |f|=1, d/f = d*f. + return (d * f) % M +``` + +Also note that this approach to track *d* and *e* throughout the computation to determine the inverse +is different from the paper. There (see paragraph 12.1 in the paper) a transition matrix for the +entire computation is determined (see section 3 below) and the inverse is computed from that. +The approach here avoids the need for 2x2 matrix multiplications of various sizes, and appears to +be faster at the level of optimization we're able to do in C. + + +## 3. Batching multiple divsteps + +Every divstep can be expressed as a matrix multiplication, applying a transition matrix *(1/2 t)* +to both vectors *[f, g]* and *[d, e]* (see paragraph 8.1 in the paper): + +``` + t = [ u, v ] + [ q, r ] + + [ out_f ] = (1/2 * t) * [ in_f ] + [ out_g ] = [ in_g ] + + [ out_d ] = (1/2 * t) * [ in_d ] (mod M) + [ out_e ] [ in_e ] +``` + +where *(u, v, q, r)* is *(0, 2, -1, 1)*, *(2, 0, 1, 1)*, or *(2, 0, 0, 1)*, depending on which branch is +taken. As above, the resulting *f* and *g* are always integers. + +Performing multiple divsteps corresponds to a multiplication with the product of all the +individual divsteps' transition matrices. As each transition matrix consists of integers +divided by *2*, the product of these matrices will consist of integers divided by *2N* (see also +theorem 9.2 in the paper). These divisions are expensive when updating *d* and *e*, so we delay +them: we compute the integer coefficients of the combined transition matrix scaled by *2N*, and +do one division by *2N* as a final step: + +```python +def divsteps_n_matrix(delta, f, g): + """Compute delta and transition matrix t after N divsteps (multiplied by 2^N).""" + u, v, q, r = 1, 0, 0, 1 # start with identity matrix + for _ in range(N): + if delta > 0 and g & 1: + delta, f, g, u, v, q, r = 1 - delta, g, (g - f) // 2, 2*q, 2*r, q-u, r-v + elif g & 1: + delta, f, g, u, v, q, r = 1 + delta, f, (g + f) // 2, 2*u, 2*v, q+u, r+v + else: + delta, f, g, u, v, q, r = 1 + delta, f, (g ) // 2, 2*u, 2*v, q , r + return delta, (u, v, q, r) +``` + +As the branches in the divsteps are completely determined by the bottom *N* bits of *f* and *g*, this +function to compute the transition matrix only needs to see those bottom bits. Furthermore all +intermediate results and outputs fit in *(N+1)*-bit numbers (unsigned for *f* and *g*; signed for *u*, *v*, +*q*, and *r*) (see also paragraph 8.3 in the paper). This means that an implementation using 64-bit +integers could set *N=62* and compute the full transition matrix for 62 steps at once without any +big integer arithmetic at all. This is the reason why this algorithm is efficient: it only needs +to update the full-size *f*, *g*, *d*, and *e* numbers once every *N* steps. + +We still need functions to compute: + +``` + [ out_f ] = (1/2^N * [ u, v ]) * [ in_f ] + [ out_g ] ( [ q, r ]) [ in_g ] + + [ out_d ] = (1/2^N * [ u, v ]) * [ in_d ] (mod M) + [ out_e ] ( [ q, r ]) [ in_e ] +``` + +Because the divsteps transformation only ever divides even numbers by two, the result of *t [f,g]* is always even. When *t* is a composition of *N* divsteps, it follows that the resulting *f* +and *g* will be multiple of *2N*, and division by *2N* is simply shifting them down: + +```python +def update_fg(f, g, t): + """Multiply matrix t/2^N with [f, g].""" + u, v, q, r = t + cf, cg = u*f + v*g, q*f + r*g + # (t / 2^N) should cleanly apply to [f,g] so the result of t*[f,g] should have N zero + # bottom bits. + assert cf % 2**N == 0 + assert cg % 2**N == 0 + return cf >> N, cg >> N +``` + +The same is not true for *d* and *e*, and we need an equivalent of the `div2` function for division by *2N mod M*. +This is easy if we have precomputed *1/M mod 2N* (which always exists for odd *M*): + +```python +def div2n(M, Mi, x): + """Compute x/2^N mod M, given Mi = 1/M mod 2^N.""" + assert (M * Mi) % 2**N == 1 + # Find a factor m such that m*M has the same bottom N bits as x. We want: + # (m * M) mod 2^N = x mod 2^N + # <=> m mod 2^N = (x / M) mod 2^N + # <=> m mod 2^N = (x * Mi) mod 2^N + m = (Mi * x) % 2**N + # Subtract that multiple from x, cancelling its bottom N bits. + x -= m * M + # Now a clean division by 2^N is possible. + assert x % 2**N == 0 + return (x >> N) % M + +def update_de(d, e, t, M, Mi): + """Multiply matrix t/2^N with [d, e], modulo M.""" + u, v, q, r = t + cd, ce = u*d + v*e, q*d + r*e + return div2n(M, Mi, cd), div2n(M, Mi, ce) +``` + +With all of those, we can write a version of `modinv` that performs *N* divsteps at once: + +```python3 +def modinv(M, Mi, x): + """Compute the modular inverse of x mod M, given Mi=1/M mod 2^N.""" + assert M & 1 + delta, f, g, d, e = 1, M, x, 0, 1 + while g != 0: + # Compute the delta and transition matrix t for the next N divsteps (this only needs + # (N+1)-bit signed integer arithmetic). + delta, t = divsteps_n_matrix(delta, f % 2**N, g % 2**N) + # Apply the transition matrix t to [f, g]: + f, g = update_fg(f, g, t) + # Apply the transition matrix t to [d, e]: + d, e = update_de(d, e, t, M, Mi) + return (d * f) % M +``` + +This means that in practice we'll always perform a multiple of *N* divsteps. This is not a problem +because once *g=0*, further divsteps do not affect *f*, *g*, *d*, or *e* anymore (only *δ* keeps +increasing). For variable time code such excess iterations will be mostly optimized away in +section 6. + + +## 4. Avoiding modulus operations + +So far, there are two places where we compute a remainder of big numbers modulo *M*: at the end of +`div2n` in every `update_de`, and at the very end of `modinv` after potentially negating *d* due to the +sign of *f*. These are relatively expensive operations when done generically. + +To deal with the modulus operation in `div2n`, we simply stop requiring *d* and *e* to be in range +*[0,M)* all the time. Let's start by inlining `div2n` into `update_de`, and dropping the modulus +operation at the end: + +```python +def update_de(d, e, t, M, Mi): + """Multiply matrix t/2^N with [d, e] mod M, given Mi=1/M mod 2^N.""" + u, v, q, r = t + cd, ce = u*d + v*e, q*d + r*e + # Cancel out bottom N bits of cd and ce. + md = -((Mi * cd) % 2**N) + me = -((Mi * ce) % 2**N) + cd += md * M + ce += me * M + # And cleanly divide by 2**N. + return cd >> N, ce >> N +``` + +Let's look at bounds on the ranges of these numbers. It can be shown that *|u|+|v|* and *|q|+|r|* +never exceed *2N* (see paragraph 8.3 in the paper), and thus a multiplication with *t* will have +outputs whose absolute values are at most *2N* times the maximum absolute input value. In case the +inputs *d* and *e* are in *(-M,M)*, which is certainly true for the initial values *d=0* and *e=1* assuming +*M > 1*, the multiplication results in numbers in range *(-2NM,2NM)*. Subtracting less than *2N* +times *M* to cancel out *N* bits brings that up to *(-2N+1M,2NM)*, and +dividing by *2N* at the end takes it to *(-2M,M)*. Another application of `update_de` would take that +to *(-3M,2M)*, and so forth. This progressive expansion of the variables' ranges can be +counteracted by incrementing *d* and *e* by *M* whenever they're negative: + +```python + ... + if d < 0: + d += M + if e < 0: + e += M + cd, ce = u*d + v*e, q*d + r*e + # Cancel out bottom N bits of cd and ce. + ... +``` + +With inputs in *(-2M,M)*, they will first be shifted into range *(-M,M)*, which means that the +output will again be in *(-2M,M)*, and this remains the case regardless of how many `update_de` +invocations there are. In what follows, we will try to make this more efficient. + +Note that increasing *d* by *M* is equal to incrementing *cd* by *u M* and *ce* by *q M*. Similarly, +increasing *e* by *M* is equal to incrementing *cd* by *v M* and *ce* by *r M*. So we could instead write: + +```python + ... + cd, ce = u*d + v*e, q*d + r*e + # Perform the equivalent of incrementing d, e by M when they're negative. + if d < 0: + cd += u*M + ce += q*M + if e < 0: + cd += v*M + ce += r*M + # Cancel out bottom N bits of cd and ce. + md = -((Mi * cd) % 2**N) + me = -((Mi * ce) % 2**N) + cd += md * M + ce += me * M + ... +``` + +Now note that we have two steps of corrections to *cd* and *ce* that add multiples of *M*: this +increment, and the decrement that cancels out bottom bits. The second one depends on the first +one, but they can still be efficiently combined by only computing the bottom bits of *cd* and *ce* +at first, and using that to compute the final *md*, *me* values: + +```python +def update_de(d, e, t, M, Mi): + """Multiply matrix t/2^N with [d, e], modulo M.""" + u, v, q, r = t + md, me = 0, 0 + # Compute what multiples of M to add to cd and ce. + if d < 0: + md += u + me += q + if e < 0: + md += v + me += r + # Compute bottom N bits of t*[d,e] + M*[md,me]. + cd, ce = (u*d + v*e + md*M) % 2**N, (q*d + r*e + me*M) % 2**N + # Correct md and me such that the bottom N bits of t*[d,e] + M*[md,me] are zero. + md -= (Mi * cd) % 2**N + me -= (Mi * ce) % 2**N + # Do the full computation. + cd, ce = u*d + v*e + md*M, q*d + r*e + me*M + # And cleanly divide by 2**N. + return cd >> N, ce >> N +``` + +One last optimization: we can avoid the *md M* and *me M* multiplications in the bottom bits of *cd* +and *ce* by moving them to the *md* and *me* correction: + +```python + ... + # Compute bottom N bits of t*[d,e]. + cd, ce = (u*d + v*e) % 2**N, (q*d + r*e) % 2**N + # Correct md and me such that the bottom N bits of t*[d,e]+M*[md,me] are zero. + # Note that this is not the same as {md = (-Mi * cd) % 2**N} etc. That would also result in N + # zero bottom bits, but isn't guaranteed to be a reduction of [0,2^N) compared to the + # previous md and me values, and thus would violate our bounds analysis. + md -= (Mi*cd + md) % 2**N + me -= (Mi*ce + me) % 2**N + ... +``` + +The resulting function takes *d* and *e* in range *(-2M,M)* as inputs, and outputs values in the same +range. That also means that the *d* value at the end of `modinv` will be in that range, while we want +a result in *[0,M)*. To do that, we need a normalization function. It's easy to integrate the +conditional negation of *d* (based on the sign of *f*) into it as well: + +```python +def normalize(sign, v, M): + """Compute sign*v mod M, where v is in range (-2*M,M); output in [0,M).""" + assert sign == 1 or sign == -1 + # v in (-2*M,M) + if v < 0: + v += M + # v in (-M,M). Now multiply v with sign (which can only be 1 or -1). + if sign == -1: + v = -v + # v in (-M,M) + if v < 0: + v += M + # v in [0,M) + return v +``` + +And calling it in `modinv` is simply: + +```python + ... + return normalize(f, d, M) +``` + + +## 5. Constant-time operation + +The primary selling point of the algorithm is fast constant-time operation. What code flow still +depends on the input data so far? + +- the number of iterations of the while *g ≠ 0* loop in `modinv` +- the branches inside `divsteps_n_matrix` +- the sign checks in `update_de` +- the sign checks in `normalize` + +To make the while loop in `modinv` constant time it can be replaced with a constant number of +iterations. The paper proves (Theorem 11.2) that *741* divsteps are sufficient for any *256*-bit +inputs, and [safegcd-bounds](https://github.com/sipa/safegcd-bounds) shows that the slightly better bound *724* is +sufficient even. Given that every loop iteration performs *N* divsteps, it will run a total of +*⌈724/N⌉* times. + +To deal with the branches in `divsteps_n_matrix` we will replace them with constant-time bitwise +operations (and hope the C compiler isn't smart enough to turn them back into branches; see +`valgrind_ctime_test.c` for automated tests that this isn't the case). To do so, observe that a +divstep can be written instead as (compare to the inner loop of `gcd` in section 1). + +```python + x = -f if delta > 0 else f # set x equal to (input) -f or f + if g & 1: + g += x # set g to (input) g-f or g+f + if delta > 0: + delta = -delta + f += g # set f to (input) g (note that g was set to g-f before) + delta += 1 + g >>= 1 +``` + +To convert the above to bitwise operations, we rely on a trick to negate conditionally: per the +definition of negative numbers in two's complement, (*-v == ~v + 1*) holds for every number *v*. As +*-1* in two's complement is all *1* bits, bitflipping can be expressed as xor with *-1*. It follows +that *-v == (v ^ -1) - (-1)*. Thus, if we have a variable *c* that takes on values *0* or *-1*, then +*(v ^ c) - c* is *v* if *c=0* and *-v* if *c=-1*. + +Using this we can write: + +```python + x = -f if delta > 0 else f +``` + +in constant-time form as: + +```python + c1 = (-delta) >> 63 + # Conditionally negate f based on c1: + x = (f ^ c1) - c1 +``` + +To use that trick, we need a helper mask variable *c1* that resolves the condition *δ>0* to *-1* +(if true) or *0* (if false). We compute *c1* using right shifting, which is equivalent to dividing by +the specified power of *2* and rounding down (in Python, and also in C under the assumption of a typical two's complement system; see +`assumptions.h` for tests that this is the case). Right shifting by *63* thus maps all +numbers in range *[-263,0)* to *-1*, and numbers in range *[0,263)* to *0*. + +Using the facts that *x&0=0* and *x&(-1)=x* (on two's complement systems again), we can write: + +```python + if g & 1: + g += x +``` + +as: + +```python + # Compute c2=0 if g is even and c2=-1 if g is odd. + c2 = -(g & 1) + # This masks out x if g is even, and leaves x be if g is odd. + g += x & c2 +``` + +Using the conditional negation trick again we can write: + +```python + if g & 1: + if delta > 0: + delta = -delta +``` + +as: + +```python + # Compute c3=-1 if g is odd and delta>0, and 0 otherwise. + c3 = c1 & c2 + # Conditionally negate delta based on c3: + delta = (delta ^ c3) - c3 +``` + +Finally: + +```python + if g & 1: + if delta > 0: + f += g +``` + +becomes: + +```python + f += g & c3 +``` + +It turns out that this can be implemented more efficiently by applying the substitution +*η=-δ*. In this representation, negating *δ* corresponds to negating *η*, and incrementing +*δ* corresponds to decrementing *η*. This allows us to remove the negation in the *c1* +computation: + +```python + # Compute a mask c1 for eta < 0, and compute the conditional negation x of f: + c1 = eta >> 63 + x = (f ^ c1) - c1 + # Compute a mask c2 for odd g, and conditionally add x to g: + c2 = -(g & 1) + g += x & c2 + # Compute a mask c for (eta < 0) and odd (input) g, and use it to conditionally negate eta, + # and add g to f: + c3 = c1 & c2 + eta = (eta ^ c3) - c3 + f += g & c3 + # Incrementing delta corresponds to decrementing eta. + eta -= 1 + g >>= 1 +``` + +By replacing the loop in `divsteps_n_matrix` with a variant of the divstep code above (extended to +also apply all *f* operations to *u*, *v* and all *g* operations to *q*, *r*), a constant-time version of +`divsteps_n_matrix` is obtained. The full code will be in section 7. + +These bit fiddling tricks can also be used to make the conditional negations and additions in +`update_de` and `normalize` constant-time. + + +## 6. Variable-time optimizations + +In section 5, we modified the `divsteps_n_matrix` function (and a few others) to be constant time. +Constant time operations are only necessary when computing modular inverses of secret data. In +other cases, it slows down calculations unnecessarily. In this section, we will construct a +faster non-constant time `divsteps_n_matrix` function. + +To do so, first consider yet another way of writing the inner loop of divstep operations in +`gcd` from section 1. This decomposition is also explained in the paper in section 8.2. + +```python +for _ in range(N): + if g & 1 and eta < 0: + eta, f, g = -eta, g, -f + if g & 1: + g += f + eta -= 1 + g >>= 1 +``` + +Whenever *g* is even, the loop only shifts *g* down and decreases *η*. When *g* ends in multiple zero +bits, these iterations can be consolidated into one step. This requires counting the bottom zero +bits efficiently, which is possible on most platforms; it is abstracted here as the function +`count_trailing_zeros`. + +```python +def count_trailing_zeros(v): + """For a non-zero value v, find z such that v=(d<>= zeros + i -= zeros + if i == 0: + break + # We know g is odd now + if eta < 0: + eta, f, g = -eta, g, -f + g += f + # g is even now, and the eta decrement and g shift will happen in the next loop. +``` + +We can now remove multiple bottom *0* bits from *g* at once, but still need a full iteration whenever +there is a bottom *1* bit. In what follows, we will get rid of multiple *1* bits simultaneously as +well. + +Observe that as long as *η ≥ 0*, the loop does not modify *f*. Instead, it cancels out bottom +bits of *g* and shifts them out, and decreases *η* and *i* accordingly - interrupting only when *η* +becomes negative, or when *i* reaches *0*. Combined, this is equivalent to adding a multiple of *f* to +*g* to cancel out multiple bottom bits, and then shifting them out. + +It is easy to find what that multiple is: we want a number *w* such that *g+w f* has a few bottom +zero bits. If that number of bits is *L*, we want *g+w f mod 2L = 0*, or *w = -g/f mod 2L*. Since *f* +is odd, such a *w* exists for any *L*. *L* cannot be more than *i* steps (as we'd finish the loop before +doing more) or more than *η+1* steps (as we'd run `eta, f, g = -eta, g, f` at that point), but +apart from that, we're only limited by the complexity of computing *w*. + +This code demonstrates how to cancel up to 4 bits per step: + +```python +NEGINV16 = [15, 5, 3, 9, 7, 13, 11, 1] # NEGINV16[n//2] = (-n)^-1 mod 16, for odd n +i = N +while True: + zeros = min(i, count_trailing_zeros(g)) + eta -= zeros + g >>= zeros + i -= zeros + if i == 0: + break + # We know g is odd now + if eta < 0: + eta, f, g = -eta, g, f + # Compute limit on number of bits to cancel + limit = min(min(eta + 1, i), 4) + # Compute w = -g/f mod 2**limit, using the table value for -1/f mod 2**4. Note that f is + # always odd, so its inverse modulo a power of two always exists. + w = (g * NEGINV16[(f & 15) // 2]) % (2**limit) + # As w = -g/f mod (2**limit), g+w*f mod 2**limit = 0 mod 2**limit. + g += w * f + assert g % (2**limit) == 0 + # The next iteration will now shift out at least limit bottom zero bits from g. +``` + +By using a bigger table more bits can be cancelled at once. The table can also be implemented +as a formula. Several formulas are known for computing modular inverses modulo powers of two; +some can be found in Hacker's Delight second edition by Henry S. Warren, Jr. pages 245-247. +Here we need the negated modular inverse, which is a simple transformation of those: + +- Instead of a 3-bit table: + - *-f* or *f ^ 6* +- Instead of a 4-bit table: + - *1 - f(f + 1)* + - *-(f + (((f + 1) & 4) << 1))* +- For larger tables the following technique can be used: if *w=-1/f mod 2L*, then *w(w f+2)* is + *-1/f mod 22L*. This allows extending the previous formulas (or tables). In particular we + have this 6-bit function (based on the 3-bit function above): + - *f(f2 - 2)* + +This loop, again extended to also handle *u*, *v*, *q*, and *r* alongside *f* and *g*, placed in +`divsteps_n_matrix`, gives a significantly faster, but non-constant time version. + + +## 7. Final Python version + +All together we need the following functions: + +- A way to compute the transition matrix in constant time, using the `divsteps_n_matrix` function + from section 2, but with its loop replaced by a variant of the constant-time divstep from + section 5, extended to handle *u*, *v*, *q*, *r*: + +```python +def divsteps_n_matrix(eta, f, g): + """Compute eta and transition matrix t after N divsteps (multiplied by 2^N).""" + u, v, q, r = 1, 0, 0, 1 # start with identity matrix + for _ in range(N): + c1 = eta >> 63 + # Compute x, y, z as conditionally-negated versions of f, u, v. + x, y, z = (f ^ c1) - c1, (u ^ c1) - c1, (v ^ c1) - c1 + c2 = -(g & 1) + # Conditionally add x, y, z to g, q, r. + g, q, r = g + (x & c2), q + (y & c2), r + (z & c2) + c1 &= c2 # reusing c1 here for the earlier c3 variable + eta = (eta ^ c1) - (c1 + 1) # inlining the unconditional eta decrement here + # Conditionally add g, q, r to f, u, v. + f, u, v = f + (g & c1), u + (q & c1), v + (r & c1) + # When shifting g down, don't shift q, r, as we construct a transition matrix multiplied + # by 2^N. Instead, shift f's coefficients u and v up. + g, u, v = g >> 1, u << 1, v << 1 + return eta, (u, v, q, r) +``` + +- The functions to update *f* and *g*, and *d* and *e*, from section 2 and section 4, with the constant-time + changes to `update_de` from section 5: + +```python +def update_fg(f, g, t): + """Multiply matrix t/2^N with [f, g].""" + u, v, q, r = t + cf, cg = u*f + v*g, q*f + r*g + return cf >> N, cg >> N + +def update_de(d, e, t, M, Mi): + """Multiply matrix t/2^N with [d, e], modulo M.""" + u, v, q, r = t + d_sign, e_sign = d >> 257, e >> 257 + md, me = (u & d_sign) + (v & e_sign), (q & d_sign) + (r & e_sign) + cd, ce = (u*d + v*e) % 2**N, (q*d + r*e) % 2**N + md -= (Mi*cd + md) % 2**N + me -= (Mi*ce + me) % 2**N + cd, ce = u*d + v*e + Mi*md, q*d + r*e + Mi*me + return cd >> N, ce >> N +``` + +- The `normalize` function from section 4, made constant time as well: + +```python +def normalize(sign, v, M): + """Compute sign*v mod M, where v in (-2*M,M); output in [0,M).""" + v_sign = v >> 257 + # Conditionally add M to v. + v += M & v_sign + c = (sign - 1) >> 1 + # Conditionally negate v. + v = (v ^ c) - c + v_sign = v >> 257 + # Conditionally add M to v again. + v += M & v_sign + return v +``` + +- And finally the `modinv` function too, adapted to use *η* instead of *δ*, and using the fixed + iteration count from section 5: + +```python +def modinv(M, Mi, x): + """Compute the modular inverse of x mod M, given Mi=1/M mod 2^N.""" + eta, f, g, d, e = -1, M, x, 0, 1 + for _ in range((724 + N - 1) // N): + eta, t = divsteps_n_matrix(-eta, f % 2**N, g % 2**N) + f, g = update_fg(f, g, t) + d, e = update_de(d, e, t, M, Mi) + return normalize(f, d, M) +``` + +- To get a variable time version, replace the `divsteps_n_matrix` function with one that uses the + divsteps loop from section 5, and a `modinv` version that calls it without the fixed iteration + count: + +```python +NEGINV16 = [15, 5, 3, 9, 7, 13, 11, 1] # NEGINV16[n//2] = (-n)^-1 mod 16, for odd n +def divsteps_n_matrix_var(eta, f, g): + """Compute eta and transition matrix t after N divsteps (multiplied by 2^N).""" + u, v, q, r = 1, 0, 0, 1 + i = N + while True: + zeros = min(i, count_trailing_zeros(g)) + eta, i = eta - zeros, i - zeros + g, u, v = g >> zeros, u << zeros, v << zeros + if i == 0: + break + if eta < 0: + eta, f, u, v, g, q, r = -eta, g, q, r, -f, -u, -v + limit = min(min(eta + 1, i), 4) + w = (g * NEGINV16[(f & 15) // 2]) % (2**limit) + g, q, r = g + w*f, q + w*u, r + w*v + return eta, (u, v, q, r) + +def modinv_var(M, Mi, x): + """Compute the modular inverse of x mod M, given Mi = 1/M mod 2^N.""" + eta, f, g, d, e = -1, M, x, 0, 1 + while g != 0: + eta, t = divsteps_n_matrix_var(eta, f % 2**N, g % 2**N) + f, g = update_fg(f, g, t) + d, e = update_de(d, e, t, M, Mi) + return normalize(f, d, Mi) +``` diff --git a/src/basic-config.h b/src/basic-config.h index bb6b582..e4b1b8b 100644 --- a/src/basic-config.h +++ b/src/basic-config.h @@ -13,19 +13,10 @@ #undef USE_ECMULT_STATIC_PRECOMPUTATION #undef USE_EXTERNAL_ASM #undef USE_EXTERNAL_DEFAULT_CALLBACKS -#undef USE_FIELD_INV_BUILTIN -#undef USE_FIELD_INV_NUM -#undef USE_NUM_GMP -#undef USE_NUM_NONE -#undef USE_SCALAR_INV_BUILTIN -#undef USE_SCALAR_INV_NUM #undef USE_FORCE_WIDEMUL_INT64 #undef USE_FORCE_WIDEMUL_INT128 #undef ECMULT_WINDOW_SIZE -#define USE_NUM_NONE 1 -#define USE_FIELD_INV_BUILTIN 1 -#define USE_SCALAR_INV_BUILTIN 1 #define USE_WIDEMUL_64 1 #define ECMULT_WINDOW_SIZE 15 diff --git a/src/bench_ecmult.c b/src/bench_ecmult.c index 85b9e43..204e85a 100644 --- a/src/bench_ecmult.c +++ b/src/bench_ecmult.c @@ -9,7 +9,6 @@ #include "util.h" #include "hash_impl.h" -#include "num_impl.h" #include "field_impl.h" #include "group_impl.h" #include "scalar_impl.h" diff --git a/src/bench_internal.c b/src/bench_internal.c index 7fa6882..73b8a24 100644 --- a/src/bench_internal.c +++ b/src/bench_internal.c @@ -10,7 +10,6 @@ #include "assumptions.h" #include "util.h" #include "hash_impl.h" -#include "num_impl.h" #include "field_impl.h" #include "group_impl.h" #include "scalar_impl.h" @@ -99,15 +98,6 @@ void bench_scalar_negate(void* arg, int iters) { } } -void bench_scalar_sqr(void* arg, int iters) { - int i; - bench_inv *data = (bench_inv*)arg; - - for (i = 0; i < iters; i++) { - secp256k1_scalar_sqr(&data->scalar[0], &data->scalar[0]); - } -} - void bench_scalar_mul(void* arg, int iters) { int i; bench_inv *data = (bench_inv*)arg; @@ -255,26 +245,6 @@ void bench_group_add_affine_var(void* arg, int iters) { } } -void bench_group_jacobi_var(void* arg, int iters) { - int i, j = 0; - bench_inv *data = (bench_inv*)arg; - - for (i = 0; i < iters; i++) { - j += secp256k1_gej_has_quad_y_var(&data->gej[0]); - /* Vary the Y and Z coordinates of the input (the X coordinate doesn't matter to - secp256k1_gej_has_quad_y_var). Note that the resulting coordinates will - generally not correspond to a point on the curve, but this is not a problem - for the code being benchmarked here. Adding and normalizing have less - overhead than EC operations (which could guarantee the point remains on the - curve). */ - secp256k1_fe_add(&data->gej[0].y, &data->fe[1]); - secp256k1_fe_add(&data->gej[0].z, &data->fe[2]); - secp256k1_fe_normalize_var(&data->gej[0].y); - secp256k1_fe_normalize_var(&data->gej[0].z); - } - CHECK(j <= iters); -} - void bench_group_to_affine_var(void* arg, int iters) { int i; bench_inv *data = (bench_inv*)arg; @@ -282,8 +252,10 @@ void bench_group_to_affine_var(void* arg, int iters) { for (i = 0; i < iters; ++i) { secp256k1_ge_set_gej_var(&data->ge[1], &data->gej[0]); /* Use the output affine X/Y coordinates to vary the input X/Y/Z coordinates. - Similar to bench_group_jacobi_var, this approach does not result in - coordinates of points on the curve. */ + Note that the resulting coordinates will generally not correspond to a point + on the curve, but this is not a problem for the code being benchmarked here. + Adding and normalizing have less overhead than EC operations (which could + guarantee the point remains on the curve). */ secp256k1_fe_add(&data->gej[0].x, &data->ge[1].y); secp256k1_fe_add(&data->gej[0].y, &data->fe[2]); secp256k1_fe_add(&data->gej[0].z, &data->ge[1].x); @@ -369,35 +341,16 @@ void bench_context_sign(void* arg, int iters) { } } -#ifndef USE_NUM_NONE -void bench_num_jacobi(void* arg, int iters) { - int i, j = 0; - bench_inv *data = (bench_inv*)arg; - secp256k1_num nx, na, norder; - - secp256k1_scalar_get_num(&nx, &data->scalar[0]); - secp256k1_scalar_order_get_num(&norder); - secp256k1_scalar_get_num(&na, &data->scalar[1]); - - for (i = 0; i < iters; i++) { - j += secp256k1_num_jacobi(&nx, &norder); - secp256k1_num_add(&nx, &nx, &na); - } - CHECK(j <= iters); -} -#endif - int main(int argc, char **argv) { bench_inv data; int iters = get_iters(20000); if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "add")) run_benchmark("scalar_add", bench_scalar_add, bench_setup, NULL, &data, 10, iters*100); if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "negate")) run_benchmark("scalar_negate", bench_scalar_negate, bench_setup, NULL, &data, 10, iters*100); - if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "sqr")) run_benchmark("scalar_sqr", bench_scalar_sqr, bench_setup, NULL, &data, 10, iters*10); if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "mul")) run_benchmark("scalar_mul", bench_scalar_mul, bench_setup, NULL, &data, 10, iters*10); if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "split")) run_benchmark("scalar_split", bench_scalar_split, bench_setup, NULL, &data, 10, iters); - if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse", bench_scalar_inverse, bench_setup, NULL, &data, 10, 2000); - if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse_var", bench_scalar_inverse_var, bench_setup, NULL, &data, 10, 2000); + if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse", bench_scalar_inverse, bench_setup, NULL, &data, 10, iters); + if (have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse_var", bench_scalar_inverse_var, bench_setup, NULL, &data, 10, iters); if (have_flag(argc, argv, "field") || have_flag(argc, argv, "normalize")) run_benchmark("field_normalize", bench_field_normalize, bench_setup, NULL, &data, 10, iters*100); if (have_flag(argc, argv, "field") || have_flag(argc, argv, "normalize")) run_benchmark("field_normalize_weak", bench_field_normalize_weak, bench_setup, NULL, &data, 10, iters*100); @@ -411,7 +364,6 @@ int main(int argc, char **argv) { if (have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_var", bench_group_add_var, bench_setup, NULL, &data, 10, iters*10); if (have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine", bench_group_add_affine, bench_setup, NULL, &data, 10, iters*10); if (have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine_var", bench_group_add_affine_var, bench_setup, NULL, &data, 10, iters*10); - if (have_flag(argc, argv, "group") || have_flag(argc, argv, "jacobi")) run_benchmark("group_jacobi_var", bench_group_jacobi_var, bench_setup, NULL, &data, 10, iters); if (have_flag(argc, argv, "group") || have_flag(argc, argv, "to_affine")) run_benchmark("group_to_affine_var", bench_group_to_affine_var, bench_setup, NULL, &data, 10, iters); if (have_flag(argc, argv, "ecmult") || have_flag(argc, argv, "wnaf")) run_benchmark("wnaf_const", bench_wnaf_const, bench_setup, NULL, &data, 10, iters); @@ -424,8 +376,5 @@ int main(int argc, char **argv) { if (have_flag(argc, argv, "context") || have_flag(argc, argv, "verify")) run_benchmark("context_verify", bench_context_verify, bench_setup, NULL, &data, 10, 1 + iters/1000); if (have_flag(argc, argv, "context") || have_flag(argc, argv, "sign")) run_benchmark("context_sign", bench_context_sign, bench_setup, NULL, &data, 10, 1 + iters/100); -#ifndef USE_NUM_NONE - if (have_flag(argc, argv, "num") || have_flag(argc, argv, "jacobi")) run_benchmark("num_jacobi", bench_num_jacobi, bench_setup, NULL, &data, 10, iters*10); -#endif return 0; } diff --git a/src/ecmult.h b/src/ecmult.h index 7aa394a..7ab617e 100644 --- a/src/ecmult.h +++ b/src/ecmult.h @@ -7,7 +7,6 @@ #ifndef SECP256K1_ECMULT_H #define SECP256K1_ECMULT_H -#include "num.h" #include "group.h" #include "scalar.h" #include "scratch.h" diff --git a/src/field.h b/src/field.h index ee222ee..c58554b 100644 --- a/src/field.h +++ b/src/field.h @@ -104,9 +104,6 @@ static void secp256k1_fe_sqr(secp256k1_fe *r, const secp256k1_fe *a); * itself. */ static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a); -/** Checks whether a field element is a quadratic residue. */ -static int secp256k1_fe_is_quad_var(const secp256k1_fe *a); - /** Sets a field element to be the (modular) inverse of another. Requires the input's magnitude to be * at most 8. The output magnitude is 1 (but not guaranteed to be normalized). */ static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a); diff --git a/src/field_10x26_impl.h b/src/field_10x26_impl.h index 62bffdc..c280251 100644 --- a/src/field_10x26_impl.h +++ b/src/field_10x26_impl.h @@ -9,6 +9,7 @@ #include "util.h" #include "field.h" +#include "modinv32_impl.h" #ifdef VERIFY static void secp256k1_fe_verify(const secp256k1_fe *a) { @@ -1164,4 +1165,92 @@ static SECP256K1_INLINE void secp256k1_fe_from_storage(secp256k1_fe *r, const se #endif } +static void secp256k1_fe_from_signed30(secp256k1_fe *r, const secp256k1_modinv32_signed30 *a) { + const uint32_t M26 = UINT32_MAX >> 6; + const uint32_t a0 = a->v[0], a1 = a->v[1], a2 = a->v[2], a3 = a->v[3], a4 = a->v[4], + a5 = a->v[5], a6 = a->v[6], a7 = a->v[7], a8 = a->v[8]; + + /* The output from secp256k1_modinv32{_var} should be normalized to range [0,modulus), and + * have limbs in [0,2^30). The modulus is < 2^256, so the top limb must be below 2^(256-30*8). + */ + VERIFY_CHECK(a0 >> 30 == 0); + VERIFY_CHECK(a1 >> 30 == 0); + VERIFY_CHECK(a2 >> 30 == 0); + VERIFY_CHECK(a3 >> 30 == 0); + VERIFY_CHECK(a4 >> 30 == 0); + VERIFY_CHECK(a5 >> 30 == 0); + VERIFY_CHECK(a6 >> 30 == 0); + VERIFY_CHECK(a7 >> 30 == 0); + VERIFY_CHECK(a8 >> 16 == 0); + + r->n[0] = a0 & M26; + r->n[1] = (a0 >> 26 | a1 << 4) & M26; + r->n[2] = (a1 >> 22 | a2 << 8) & M26; + r->n[3] = (a2 >> 18 | a3 << 12) & M26; + r->n[4] = (a3 >> 14 | a4 << 16) & M26; + r->n[5] = (a4 >> 10 | a5 << 20) & M26; + r->n[6] = (a5 >> 6 | a6 << 24) & M26; + r->n[7] = (a6 >> 2 ) & M26; + r->n[8] = (a6 >> 28 | a7 << 2) & M26; + r->n[9] = (a7 >> 24 | a8 << 6); + +#ifdef VERIFY + r->magnitude = 1; + r->normalized = 1; + secp256k1_fe_verify(r); +#endif +} + +static void secp256k1_fe_to_signed30(secp256k1_modinv32_signed30 *r, const secp256k1_fe *a) { + const uint32_t M30 = UINT32_MAX >> 2; + const uint64_t a0 = a->n[0], a1 = a->n[1], a2 = a->n[2], a3 = a->n[3], a4 = a->n[4], + a5 = a->n[5], a6 = a->n[6], a7 = a->n[7], a8 = a->n[8], a9 = a->n[9]; + +#ifdef VERIFY + VERIFY_CHECK(a->normalized); +#endif + + r->v[0] = (a0 | a1 << 26) & M30; + r->v[1] = (a1 >> 4 | a2 << 22) & M30; + r->v[2] = (a2 >> 8 | a3 << 18) & M30; + r->v[3] = (a3 >> 12 | a4 << 14) & M30; + r->v[4] = (a4 >> 16 | a5 << 10) & M30; + r->v[5] = (a5 >> 20 | a6 << 6) & M30; + r->v[6] = (a6 >> 24 | a7 << 2 + | a8 << 28) & M30; + r->v[7] = (a8 >> 2 | a9 << 24) & M30; + r->v[8] = a9 >> 6; +} + +static const secp256k1_modinv32_modinfo secp256k1_const_modinfo_fe = { + {{-0x3D1, -4, 0, 0, 0, 0, 0, 0, 65536}}, + 0x2DDACACFL +}; + +static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *x) { + secp256k1_fe tmp; + secp256k1_modinv32_signed30 s; + + tmp = *x; + secp256k1_fe_normalize(&tmp); + secp256k1_fe_to_signed30(&s, &tmp); + secp256k1_modinv32(&s, &secp256k1_const_modinfo_fe); + secp256k1_fe_from_signed30(r, &s); + + VERIFY_CHECK(secp256k1_fe_normalizes_to_zero(r) == secp256k1_fe_normalizes_to_zero(&tmp)); +} + +static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *x) { + secp256k1_fe tmp; + secp256k1_modinv32_signed30 s; + + tmp = *x; + secp256k1_fe_normalize_var(&tmp); + secp256k1_fe_to_signed30(&s, &tmp); + secp256k1_modinv32_var(&s, &secp256k1_const_modinfo_fe); + secp256k1_fe_from_signed30(r, &s); + + VERIFY_CHECK(secp256k1_fe_normalizes_to_zero(r) == secp256k1_fe_normalizes_to_zero(&tmp)); +} + #endif /* SECP256K1_FIELD_REPR_IMPL_H */ diff --git a/src/field_5x52_impl.h b/src/field_5x52_impl.h index 3465ea3..b73cfea 100644 --- a/src/field_5x52_impl.h +++ b/src/field_5x52_impl.h @@ -13,6 +13,7 @@ #include "util.h" #include "field.h" +#include "modinv64_impl.h" #if defined(USE_ASM_X86_64) #include "field_5x52_asm_impl.h" @@ -498,4 +499,80 @@ static SECP256K1_INLINE void secp256k1_fe_from_storage(secp256k1_fe *r, const se #endif } +static void secp256k1_fe_from_signed62(secp256k1_fe *r, const secp256k1_modinv64_signed62 *a) { + const uint64_t M52 = UINT64_MAX >> 12; + const uint64_t a0 = a->v[0], a1 = a->v[1], a2 = a->v[2], a3 = a->v[3], a4 = a->v[4]; + + /* The output from secp256k1_modinv64{_var} should be normalized to range [0,modulus), and + * have limbs in [0,2^62). The modulus is < 2^256, so the top limb must be below 2^(256-62*4). + */ + VERIFY_CHECK(a0 >> 62 == 0); + VERIFY_CHECK(a1 >> 62 == 0); + VERIFY_CHECK(a2 >> 62 == 0); + VERIFY_CHECK(a3 >> 62 == 0); + VERIFY_CHECK(a4 >> 8 == 0); + + r->n[0] = a0 & M52; + r->n[1] = (a0 >> 52 | a1 << 10) & M52; + r->n[2] = (a1 >> 42 | a2 << 20) & M52; + r->n[3] = (a2 >> 32 | a3 << 30) & M52; + r->n[4] = (a3 >> 22 | a4 << 40); + +#ifdef VERIFY + r->magnitude = 1; + r->normalized = 1; + secp256k1_fe_verify(r); +#endif +} + +static void secp256k1_fe_to_signed62(secp256k1_modinv64_signed62 *r, const secp256k1_fe *a) { + const uint64_t M62 = UINT64_MAX >> 2; + const uint64_t a0 = a->n[0], a1 = a->n[1], a2 = a->n[2], a3 = a->n[3], a4 = a->n[4]; + +#ifdef VERIFY + VERIFY_CHECK(a->normalized); +#endif + + r->v[0] = (a0 | a1 << 52) & M62; + r->v[1] = (a1 >> 10 | a2 << 42) & M62; + r->v[2] = (a2 >> 20 | a3 << 32) & M62; + r->v[3] = (a3 >> 30 | a4 << 22) & M62; + r->v[4] = a4 >> 40; +} + +static const secp256k1_modinv64_modinfo secp256k1_const_modinfo_fe = { + {{-0x1000003D1LL, 0, 0, 0, 256}}, + 0x27C7F6E22DDACACFLL +}; + +static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *x) { + secp256k1_fe tmp; + secp256k1_modinv64_signed62 s; + + tmp = *x; + secp256k1_fe_normalize(&tmp); + secp256k1_fe_to_signed62(&s, &tmp); + secp256k1_modinv64(&s, &secp256k1_const_modinfo_fe); + secp256k1_fe_from_signed62(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_fe_normalizes_to_zero(r) == secp256k1_fe_normalizes_to_zero(&tmp)); +#endif +} + +static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *x) { + secp256k1_fe tmp; + secp256k1_modinv64_signed62 s; + + tmp = *x; + secp256k1_fe_normalize_var(&tmp); + secp256k1_fe_to_signed62(&s, &tmp); + secp256k1_modinv64_var(&s, &secp256k1_const_modinfo_fe); + secp256k1_fe_from_signed62(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_fe_normalizes_to_zero(r) == secp256k1_fe_normalizes_to_zero(&tmp)); +#endif +} + #endif /* SECP256K1_FIELD_REPR_IMPL_H */ diff --git a/src/field_impl.h b/src/field_impl.h index f0096f6..374284a 100644 --- a/src/field_impl.h +++ b/src/field_impl.h @@ -12,7 +12,6 @@ #endif #include "util.h" -#include "num.h" #if defined(SECP256K1_WIDEMUL_INT128) #include "field_5x52_impl.h" @@ -136,158 +135,6 @@ static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) { return secp256k1_fe_equal(&t1, a); } -static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a) { - secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1; - int j; - - /** The binary representation of (p - 2) has 5 blocks of 1s, with lengths in - * { 1, 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block: - * [1], [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223] - */ - - secp256k1_fe_sqr(&x2, a); - secp256k1_fe_mul(&x2, &x2, a); - - secp256k1_fe_sqr(&x3, &x2); - secp256k1_fe_mul(&x3, &x3, a); - - x6 = x3; - for (j=0; j<3; j++) { - secp256k1_fe_sqr(&x6, &x6); - } - secp256k1_fe_mul(&x6, &x6, &x3); - - x9 = x6; - for (j=0; j<3; j++) { - secp256k1_fe_sqr(&x9, &x9); - } - secp256k1_fe_mul(&x9, &x9, &x3); - - x11 = x9; - for (j=0; j<2; j++) { - secp256k1_fe_sqr(&x11, &x11); - } - secp256k1_fe_mul(&x11, &x11, &x2); - - x22 = x11; - for (j=0; j<11; j++) { - secp256k1_fe_sqr(&x22, &x22); - } - secp256k1_fe_mul(&x22, &x22, &x11); - - x44 = x22; - for (j=0; j<22; j++) { - secp256k1_fe_sqr(&x44, &x44); - } - secp256k1_fe_mul(&x44, &x44, &x22); - - x88 = x44; - for (j=0; j<44; j++) { - secp256k1_fe_sqr(&x88, &x88); - } - secp256k1_fe_mul(&x88, &x88, &x44); - - x176 = x88; - for (j=0; j<88; j++) { - secp256k1_fe_sqr(&x176, &x176); - } - secp256k1_fe_mul(&x176, &x176, &x88); - - x220 = x176; - for (j=0; j<44; j++) { - secp256k1_fe_sqr(&x220, &x220); - } - secp256k1_fe_mul(&x220, &x220, &x44); - - x223 = x220; - for (j=0; j<3; j++) { - secp256k1_fe_sqr(&x223, &x223); - } - secp256k1_fe_mul(&x223, &x223, &x3); - - /* The final result is then assembled using a sliding window over the blocks. */ - - t1 = x223; - for (j=0; j<23; j++) { - secp256k1_fe_sqr(&t1, &t1); - } - secp256k1_fe_mul(&t1, &t1, &x22); - for (j=0; j<5; j++) { - secp256k1_fe_sqr(&t1, &t1); - } - secp256k1_fe_mul(&t1, &t1, a); - for (j=0; j<3; j++) { - secp256k1_fe_sqr(&t1, &t1); - } - secp256k1_fe_mul(&t1, &t1, &x2); - for (j=0; j<2; j++) { - secp256k1_fe_sqr(&t1, &t1); - } - secp256k1_fe_mul(r, a, &t1); -} - -static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a) { -#if defined(USE_FIELD_INV_BUILTIN) - secp256k1_fe_inv(r, a); -#elif defined(USE_FIELD_INV_NUM) - secp256k1_num n, m; - static const secp256k1_fe negone = SECP256K1_FE_CONST( - 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, - 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL, 0xFFFFFC2EUL - ); - /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */ - static const unsigned char prime[32] = { - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F - }; - unsigned char b[32]; - int res; - secp256k1_fe c = *a; - secp256k1_fe_normalize_var(&c); - secp256k1_fe_get_b32(b, &c); - secp256k1_num_set_bin(&n, b, 32); - secp256k1_num_set_bin(&m, prime, 32); - secp256k1_num_mod_inverse(&n, &n, &m); - secp256k1_num_get_bin(b, 32, &n); - res = secp256k1_fe_set_b32(r, b); - (void)res; - VERIFY_CHECK(res); - /* Verify the result is the (unique) valid inverse using non-GMP code. */ - secp256k1_fe_mul(&c, &c, r); - secp256k1_fe_add(&c, &negone); - CHECK(secp256k1_fe_normalizes_to_zero_var(&c)); -#else -#error "Please select field inverse implementation" -#endif -} - -static int secp256k1_fe_is_quad_var(const secp256k1_fe *a) { -#ifndef USE_NUM_NONE - unsigned char b[32]; - secp256k1_num n; - secp256k1_num m; - /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */ - static const unsigned char prime[32] = { - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F - }; - - secp256k1_fe c = *a; - secp256k1_fe_normalize_var(&c); - secp256k1_fe_get_b32(b, &c); - secp256k1_num_set_bin(&n, b, 32); - secp256k1_num_set_bin(&m, prime, 32); - return secp256k1_num_jacobi(&n, &m) >= 0; -#else - secp256k1_fe r; - return secp256k1_fe_sqrt(&r, a); -#endif -} - static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1); #endif /* SECP256K1_FIELD_IMPL_H */ diff --git a/src/group.h b/src/group.h index 40bf961..b9cd334 100644 --- a/src/group.h +++ b/src/group.h @@ -7,7 +7,6 @@ #ifndef SECP256K1_GROUP_H #define SECP256K1_GROUP_H -#include "num.h" #include "field.h" /** A group element of the secp256k1 curve, in affine coordinates. */ @@ -43,12 +42,6 @@ typedef struct { /** Set a group element equal to the point with given X and Y coordinates */ static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y); -/** Set a group element (affine) equal to the point with the given X coordinate - * and a Y coordinate that is a quadratic residue modulo p. The return value - * is true iff a coordinate with the given X coordinate exists. - */ -static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x); - /** Set a group element (affine) equal to the point with the given X coordinate, and given oddness * for Y. Return value indicates whether the result is valid. */ static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd); @@ -96,9 +89,6 @@ static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a); /** Check whether a group element is the point at infinity. */ static int secp256k1_gej_is_infinity(const secp256k1_gej *a); -/** Check whether a group element's y coordinate is a quadratic residue. */ -static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a); - /** Set r equal to the double of a. Constant time. */ static void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a); diff --git a/src/group_impl.h b/src/group_impl.h index b7094c5..19ebd8f 100644 --- a/src/group_impl.h +++ b/src/group_impl.h @@ -7,7 +7,6 @@ #ifndef SECP256K1_GROUP_IMPL_H #define SECP256K1_GROUP_IMPL_H -#include "num.h" #include "field.h" #include "group.h" @@ -207,18 +206,14 @@ static void secp256k1_ge_clear(secp256k1_ge *r) { secp256k1_fe_clear(&r->y); } -static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x) { +static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) { secp256k1_fe x2, x3; r->x = *x; secp256k1_fe_sqr(&x2, x); secp256k1_fe_mul(&x3, x, &x2); r->infinity = 0; secp256k1_fe_add(&x3, &secp256k1_fe_const_b); - return secp256k1_fe_sqrt(&r->y, &x3); -} - -static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) { - if (!secp256k1_ge_set_xquad(r, x)) { + if (!secp256k1_fe_sqrt(&r->y, &x3)) { return 0; } secp256k1_fe_normalize_var(&r->y); @@ -655,20 +650,6 @@ static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) { secp256k1_fe_mul(&r->x, &r->x, &beta); } -static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a) { - secp256k1_fe yz; - - if (a->infinity) { - return 0; - } - - /* We rely on the fact that the Jacobi symbol of 1 / a->z^3 is the same as - * that of a->z. Thus a->y / a->z^3 is a quadratic residue iff a->y * a->z - is */ - secp256k1_fe_mul(&yz, &a->y, &a->z); - return secp256k1_fe_is_quad_var(&yz); -} - static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) { #ifdef EXHAUSTIVE_TEST_ORDER secp256k1_gej out; diff --git a/src/modinv32.h b/src/modinv32.h new file mode 100644 index 0000000..0efdda9 --- /dev/null +++ b/src/modinv32.h @@ -0,0 +1,42 @@ +/*********************************************************************** + * Copyright (c) 2020 Peter Dettman * + * Distributed under the MIT software license, see the accompanying * + * file COPYING or https://www.opensource.org/licenses/mit-license.php.* + **********************************************************************/ + +#ifndef SECP256K1_MODINV32_H +#define SECP256K1_MODINV32_H + +#if defined HAVE_CONFIG_H +#include "libsecp256k1-config.h" +#endif + +#include "util.h" + +/* A signed 30-bit limb representation of integers. + * + * Its value is sum(v[i] * 2^(30*i), i=0..8). */ +typedef struct { + int32_t v[9]; +} secp256k1_modinv32_signed30; + +typedef struct { + /* The modulus in signed30 notation, must be odd and in [3, 2^256]. */ + secp256k1_modinv32_signed30 modulus; + + /* modulus^{-1} mod 2^30 */ + uint32_t modulus_inv30; +} secp256k1_modinv32_modinfo; + +/* Replace x with its modular inverse mod modinfo->modulus. x must be in range [0, modulus). + * If x is zero, the result will be zero as well. If not, the inverse must exist (i.e., the gcd of + * x and modulus must be 1). These rules are automatically satisfied if the modulus is prime. + * + * On output, all of x's limbs will be in [0, 2^30). + */ +static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo); + +/* Same as secp256k1_modinv32_var, but constant time in x (not in the modulus). */ +static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo); + +#endif /* SECP256K1_MODINV32_H */ diff --git a/src/modinv32_impl.h b/src/modinv32_impl.h new file mode 100644 index 0000000..aa7988c --- /dev/null +++ b/src/modinv32_impl.h @@ -0,0 +1,587 @@ +/*********************************************************************** + * Copyright (c) 2020 Peter Dettman * + * Distributed under the MIT software license, see the accompanying * + * file COPYING or https://www.opensource.org/licenses/mit-license.php.* + **********************************************************************/ + +#ifndef SECP256K1_MODINV32_IMPL_H +#define SECP256K1_MODINV32_IMPL_H + +#include "modinv32.h" + +#include "util.h" + +#include + +/* This file implements modular inversion based on the paper "Fast constant-time gcd computation and + * modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. + * + * For an explanation of the algorithm, see doc/safegcd_implementation.md. This file contains an + * implementation for N=30, using 30-bit signed limbs represented as int32_t. + */ + +#ifdef VERIFY +static const secp256k1_modinv32_signed30 SECP256K1_SIGNED30_ONE = {{1}}; + +/* Compute a*factor and put it in r. All but the top limb in r will be in range [0,2^30). */ +static void secp256k1_modinv32_mul_30(secp256k1_modinv32_signed30 *r, const secp256k1_modinv32_signed30 *a, int alen, int32_t factor) { + const int32_t M30 = (int32_t)(UINT32_MAX >> 2); + int64_t c = 0; + int i; + for (i = 0; i < 8; ++i) { + if (i < alen) c += (int64_t)a->v[i] * factor; + r->v[i] = (int32_t)c & M30; c >>= 30; + } + if (8 < alen) c += (int64_t)a->v[8] * factor; + VERIFY_CHECK(c == (int32_t)c); + r->v[8] = (int32_t)c; +} + +/* Return -1 for ab*factor. A consists of alen limbs; b has 9. */ +static int secp256k1_modinv32_mul_cmp_30(const secp256k1_modinv32_signed30 *a, int alen, const secp256k1_modinv32_signed30 *b, int32_t factor) { + int i; + secp256k1_modinv32_signed30 am, bm; + secp256k1_modinv32_mul_30(&am, a, alen, 1); /* Normalize all but the top limb of a. */ + secp256k1_modinv32_mul_30(&bm, b, 9, factor); + for (i = 0; i < 8; ++i) { + /* Verify that all but the top limb of a and b are normalized. */ + VERIFY_CHECK(am.v[i] >> 30 == 0); + VERIFY_CHECK(bm.v[i] >> 30 == 0); + } + for (i = 8; i >= 0; --i) { + if (am.v[i] < bm.v[i]) return -1; + if (am.v[i] > bm.v[i]) return 1; + } + return 0; +} +#endif + +/* Take as input a signed30 number in range (-2*modulus,modulus), and add a multiple of the modulus + * to it to bring it to range [0,modulus). If sign < 0, the input will also be negated in the + * process. The input must have limbs in range (-2^30,2^30). The output will have limbs in range + * [0,2^30). */ +static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int32_t sign, const secp256k1_modinv32_modinfo *modinfo) { + const int32_t M30 = (int32_t)(UINT32_MAX >> 2); + int32_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4], + r5 = r->v[5], r6 = r->v[6], r7 = r->v[7], r8 = r->v[8]; + int32_t cond_add, cond_negate; + +#ifdef VERIFY + /* Verify that all limbs are in range (-2^30,2^30). */ + int i; + for (i = 0; i < 9; ++i) { + VERIFY_CHECK(r->v[i] >= -M30); + VERIFY_CHECK(r->v[i] <= M30); + } + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(r, 9, &modinfo->modulus, -2) > 0); /* r > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(r, 9, &modinfo->modulus, 1) < 0); /* r < modulus */ +#endif + + /* In a first step, add the modulus if the input is negative, and then negate if requested. + * This brings r from range (-2*modulus,modulus) to range (-modulus,modulus). As all input + * limbs are in range (-2^30,2^30), this cannot overflow an int32_t. Note that the right + * shifts below are signed sign-extending shifts (see assumptions.h for tests that that is + * indeed the behavior of the right shift operator). */ + cond_add = r8 >> 31; + r0 += modinfo->modulus.v[0] & cond_add; + r1 += modinfo->modulus.v[1] & cond_add; + r2 += modinfo->modulus.v[2] & cond_add; + r3 += modinfo->modulus.v[3] & cond_add; + r4 += modinfo->modulus.v[4] & cond_add; + r5 += modinfo->modulus.v[5] & cond_add; + r6 += modinfo->modulus.v[6] & cond_add; + r7 += modinfo->modulus.v[7] & cond_add; + r8 += modinfo->modulus.v[8] & cond_add; + cond_negate = sign >> 31; + r0 = (r0 ^ cond_negate) - cond_negate; + r1 = (r1 ^ cond_negate) - cond_negate; + r2 = (r2 ^ cond_negate) - cond_negate; + r3 = (r3 ^ cond_negate) - cond_negate; + r4 = (r4 ^ cond_negate) - cond_negate; + r5 = (r5 ^ cond_negate) - cond_negate; + r6 = (r6 ^ cond_negate) - cond_negate; + r7 = (r7 ^ cond_negate) - cond_negate; + r8 = (r8 ^ cond_negate) - cond_negate; + /* Propagate the top bits, to bring limbs back to range (-2^30,2^30). */ + r1 += r0 >> 30; r0 &= M30; + r2 += r1 >> 30; r1 &= M30; + r3 += r2 >> 30; r2 &= M30; + r4 += r3 >> 30; r3 &= M30; + r5 += r4 >> 30; r4 &= M30; + r6 += r5 >> 30; r5 &= M30; + r7 += r6 >> 30; r6 &= M30; + r8 += r7 >> 30; r7 &= M30; + + /* In a second step add the modulus again if the result is still negative, bringing r to range + * [0,modulus). */ + cond_add = r8 >> 31; + r0 += modinfo->modulus.v[0] & cond_add; + r1 += modinfo->modulus.v[1] & cond_add; + r2 += modinfo->modulus.v[2] & cond_add; + r3 += modinfo->modulus.v[3] & cond_add; + r4 += modinfo->modulus.v[4] & cond_add; + r5 += modinfo->modulus.v[5] & cond_add; + r6 += modinfo->modulus.v[6] & cond_add; + r7 += modinfo->modulus.v[7] & cond_add; + r8 += modinfo->modulus.v[8] & cond_add; + /* And propagate again. */ + r1 += r0 >> 30; r0 &= M30; + r2 += r1 >> 30; r1 &= M30; + r3 += r2 >> 30; r2 &= M30; + r4 += r3 >> 30; r3 &= M30; + r5 += r4 >> 30; r4 &= M30; + r6 += r5 >> 30; r5 &= M30; + r7 += r6 >> 30; r6 &= M30; + r8 += r7 >> 30; r7 &= M30; + + r->v[0] = r0; + r->v[1] = r1; + r->v[2] = r2; + r->v[3] = r3; + r->v[4] = r4; + r->v[5] = r5; + r->v[6] = r6; + r->v[7] = r7; + r->v[8] = r8; + +#ifdef VERIFY + VERIFY_CHECK(r0 >> 30 == 0); + VERIFY_CHECK(r1 >> 30 == 0); + VERIFY_CHECK(r2 >> 30 == 0); + VERIFY_CHECK(r3 >> 30 == 0); + VERIFY_CHECK(r4 >> 30 == 0); + VERIFY_CHECK(r5 >> 30 == 0); + VERIFY_CHECK(r6 >> 30 == 0); + VERIFY_CHECK(r7 >> 30 == 0); + VERIFY_CHECK(r8 >> 30 == 0); + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(r, 9, &modinfo->modulus, 0) >= 0); /* r >= 0 */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(r, 9, &modinfo->modulus, 1) < 0); /* r < modulus */ +#endif +} + +/* Data type for transition matrices (see section 3 of explanation). + * + * t = [ u v ] + * [ q r ] + */ +typedef struct { + int32_t u, v, q, r; +} secp256k1_modinv32_trans2x2; + +/* Compute the transition matrix and eta for 30 divsteps. + * + * Input: eta: initial eta + * f0: bottom limb of initial f + * g0: bottom limb of initial g + * Output: t: transition matrix + * Return: final eta + * + * Implements the divsteps_n_matrix function from the explanation. + */ +static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) { + /* u,v,q,r are the elements of the transformation matrix being built up, + * starting with the identity matrix. Semantically they are signed integers + * in range [-2^30,2^30], but here represented as unsigned mod 2^32. This + * permits left shifting (which is UB for negative numbers). The range + * being inside [-2^31,2^31) means that casting to signed works correctly. + */ + uint32_t u = 1, v = 0, q = 0, r = 1; + uint32_t c1, c2, f = f0, g = g0, x, y, z; + int i; + + for (i = 0; i < 30; ++i) { + VERIFY_CHECK((f & 1) == 1); /* f must always be odd */ + VERIFY_CHECK((u * f0 + v * g0) == f << i); + VERIFY_CHECK((q * f0 + r * g0) == g << i); + /* Compute conditional masks for (eta < 0) and for (g & 1). */ + c1 = eta >> 31; + c2 = -(g & 1); + /* Compute x,y,z, conditionally negated versions of f,u,v. */ + x = (f ^ c1) - c1; + y = (u ^ c1) - c1; + z = (v ^ c1) - c1; + /* Conditionally add x,y,z to g,q,r. */ + g += x & c2; + q += y & c2; + r += z & c2; + /* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */ + c1 &= c2; + /* Conditionally negate eta, and unconditionally subtract 1. */ + eta = (eta ^ c1) - (c1 + 1); + /* Conditionally add g,q,r to f,u,v. */ + f += g & c1; + u += q & c1; + v += r & c1; + /* Shifts */ + g >>= 1; + u <<= 1; + v <<= 1; + /* Bounds on eta that follow from the bounds on iteration count (max 25*30 divsteps). */ + VERIFY_CHECK(eta >= -751 && eta <= 751); + } + /* Return data in t and return value. */ + t->u = (int32_t)u; + t->v = (int32_t)v; + t->q = (int32_t)q; + t->r = (int32_t)r; + /* The determinant of t must be a power of two. This guarantees that multiplication with t + * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which + * will be divided out again). As each divstep's individual matrix has determinant 2, the + * aggregate of 30 of them will have determinant 2^30. */ + VERIFY_CHECK((int64_t)t->u * t->r - (int64_t)t->v * t->q == ((int64_t)1) << 30); + return eta; +} + +/* Compute the transition matrix and eta for 30 divsteps (variable time). + * + * Input: eta: initial eta + * f0: bottom limb of initial f + * g0: bottom limb of initial g + * Output: t: transition matrix + * Return: final eta + * + * Implements the divsteps_n_matrix_var function from the explanation. + */ +static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) { + /* inv256[i] = -(2*i+1)^-1 (mod 256) */ + static const uint8_t inv256[128] = { + 0xFF, 0x55, 0x33, 0x49, 0xC7, 0x5D, 0x3B, 0x11, 0x0F, 0xE5, 0xC3, 0x59, + 0xD7, 0xED, 0xCB, 0x21, 0x1F, 0x75, 0x53, 0x69, 0xE7, 0x7D, 0x5B, 0x31, + 0x2F, 0x05, 0xE3, 0x79, 0xF7, 0x0D, 0xEB, 0x41, 0x3F, 0x95, 0x73, 0x89, + 0x07, 0x9D, 0x7B, 0x51, 0x4F, 0x25, 0x03, 0x99, 0x17, 0x2D, 0x0B, 0x61, + 0x5F, 0xB5, 0x93, 0xA9, 0x27, 0xBD, 0x9B, 0x71, 0x6F, 0x45, 0x23, 0xB9, + 0x37, 0x4D, 0x2B, 0x81, 0x7F, 0xD5, 0xB3, 0xC9, 0x47, 0xDD, 0xBB, 0x91, + 0x8F, 0x65, 0x43, 0xD9, 0x57, 0x6D, 0x4B, 0xA1, 0x9F, 0xF5, 0xD3, 0xE9, + 0x67, 0xFD, 0xDB, 0xB1, 0xAF, 0x85, 0x63, 0xF9, 0x77, 0x8D, 0x6B, 0xC1, + 0xBF, 0x15, 0xF3, 0x09, 0x87, 0x1D, 0xFB, 0xD1, 0xCF, 0xA5, 0x83, 0x19, + 0x97, 0xAD, 0x8B, 0xE1, 0xDF, 0x35, 0x13, 0x29, 0xA7, 0x3D, 0x1B, 0xF1, + 0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01 + }; + + /* Transformation matrix; see comments in secp256k1_modinv32_divsteps_30. */ + uint32_t u = 1, v = 0, q = 0, r = 1; + uint32_t f = f0, g = g0, m; + uint16_t w; + int i = 30, limit, zeros; + + for (;;) { + /* Use a sentinel bit to count zeros only up to i. */ + zeros = secp256k1_ctz32_var(g | (UINT32_MAX << i)); + /* Perform zeros divsteps at once; they all just divide g by two. */ + g >>= zeros; + u <<= zeros; + v <<= zeros; + eta -= zeros; + i -= zeros; + /* We're done once we've done 30 divsteps. */ + if (i == 0) break; + VERIFY_CHECK((f & 1) == 1); + VERIFY_CHECK((g & 1) == 1); + VERIFY_CHECK((u * f0 + v * g0) == f << (30 - i)); + VERIFY_CHECK((q * f0 + r * g0) == g << (30 - i)); + /* Bounds on eta that follow from the bounds on iteration count (max 25*30 divsteps). */ + VERIFY_CHECK(eta >= -751 && eta <= 751); + /* If eta is negative, negate it and replace f,g with g,-f. */ + if (eta < 0) { + uint32_t tmp; + eta = -eta; + tmp = f; f = g; g = -tmp; + tmp = u; u = q; q = -tmp; + tmp = v; v = r; r = -tmp; + } + /* eta is now >= 0. In what follows we're going to cancel out the bottom bits of g. No more + * than i can be cancelled out (as we'd be done before that point), and no more than eta+1 + * can be done as its sign will flip once that happens. */ + limit = ((int)eta + 1) > i ? i : ((int)eta + 1); + /* m is a mask for the bottom min(limit, 8) bits (our table only supports 8 bits). */ + VERIFY_CHECK(limit > 0 && limit <= 30); + m = (UINT32_MAX >> (32 - limit)) & 255U; + /* Find what multiple of f must be added to g to cancel its bottom min(limit, 8) bits. */ + w = (g * inv256[(f >> 1) & 127]) & m; + /* Do so. */ + g += f * w; + q += u * w; + r += v * w; + VERIFY_CHECK((g & m) == 0); + } + /* Return data in t and return value. */ + t->u = (int32_t)u; + t->v = (int32_t)v; + t->q = (int32_t)q; + t->r = (int32_t)r; + /* The determinant of t must be a power of two. This guarantees that multiplication with t + * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which + * will be divided out again). As each divstep's individual matrix has determinant 2, the + * aggregate of 30 of them will have determinant 2^30. */ + VERIFY_CHECK((int64_t)t->u * t->r - (int64_t)t->v * t->q == ((int64_t)1) << 30); + return eta; +} + +/* Compute (t/2^30) * [d, e] mod modulus, where t is a transition matrix for 30 divsteps. + * + * On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range + * (-2^30,2^30). + * + * This implements the update_de function from the explanation. + */ +static void secp256k1_modinv32_update_de_30(secp256k1_modinv32_signed30 *d, secp256k1_modinv32_signed30 *e, const secp256k1_modinv32_trans2x2 *t, const secp256k1_modinv32_modinfo* modinfo) { + const int32_t M30 = (int32_t)(UINT32_MAX >> 2); + const int32_t u = t->u, v = t->v, q = t->q, r = t->r; + int32_t di, ei, md, me, sd, se; + int64_t cd, ce; + int i; +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(d, 9, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(d, 9, &modinfo->modulus, 1) < 0); /* d < modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(e, 9, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(e, 9, &modinfo->modulus, 1) < 0); /* e < modulus */ + VERIFY_CHECK((labs(u) + labs(v)) >= 0); /* |u|+|v| doesn't overflow */ + VERIFY_CHECK((labs(q) + labs(r)) >= 0); /* |q|+|r| doesn't overflow */ + VERIFY_CHECK((labs(u) + labs(v)) <= M30 + 1); /* |u|+|v| <= 2^30 */ + VERIFY_CHECK((labs(q) + labs(r)) <= M30 + 1); /* |q|+|r| <= 2^30 */ +#endif + /* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */ + sd = d->v[8] >> 31; + se = e->v[8] >> 31; + md = (u & sd) + (v & se); + me = (q & sd) + (r & se); + /* Begin computing t*[d,e]. */ + di = d->v[0]; + ei = e->v[0]; + cd = (int64_t)u * di + (int64_t)v * ei; + ce = (int64_t)q * di + (int64_t)r * ei; + /* Correct md,me so that t*[d,e]+modulus*[md,me] has 30 zero bottom bits. */ + md -= (modinfo->modulus_inv30 * (uint32_t)cd + md) & M30; + me -= (modinfo->modulus_inv30 * (uint32_t)ce + me) & M30; + /* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */ + cd += (int64_t)modinfo->modulus.v[0] * md; + ce += (int64_t)modinfo->modulus.v[0] * me; + /* Verify that the low 30 bits of the computation are indeed zero, and then throw them away. */ + VERIFY_CHECK(((int32_t)cd & M30) == 0); cd >>= 30; + VERIFY_CHECK(((int32_t)ce & M30) == 0); ce >>= 30; + /* Now iteratively compute limb i=1..8 of t*[d,e]+modulus*[md,me], and store them in output + * limb i-1 (shifting down by 30 bits). */ + for (i = 1; i < 9; ++i) { + di = d->v[i]; + ei = e->v[i]; + cd += (int64_t)u * di + (int64_t)v * ei; + ce += (int64_t)q * di + (int64_t)r * ei; + cd += (int64_t)modinfo->modulus.v[i] * md; + ce += (int64_t)modinfo->modulus.v[i] * me; + d->v[i - 1] = (int32_t)cd & M30; cd >>= 30; + e->v[i - 1] = (int32_t)ce & M30; ce >>= 30; + } + /* What remains is limb 9 of t*[d,e]+modulus*[md,me]; store it as output limb 8. */ + d->v[8] = (int32_t)cd; + e->v[8] = (int32_t)ce; +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(d, 9, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(d, 9, &modinfo->modulus, 1) < 0); /* d < modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(e, 9, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(e, 9, &modinfo->modulus, 1) < 0); /* e < modulus */ +#endif +} + +/* Compute (t/2^30) * [f, g], where t is a transition matrix for 30 divsteps. + * + * This implements the update_fg function from the explanation. + */ +static void secp256k1_modinv32_update_fg_30(secp256k1_modinv32_signed30 *f, secp256k1_modinv32_signed30 *g, const secp256k1_modinv32_trans2x2 *t) { + const int32_t M30 = (int32_t)(UINT32_MAX >> 2); + const int32_t u = t->u, v = t->v, q = t->q, r = t->r; + int32_t fi, gi; + int64_t cf, cg; + int i; + /* Start computing t*[f,g]. */ + fi = f->v[0]; + gi = g->v[0]; + cf = (int64_t)u * fi + (int64_t)v * gi; + cg = (int64_t)q * fi + (int64_t)r * gi; + /* Verify that the bottom 30 bits of the result are zero, and then throw them away. */ + VERIFY_CHECK(((int32_t)cf & M30) == 0); cf >>= 30; + VERIFY_CHECK(((int32_t)cg & M30) == 0); cg >>= 30; + /* Now iteratively compute limb i=1..8 of t*[f,g], and store them in output limb i-1 (shifting + * down by 30 bits). */ + for (i = 1; i < 9; ++i) { + fi = f->v[i]; + gi = g->v[i]; + cf += (int64_t)u * fi + (int64_t)v * gi; + cg += (int64_t)q * fi + (int64_t)r * gi; + f->v[i - 1] = (int32_t)cf & M30; cf >>= 30; + g->v[i - 1] = (int32_t)cg & M30; cg >>= 30; + } + /* What remains is limb 9 of t*[f,g]; store it as output limb 8. */ + f->v[8] = (int32_t)cf; + g->v[8] = (int32_t)cg; +} + +/* Compute (t/2^30) * [f, g], where t is a transition matrix for 30 divsteps. + * + * Version that operates on a variable number of limbs in f and g. + * + * This implements the update_fg function from the explanation in modinv64_impl.h. + */ +static void secp256k1_modinv32_update_fg_30_var(int len, secp256k1_modinv32_signed30 *f, secp256k1_modinv32_signed30 *g, const secp256k1_modinv32_trans2x2 *t) { + const int32_t M30 = (int32_t)(UINT32_MAX >> 2); + const int32_t u = t->u, v = t->v, q = t->q, r = t->r; + int32_t fi, gi; + int64_t cf, cg; + int i; + VERIFY_CHECK(len > 0); + /* Start computing t*[f,g]. */ + fi = f->v[0]; + gi = g->v[0]; + cf = (int64_t)u * fi + (int64_t)v * gi; + cg = (int64_t)q * fi + (int64_t)r * gi; + /* Verify that the bottom 62 bits of the result are zero, and then throw them away. */ + VERIFY_CHECK(((int32_t)cf & M30) == 0); cf >>= 30; + VERIFY_CHECK(((int32_t)cg & M30) == 0); cg >>= 30; + /* Now iteratively compute limb i=1..len of t*[f,g], and store them in output limb i-1 (shifting + * down by 30 bits). */ + for (i = 1; i < len; ++i) { + fi = f->v[i]; + gi = g->v[i]; + cf += (int64_t)u * fi + (int64_t)v * gi; + cg += (int64_t)q * fi + (int64_t)r * gi; + f->v[i - 1] = (int32_t)cf & M30; cf >>= 30; + g->v[i - 1] = (int32_t)cg & M30; cg >>= 30; + } + /* What remains is limb (len) of t*[f,g]; store it as output limb (len-1). */ + f->v[len - 1] = (int32_t)cf; + g->v[len - 1] = (int32_t)cg; +} + +/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */ +static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) { + /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ + secp256k1_modinv32_signed30 d = {{0}}; + secp256k1_modinv32_signed30 e = {{1}}; + secp256k1_modinv32_signed30 f = modinfo->modulus; + secp256k1_modinv32_signed30 g = *x; + int i; + int32_t eta = -1; + + /* Do 25 iterations of 30 divsteps each = 750 divsteps. 724 suffices for 256-bit inputs. */ + for (i = 0; i < 25; ++i) { + /* Compute transition matrix and new eta after 30 divsteps. */ + secp256k1_modinv32_trans2x2 t; + eta = secp256k1_modinv32_divsteps_30(eta, f.v[0], g.v[0], &t); + /* Update d,e using that transition matrix. */ + secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo); + /* Update f,g using that transition matrix. */ +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, 9, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, 9, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + secp256k1_modinv32_update_fg_30(&f, &g, &t); +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, 9, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, 9, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + } + + /* At this point sufficient iterations have been performed that g must have reached 0 + * and (if g was not originally 0) f must now equal +/- GCD of the initial f, g + * values i.e. +/- 1, and d now contains +/- the modular inverse. */ +#ifdef VERIFY + /* g == 0 */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, 9, &SECP256K1_SIGNED30_ONE, 0) == 0); + /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, 9, &SECP256K1_SIGNED30_ONE, -1) == 0 || + secp256k1_modinv32_mul_cmp_30(&f, 9, &SECP256K1_SIGNED30_ONE, 1) == 0 || + (secp256k1_modinv32_mul_cmp_30(x, 9, &SECP256K1_SIGNED30_ONE, 0) == 0 && + secp256k1_modinv32_mul_cmp_30(&d, 9, &SECP256K1_SIGNED30_ONE, 0) == 0 && + (secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, 1) == 0 || + secp256k1_modinv32_mul_cmp_30(&f, 9, &modinfo->modulus, -1) == 0))); +#endif + + /* Optionally negate d, normalize to [0,modulus), and return it. */ + secp256k1_modinv32_normalize_30(&d, f.v[8], modinfo); + *x = d; +} + +/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */ +static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) { + /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ + secp256k1_modinv32_signed30 d = {{0, 0, 0, 0, 0, 0, 0, 0, 0}}; + secp256k1_modinv32_signed30 e = {{1, 0, 0, 0, 0, 0, 0, 0, 0}}; + secp256k1_modinv32_signed30 f = modinfo->modulus; + secp256k1_modinv32_signed30 g = *x; +#ifdef VERIFY + int i = 0; +#endif + int j, len = 9; + int32_t eta = -1; + int32_t cond, fn, gn; + + /* Do iterations of 30 divsteps each until g=0. */ + while (1) { + /* Compute transition matrix and new eta after 30 divsteps. */ + secp256k1_modinv32_trans2x2 t; + eta = secp256k1_modinv32_divsteps_30_var(eta, f.v[0], g.v[0], &t); + /* Update d,e using that transition matrix. */ + secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo); + /* Update f,g using that transition matrix. */ +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + secp256k1_modinv32_update_fg_30_var(len, &f, &g, &t); + /* If the bottom limb of g is 0, there is a chance g=0. */ + if (g.v[0] == 0) { + cond = 0; + /* Check if all other limbs are also 0. */ + for (j = 1; j < len; ++j) { + cond |= g.v[j]; + } + /* If so, we're done. */ + if (cond == 0) break; + } + + /* Determine if len>1 and limb (len-1) of both f and g is 0 or -1. */ + fn = f.v[len - 1]; + gn = g.v[len - 1]; + cond = ((int32_t)len - 2) >> 31; + cond |= fn ^ (fn >> 31); + cond |= gn ^ (gn >> 31); + /* If so, reduce length, propagating the sign of f and g's top limb into the one below. */ + if (cond == 0) { + f.v[len - 2] |= (uint32_t)fn << 30; + g.v[len - 2] |= (uint32_t)gn << 30; + --len; + } +#ifdef VERIFY + VERIFY_CHECK(++i < 25); /* We should never need more than 25*30 = 750 divsteps */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + } + + /* At this point g is 0 and (if g was not originally 0) f must now equal +/- GCD of + * the initial f, g values i.e. +/- 1, and d now contains +/- the modular inverse. */ +#ifdef VERIFY + /* g == 0 */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&g, len, &SECP256K1_SIGNED30_ONE, 0) == 0); + /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ + VERIFY_CHECK(secp256k1_modinv32_mul_cmp_30(&f, len, &SECP256K1_SIGNED30_ONE, -1) == 0 || + secp256k1_modinv32_mul_cmp_30(&f, len, &SECP256K1_SIGNED30_ONE, 1) == 0 || + (secp256k1_modinv32_mul_cmp_30(x, 9, &SECP256K1_SIGNED30_ONE, 0) == 0 && + secp256k1_modinv32_mul_cmp_30(&d, 9, &SECP256K1_SIGNED30_ONE, 0) == 0 && + (secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, 1) == 0 || + secp256k1_modinv32_mul_cmp_30(&f, len, &modinfo->modulus, -1) == 0))); +#endif + + /* Optionally negate d, normalize to [0,modulus), and return it. */ + secp256k1_modinv32_normalize_30(&d, f.v[len - 1], modinfo); + *x = d; +} + +#endif /* SECP256K1_MODINV32_IMPL_H */ diff --git a/src/modinv64.h b/src/modinv64.h new file mode 100644 index 0000000..da506df --- /dev/null +++ b/src/modinv64.h @@ -0,0 +1,46 @@ +/*********************************************************************** + * Copyright (c) 2020 Peter Dettman * + * Distributed under the MIT software license, see the accompanying * + * file COPYING or https://www.opensource.org/licenses/mit-license.php.* + **********************************************************************/ + +#ifndef SECP256K1_MODINV64_H +#define SECP256K1_MODINV64_H + +#if defined HAVE_CONFIG_H +#include "libsecp256k1-config.h" +#endif + +#include "util.h" + +#ifndef SECP256K1_WIDEMUL_INT128 +#error "modinv64 requires 128-bit wide multiplication support" +#endif + +/* A signed 62-bit limb representation of integers. + * + * Its value is sum(v[i] * 2^(62*i), i=0..4). */ +typedef struct { + int64_t v[5]; +} secp256k1_modinv64_signed62; + +typedef struct { + /* The modulus in signed62 notation, must be odd and in [3, 2^256]. */ + secp256k1_modinv64_signed62 modulus; + + /* modulus^{-1} mod 2^62 */ + uint64_t modulus_inv62; +} secp256k1_modinv64_modinfo; + +/* Replace x with its modular inverse mod modinfo->modulus. x must be in range [0, modulus). + * If x is zero, the result will be zero as well. If not, the inverse must exist (i.e., the gcd of + * x and modulus must be 1). These rules are automatically satisfied if the modulus is prime. + * + * On output, all of x's limbs will be in [0, 2^62). + */ +static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo); + +/* Same as secp256k1_modinv64_var, but constant time in x (not in the modulus). */ +static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo); + +#endif /* SECP256K1_MODINV64_H */ diff --git a/src/modinv64_impl.h b/src/modinv64_impl.h new file mode 100644 index 0000000..78505fa --- /dev/null +++ b/src/modinv64_impl.h @@ -0,0 +1,589 @@ +/*********************************************************************** + * Copyright (c) 2020 Peter Dettman * + * Distributed under the MIT software license, see the accompanying * + * file COPYING or https://www.opensource.org/licenses/mit-license.php.* + **********************************************************************/ + +#ifndef SECP256K1_MODINV64_IMPL_H +#define SECP256K1_MODINV64_IMPL_H + +#include "modinv64.h" + +#include "util.h" + +/* This file implements modular inversion based on the paper "Fast constant-time gcd computation and + * modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. + * + * For an explanation of the algorithm, see doc/safegcd_implementation.md. This file contains an + * implementation for N=62, using 62-bit signed limbs represented as int64_t. + */ + +#ifdef VERIFY +/* Helper function to compute the absolute value of an int64_t. + * (we don't use abs/labs/llabs as it depends on the int sizes). */ +static int64_t secp256k1_modinv64_abs(int64_t v) { + VERIFY_CHECK(v > INT64_MIN); + if (v < 0) return -v; + return v; +} + +static const secp256k1_modinv64_signed62 SECP256K1_SIGNED62_ONE = {{1}}; + +/* Compute a*factor and put it in r. All but the top limb in r will be in range [0,2^62). */ +static void secp256k1_modinv64_mul_62(secp256k1_modinv64_signed62 *r, const secp256k1_modinv64_signed62 *a, int alen, int64_t factor) { + const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + int128_t c = 0; + int i; + for (i = 0; i < 4; ++i) { + if (i < alen) c += (int128_t)a->v[i] * factor; + r->v[i] = (int64_t)c & M62; c >>= 62; + } + if (4 < alen) c += (int128_t)a->v[4] * factor; + VERIFY_CHECK(c == (int64_t)c); + r->v[4] = (int64_t)c; +} + +/* Return -1 for ab*factor. A has alen limbs; b has 5. */ +static int secp256k1_modinv64_mul_cmp_62(const secp256k1_modinv64_signed62 *a, int alen, const secp256k1_modinv64_signed62 *b, int64_t factor) { + int i; + secp256k1_modinv64_signed62 am, bm; + secp256k1_modinv64_mul_62(&am, a, alen, 1); /* Normalize all but the top limb of a. */ + secp256k1_modinv64_mul_62(&bm, b, 5, factor); + for (i = 0; i < 4; ++i) { + /* Verify that all but the top limb of a and b are normalized. */ + VERIFY_CHECK(am.v[i] >> 62 == 0); + VERIFY_CHECK(bm.v[i] >> 62 == 0); + } + for (i = 4; i >= 0; --i) { + if (am.v[i] < bm.v[i]) return -1; + if (am.v[i] > bm.v[i]) return 1; + } + return 0; +} +#endif + +/* Take as input a signed62 number in range (-2*modulus,modulus), and add a multiple of the modulus + * to it to bring it to range [0,modulus). If sign < 0, the input will also be negated in the + * process. The input must have limbs in range (-2^62,2^62). The output will have limbs in range + * [0,2^62). */ +static void secp256k1_modinv64_normalize_62(secp256k1_modinv64_signed62 *r, int64_t sign, const secp256k1_modinv64_modinfo *modinfo) { + const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + int64_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4]; + int64_t cond_add, cond_negate; + +#ifdef VERIFY + /* Verify that all limbs are in range (-2^62,2^62). */ + int i; + for (i = 0; i < 5; ++i) { + VERIFY_CHECK(r->v[i] >= -M62); + VERIFY_CHECK(r->v[i] <= M62); + } + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, -2) > 0); /* r > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 1) < 0); /* r < modulus */ +#endif + + /* In a first step, add the modulus if the input is negative, and then negate if requested. + * This brings r from range (-2*modulus,modulus) to range (-modulus,modulus). As all input + * limbs are in range (-2^62,2^62), this cannot overflow an int64_t. Note that the right + * shifts below are signed sign-extending shifts (see assumptions.h for tests that that is + * indeed the behavior of the right shift operator). */ + cond_add = r4 >> 63; + r0 += modinfo->modulus.v[0] & cond_add; + r1 += modinfo->modulus.v[1] & cond_add; + r2 += modinfo->modulus.v[2] & cond_add; + r3 += modinfo->modulus.v[3] & cond_add; + r4 += modinfo->modulus.v[4] & cond_add; + cond_negate = sign >> 63; + r0 = (r0 ^ cond_negate) - cond_negate; + r1 = (r1 ^ cond_negate) - cond_negate; + r2 = (r2 ^ cond_negate) - cond_negate; + r3 = (r3 ^ cond_negate) - cond_negate; + r4 = (r4 ^ cond_negate) - cond_negate; + /* Propagate the top bits, to bring limbs back to range (-2^62,2^62). */ + r1 += r0 >> 62; r0 &= M62; + r2 += r1 >> 62; r1 &= M62; + r3 += r2 >> 62; r2 &= M62; + r4 += r3 >> 62; r3 &= M62; + + /* In a second step add the modulus again if the result is still negative, bringing + * r to range [0,modulus). */ + cond_add = r4 >> 63; + r0 += modinfo->modulus.v[0] & cond_add; + r1 += modinfo->modulus.v[1] & cond_add; + r2 += modinfo->modulus.v[2] & cond_add; + r3 += modinfo->modulus.v[3] & cond_add; + r4 += modinfo->modulus.v[4] & cond_add; + /* And propagate again. */ + r1 += r0 >> 62; r0 &= M62; + r2 += r1 >> 62; r1 &= M62; + r3 += r2 >> 62; r2 &= M62; + r4 += r3 >> 62; r3 &= M62; + + r->v[0] = r0; + r->v[1] = r1; + r->v[2] = r2; + r->v[3] = r3; + r->v[4] = r4; + +#ifdef VERIFY + VERIFY_CHECK(r0 >> 62 == 0); + VERIFY_CHECK(r1 >> 62 == 0); + VERIFY_CHECK(r2 >> 62 == 0); + VERIFY_CHECK(r3 >> 62 == 0); + VERIFY_CHECK(r4 >> 62 == 0); + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 0) >= 0); /* r >= 0 */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(r, 5, &modinfo->modulus, 1) < 0); /* r < modulus */ +#endif +} + +/* Data type for transition matrices (see section 3 of explanation). + * + * t = [ u v ] + * [ q r ] + */ +typedef struct { + int64_t u, v, q, r; +} secp256k1_modinv64_trans2x2; + +/* Compute the transition matrix and eta for 62 divsteps. + * + * Input: eta: initial eta + * f0: bottom limb of initial f + * g0: bottom limb of initial g + * Output: t: transition matrix + * Return: final eta + * + * Implements the divsteps_n_matrix function from the explanation. + */ +static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) { + /* u,v,q,r are the elements of the transformation matrix being built up, + * starting with the identity matrix. Semantically they are signed integers + * in range [-2^62,2^62], but here represented as unsigned mod 2^64. This + * permits left shifting (which is UB for negative numbers). The range + * being inside [-2^63,2^63) means that casting to signed works correctly. + */ + uint64_t u = 1, v = 0, q = 0, r = 1; + uint64_t c1, c2, f = f0, g = g0, x, y, z; + int i; + + for (i = 0; i < 62; ++i) { + VERIFY_CHECK((f & 1) == 1); /* f must always be odd */ + VERIFY_CHECK((u * f0 + v * g0) == f << i); + VERIFY_CHECK((q * f0 + r * g0) == g << i); + /* Compute conditional masks for (eta < 0) and for (g & 1). */ + c1 = eta >> 63; + c2 = -(g & 1); + /* Compute x,y,z, conditionally negated versions of f,u,v. */ + x = (f ^ c1) - c1; + y = (u ^ c1) - c1; + z = (v ^ c1) - c1; + /* Conditionally add x,y,z to g,q,r. */ + g += x & c2; + q += y & c2; + r += z & c2; + /* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */ + c1 &= c2; + /* Conditionally negate eta, and unconditionally subtract 1. */ + eta = (eta ^ c1) - (c1 + 1); + /* Conditionally add g,q,r to f,u,v. */ + f += g & c1; + u += q & c1; + v += r & c1; + /* Shifts */ + g >>= 1; + u <<= 1; + v <<= 1; + /* Bounds on eta that follow from the bounds on iteration count (max 12*62 divsteps). */ + VERIFY_CHECK(eta >= -745 && eta <= 745); + } + /* Return data in t and return value. */ + t->u = (int64_t)u; + t->v = (int64_t)v; + t->q = (int64_t)q; + t->r = (int64_t)r; + /* The determinant of t must be a power of two. This guarantees that multiplication with t + * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which + * will be divided out again). As each divstep's individual matrix has determinant 2, the + * aggregate of 62 of them will have determinant 2^62. */ + VERIFY_CHECK((int128_t)t->u * t->r - (int128_t)t->v * t->q == ((int128_t)1) << 62); + return eta; +} + +/* Compute the transition matrix and eta for 62 divsteps (variable time). + * + * Input: eta: initial eta + * f0: bottom limb of initial f + * g0: bottom limb of initial g + * Output: t: transition matrix + * Return: final eta + * + * Implements the divsteps_n_matrix_var function from the explanation. + */ +static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) { + /* Transformation matrix; see comments in secp256k1_modinv64_divsteps_62. */ + uint64_t u = 1, v = 0, q = 0, r = 1; + uint64_t f = f0, g = g0, m; + uint32_t w; + int i = 62, limit, zeros; + + for (;;) { + /* Use a sentinel bit to count zeros only up to i. */ + zeros = secp256k1_ctz64_var(g | (UINT64_MAX << i)); + /* Perform zeros divsteps at once; they all just divide g by two. */ + g >>= zeros; + u <<= zeros; + v <<= zeros; + eta -= zeros; + i -= zeros; + /* We're done once we've done 62 divsteps. */ + if (i == 0) break; + VERIFY_CHECK((f & 1) == 1); + VERIFY_CHECK((g & 1) == 1); + VERIFY_CHECK((u * f0 + v * g0) == f << (62 - i)); + VERIFY_CHECK((q * f0 + r * g0) == g << (62 - i)); + /* Bounds on eta that follow from the bounds on iteration count (max 12*62 divsteps). */ + VERIFY_CHECK(eta >= -745 && eta <= 745); + /* If eta is negative, negate it and replace f,g with g,-f. */ + if (eta < 0) { + uint64_t tmp; + eta = -eta; + tmp = f; f = g; g = -tmp; + tmp = u; u = q; q = -tmp; + tmp = v; v = r; r = -tmp; + /* Use a formula to cancel out up to 6 bits of g. Also, no more than i can be cancelled + * out (as we'd be done before that point), and no more than eta+1 can be done as its + * will flip again once that happens. */ + limit = ((int)eta + 1) > i ? i : ((int)eta + 1); + VERIFY_CHECK(limit > 0 && limit <= 62); + /* m is a mask for the bottom min(limit, 6) bits. */ + m = (UINT64_MAX >> (64 - limit)) & 63U; + /* Find what multiple of f must be added to g to cancel its bottom min(limit, 6) + * bits. */ + w = (f * g * (f * f - 2)) & m; + } else { + /* In this branch, use a simpler formula that only lets us cancel up to 4 bits of g, as + * eta tends to be smaller here. */ + limit = ((int)eta + 1) > i ? i : ((int)eta + 1); + VERIFY_CHECK(limit > 0 && limit <= 62); + /* m is a mask for the bottom min(limit, 4) bits. */ + m = (UINT64_MAX >> (64 - limit)) & 15U; + /* Find what multiple of f must be added to g to cancel its bottom min(limit, 4) + * bits. */ + w = f + (((f + 1) & 4) << 1); + w = (-w * g) & m; + } + g += f * w; + q += u * w; + r += v * w; + VERIFY_CHECK((g & m) == 0); + } + /* Return data in t and return value. */ + t->u = (int64_t)u; + t->v = (int64_t)v; + t->q = (int64_t)q; + t->r = (int64_t)r; + /* The determinant of t must be a power of two. This guarantees that multiplication with t + * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which + * will be divided out again). As each divstep's individual matrix has determinant 2, the + * aggregate of 62 of them will have determinant 2^62. */ + VERIFY_CHECK((int128_t)t->u * t->r - (int128_t)t->v * t->q == ((int128_t)1) << 62); + return eta; +} + +/* Compute (t/2^62) * [d, e] mod modulus, where t is a transition matrix for 62 divsteps. + * + * On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range + * (-2^62,2^62). + * + * This implements the update_de function from the explanation. + */ +static void secp256k1_modinv64_update_de_62(secp256k1_modinv64_signed62 *d, secp256k1_modinv64_signed62 *e, const secp256k1_modinv64_trans2x2 *t, const secp256k1_modinv64_modinfo* modinfo) { + const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + const int64_t d0 = d->v[0], d1 = d->v[1], d2 = d->v[2], d3 = d->v[3], d4 = d->v[4]; + const int64_t e0 = e->v[0], e1 = e->v[1], e2 = e->v[2], e3 = e->v[3], e4 = e->v[4]; + const int64_t u = t->u, v = t->v, q = t->q, r = t->r; + int64_t md, me, sd, se; + int128_t cd, ce; +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, 1) < 0); /* d < modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, 1) < 0); /* e < modulus */ + VERIFY_CHECK((secp256k1_modinv64_abs(u) + secp256k1_modinv64_abs(v)) >= 0); /* |u|+|v| doesn't overflow */ + VERIFY_CHECK((secp256k1_modinv64_abs(q) + secp256k1_modinv64_abs(r)) >= 0); /* |q|+|r| doesn't overflow */ + VERIFY_CHECK((secp256k1_modinv64_abs(u) + secp256k1_modinv64_abs(v)) <= M62 + 1); /* |u|+|v| <= 2^62 */ + VERIFY_CHECK((secp256k1_modinv64_abs(q) + secp256k1_modinv64_abs(r)) <= M62 + 1); /* |q|+|r| <= 2^62 */ +#endif + /* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */ + sd = d4 >> 63; + se = e4 >> 63; + md = (u & sd) + (v & se); + me = (q & sd) + (r & se); + /* Begin computing t*[d,e]. */ + cd = (int128_t)u * d0 + (int128_t)v * e0; + ce = (int128_t)q * d0 + (int128_t)r * e0; + /* Correct md,me so that t*[d,e]+modulus*[md,me] has 62 zero bottom bits. */ + md -= (modinfo->modulus_inv62 * (uint64_t)cd + md) & M62; + me -= (modinfo->modulus_inv62 * (uint64_t)ce + me) & M62; + /* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */ + cd += (int128_t)modinfo->modulus.v[0] * md; + ce += (int128_t)modinfo->modulus.v[0] * me; + /* Verify that the low 62 bits of the computation are indeed zero, and then throw them away. */ + VERIFY_CHECK(((int64_t)cd & M62) == 0); cd >>= 62; + VERIFY_CHECK(((int64_t)ce & M62) == 0); ce >>= 62; + /* Compute limb 1 of t*[d,e]+modulus*[md,me], and store it as output limb 0 (= down shift). */ + cd += (int128_t)u * d1 + (int128_t)v * e1; + ce += (int128_t)q * d1 + (int128_t)r * e1; + if (modinfo->modulus.v[1]) { /* Optimize for the case where limb of modulus is zero. */ + cd += (int128_t)modinfo->modulus.v[1] * md; + ce += (int128_t)modinfo->modulus.v[1] * me; + } + d->v[0] = (int64_t)cd & M62; cd >>= 62; + e->v[0] = (int64_t)ce & M62; ce >>= 62; + /* Compute limb 2 of t*[d,e]+modulus*[md,me], and store it as output limb 1. */ + cd += (int128_t)u * d2 + (int128_t)v * e2; + ce += (int128_t)q * d2 + (int128_t)r * e2; + if (modinfo->modulus.v[2]) { /* Optimize for the case where limb of modulus is zero. */ + cd += (int128_t)modinfo->modulus.v[2] * md; + ce += (int128_t)modinfo->modulus.v[2] * me; + } + d->v[1] = (int64_t)cd & M62; cd >>= 62; + e->v[1] = (int64_t)ce & M62; ce >>= 62; + /* Compute limb 3 of t*[d,e]+modulus*[md,me], and store it as output limb 2. */ + cd += (int128_t)u * d3 + (int128_t)v * e3; + ce += (int128_t)q * d3 + (int128_t)r * e3; + if (modinfo->modulus.v[3]) { /* Optimize for the case where limb of modulus is zero. */ + cd += (int128_t)modinfo->modulus.v[3] * md; + ce += (int128_t)modinfo->modulus.v[3] * me; + } + d->v[2] = (int64_t)cd & M62; cd >>= 62; + e->v[2] = (int64_t)ce & M62; ce >>= 62; + /* Compute limb 4 of t*[d,e]+modulus*[md,me], and store it as output limb 3. */ + cd += (int128_t)u * d4 + (int128_t)v * e4; + ce += (int128_t)q * d4 + (int128_t)r * e4; + cd += (int128_t)modinfo->modulus.v[4] * md; + ce += (int128_t)modinfo->modulus.v[4] * me; + d->v[3] = (int64_t)cd & M62; cd >>= 62; + e->v[3] = (int64_t)ce & M62; ce >>= 62; + /* What remains is limb 5 of t*[d,e]+modulus*[md,me]; store it as output limb 4. */ + d->v[4] = (int64_t)cd; + e->v[4] = (int64_t)ce; +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, -2) > 0); /* d > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(d, 5, &modinfo->modulus, 1) < 0); /* d < modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, -2) > 0); /* e > -2*modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(e, 5, &modinfo->modulus, 1) < 0); /* e < modulus */ +#endif +} + +/* Compute (t/2^62) * [f, g], where t is a transition matrix for 62 divsteps. + * + * This implements the update_fg function from the explanation. + */ +static void secp256k1_modinv64_update_fg_62(secp256k1_modinv64_signed62 *f, secp256k1_modinv64_signed62 *g, const secp256k1_modinv64_trans2x2 *t) { + const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + const int64_t f0 = f->v[0], f1 = f->v[1], f2 = f->v[2], f3 = f->v[3], f4 = f->v[4]; + const int64_t g0 = g->v[0], g1 = g->v[1], g2 = g->v[2], g3 = g->v[3], g4 = g->v[4]; + const int64_t u = t->u, v = t->v, q = t->q, r = t->r; + int128_t cf, cg; + /* Start computing t*[f,g]. */ + cf = (int128_t)u * f0 + (int128_t)v * g0; + cg = (int128_t)q * f0 + (int128_t)r * g0; + /* Verify that the bottom 62 bits of the result are zero, and then throw them away. */ + VERIFY_CHECK(((int64_t)cf & M62) == 0); cf >>= 62; + VERIFY_CHECK(((int64_t)cg & M62) == 0); cg >>= 62; + /* Compute limb 1 of t*[f,g], and store it as output limb 0 (= down shift). */ + cf += (int128_t)u * f1 + (int128_t)v * g1; + cg += (int128_t)q * f1 + (int128_t)r * g1; + f->v[0] = (int64_t)cf & M62; cf >>= 62; + g->v[0] = (int64_t)cg & M62; cg >>= 62; + /* Compute limb 2 of t*[f,g], and store it as output limb 1. */ + cf += (int128_t)u * f2 + (int128_t)v * g2; + cg += (int128_t)q * f2 + (int128_t)r * g2; + f->v[1] = (int64_t)cf & M62; cf >>= 62; + g->v[1] = (int64_t)cg & M62; cg >>= 62; + /* Compute limb 3 of t*[f,g], and store it as output limb 2. */ + cf += (int128_t)u * f3 + (int128_t)v * g3; + cg += (int128_t)q * f3 + (int128_t)r * g3; + f->v[2] = (int64_t)cf & M62; cf >>= 62; + g->v[2] = (int64_t)cg & M62; cg >>= 62; + /* Compute limb 4 of t*[f,g], and store it as output limb 3. */ + cf += (int128_t)u * f4 + (int128_t)v * g4; + cg += (int128_t)q * f4 + (int128_t)r * g4; + f->v[3] = (int64_t)cf & M62; cf >>= 62; + g->v[3] = (int64_t)cg & M62; cg >>= 62; + /* What remains is limb 5 of t*[f,g]; store it as output limb 4. */ + f->v[4] = (int64_t)cf; + g->v[4] = (int64_t)cg; +} + +/* Compute (t/2^62) * [f, g], where t is a transition matrix for 62 divsteps. + * + * Version that operates on a variable number of limbs in f and g. + * + * This implements the update_fg function from the explanation. + */ +static void secp256k1_modinv64_update_fg_62_var(int len, secp256k1_modinv64_signed62 *f, secp256k1_modinv64_signed62 *g, const secp256k1_modinv64_trans2x2 *t) { + const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + const int64_t u = t->u, v = t->v, q = t->q, r = t->r; + int64_t fi, gi; + int128_t cf, cg; + int i; + VERIFY_CHECK(len > 0); + /* Start computing t*[f,g]. */ + fi = f->v[0]; + gi = g->v[0]; + cf = (int128_t)u * fi + (int128_t)v * gi; + cg = (int128_t)q * fi + (int128_t)r * gi; + /* Verify that the bottom 62 bits of the result are zero, and then throw them away. */ + VERIFY_CHECK(((int64_t)cf & M62) == 0); cf >>= 62; + VERIFY_CHECK(((int64_t)cg & M62) == 0); cg >>= 62; + /* Now iteratively compute limb i=1..len of t*[f,g], and store them in output limb i-1 (shifting + * down by 62 bits). */ + for (i = 1; i < len; ++i) { + fi = f->v[i]; + gi = g->v[i]; + cf += (int128_t)u * fi + (int128_t)v * gi; + cg += (int128_t)q * fi + (int128_t)r * gi; + f->v[i - 1] = (int64_t)cf & M62; cf >>= 62; + g->v[i - 1] = (int64_t)cg & M62; cg >>= 62; + } + /* What remains is limb (len) of t*[f,g]; store it as output limb (len-1). */ + f->v[len - 1] = (int64_t)cf; + g->v[len - 1] = (int64_t)cg; +} + +/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */ +static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) { + /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ + secp256k1_modinv64_signed62 d = {{0, 0, 0, 0, 0}}; + secp256k1_modinv64_signed62 e = {{1, 0, 0, 0, 0}}; + secp256k1_modinv64_signed62 f = modinfo->modulus; + secp256k1_modinv64_signed62 g = *x; + int i; + int64_t eta = -1; + + /* Do 12 iterations of 62 divsteps each = 744 divsteps. 724 suffices for 256-bit inputs. */ + for (i = 0; i < 12; ++i) { + /* Compute transition matrix and new eta after 62 divsteps. */ + secp256k1_modinv64_trans2x2 t; + eta = secp256k1_modinv64_divsteps_62(eta, f.v[0], g.v[0], &t); + /* Update d,e using that transition matrix. */ + secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo); + /* Update f,g using that transition matrix. */ +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + secp256k1_modinv64_update_fg_62(&f, &g, &t); +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + } + + /* At this point sufficient iterations have been performed that g must have reached 0 + * and (if g was not originally 0) f must now equal +/- GCD of the initial f, g + * values i.e. +/- 1, and d now contains +/- the modular inverse. */ +#ifdef VERIFY + /* g == 0 */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, 5, &SECP256K1_SIGNED62_ONE, 0) == 0); + /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, 5, &SECP256K1_SIGNED62_ONE, -1) == 0 || + secp256k1_modinv64_mul_cmp_62(&f, 5, &SECP256K1_SIGNED62_ONE, 1) == 0 || + (secp256k1_modinv64_mul_cmp_62(x, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && + secp256k1_modinv64_mul_cmp_62(&d, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && + (secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, 1) == 0 || + secp256k1_modinv64_mul_cmp_62(&f, 5, &modinfo->modulus, -1) == 0))); +#endif + + /* Optionally negate d, normalize to [0,modulus), and return it. */ + secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo); + *x = d; +} + +/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */ +static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) { + /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */ + secp256k1_modinv64_signed62 d = {{0, 0, 0, 0, 0}}; + secp256k1_modinv64_signed62 e = {{1, 0, 0, 0, 0}}; + secp256k1_modinv64_signed62 f = modinfo->modulus; + secp256k1_modinv64_signed62 g = *x; +#ifdef VERIFY + int i = 0; +#endif + int j, len = 5; + int64_t eta = -1; + int64_t cond, fn, gn; + + /* Do iterations of 62 divsteps each until g=0. */ + while (1) { + /* Compute transition matrix and new eta after 62 divsteps. */ + secp256k1_modinv64_trans2x2 t; + eta = secp256k1_modinv64_divsteps_62_var(eta, f.v[0], g.v[0], &t); + /* Update d,e using that transition matrix. */ + secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo); + /* Update f,g using that transition matrix. */ +#ifdef VERIFY + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + secp256k1_modinv64_update_fg_62_var(len, &f, &g, &t); + /* If the bottom limb of g is zero, there is a chance that g=0. */ + if (g.v[0] == 0) { + cond = 0; + /* Check if the other limbs are also 0. */ + for (j = 1; j < len; ++j) { + cond |= g.v[j]; + } + /* If so, we're done. */ + if (cond == 0) break; + } + + /* Determine if len>1 and limb (len-1) of both f and g is 0 or -1. */ + fn = f.v[len - 1]; + gn = g.v[len - 1]; + cond = ((int64_t)len - 2) >> 63; + cond |= fn ^ (fn >> 63); + cond |= gn ^ (gn >> 63); + /* If so, reduce length, propagating the sign of f and g's top limb into the one below. */ + if (cond == 0) { + f.v[len - 2] |= (uint64_t)fn << 62; + g.v[len - 2] |= (uint64_t)gn << 62; + --len; + } +#ifdef VERIFY + VERIFY_CHECK(++i < 12); /* We should never need more than 12*62 = 744 divsteps */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) > 0); /* f > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) <= 0); /* f <= modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, -1) > 0); /* g > -modulus */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &modinfo->modulus, 1) < 0); /* g < modulus */ +#endif + } + + /* At this point g is 0 and (if g was not originally 0) f must now equal +/- GCD of + * the initial f, g values i.e. +/- 1, and d now contains +/- the modular inverse. */ +#ifdef VERIFY + /* g == 0 */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&g, len, &SECP256K1_SIGNED62_ONE, 0) == 0); + /* |f| == 1, or (x == 0 and d == 0 and |f|=modulus) */ + VERIFY_CHECK(secp256k1_modinv64_mul_cmp_62(&f, len, &SECP256K1_SIGNED62_ONE, -1) == 0 || + secp256k1_modinv64_mul_cmp_62(&f, len, &SECP256K1_SIGNED62_ONE, 1) == 0 || + (secp256k1_modinv64_mul_cmp_62(x, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && + secp256k1_modinv64_mul_cmp_62(&d, 5, &SECP256K1_SIGNED62_ONE, 0) == 0 && + (secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, 1) == 0 || + secp256k1_modinv64_mul_cmp_62(&f, len, &modinfo->modulus, -1) == 0))); +#endif + + /* Optionally negate d, normalize to [0,modulus), and return it. */ + secp256k1_modinv64_normalize_62(&d, f.v[len - 1], modinfo); + *x = d; +} + +#endif /* SECP256K1_MODINV64_IMPL_H */ diff --git a/src/num.h b/src/num.h deleted file mode 100644 index 59a5cf2..0000000 --- a/src/num.h +++ /dev/null @@ -1,74 +0,0 @@ -/*********************************************************************** - * Copyright (c) 2013, 2014 Pieter Wuille * - * Distributed under the MIT software license, see the accompanying * - * file COPYING or https://www.opensource.org/licenses/mit-license.php.* - ***********************************************************************/ - -#ifndef SECP256K1_NUM_H -#define SECP256K1_NUM_H - -#ifndef USE_NUM_NONE - -#if defined HAVE_CONFIG_H -#include "libsecp256k1-config.h" -#endif - -#if defined(USE_NUM_GMP) -#include "num_gmp.h" -#else -#error "Please select num implementation" -#endif - -/** Copy a number. */ -static void secp256k1_num_copy(secp256k1_num *r, const secp256k1_num *a); - -/** Convert a number's absolute value to a binary big-endian string. - * There must be enough place. */ -static void secp256k1_num_get_bin(unsigned char *r, unsigned int rlen, const secp256k1_num *a); - -/** Set a number to the value of a binary big-endian string. */ -static void secp256k1_num_set_bin(secp256k1_num *r, const unsigned char *a, unsigned int alen); - -/** Compute a modular inverse. The input must be less than the modulus. */ -static void secp256k1_num_mod_inverse(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *m); - -/** Compute the jacobi symbol (a|b). b must be positive and odd. */ -static int secp256k1_num_jacobi(const secp256k1_num *a, const secp256k1_num *b); - -/** Compare the absolute value of two numbers. */ -static int secp256k1_num_cmp(const secp256k1_num *a, const secp256k1_num *b); - -/** Test whether two number are equal (including sign). */ -static int secp256k1_num_eq(const secp256k1_num *a, const secp256k1_num *b); - -/** Add two (signed) numbers. */ -static void secp256k1_num_add(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b); - -/** Subtract two (signed) numbers. */ -static void secp256k1_num_sub(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b); - -/** Multiply two (signed) numbers. */ -static void secp256k1_num_mul(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b); - -/** Replace a number by its remainder modulo m. M's sign is ignored. The result is a number between 0 and m-1, - even if r was negative. */ -static void secp256k1_num_mod(secp256k1_num *r, const secp256k1_num *m); - -/** Right-shift the passed number by bits bits. */ -static void secp256k1_num_shift(secp256k1_num *r, int bits); - -/** Check whether a number is zero. */ -static int secp256k1_num_is_zero(const secp256k1_num *a); - -/** Check whether a number is one. */ -static int secp256k1_num_is_one(const secp256k1_num *a); - -/** Check whether a number is strictly negative. */ -static int secp256k1_num_is_neg(const secp256k1_num *a); - -/** Change a number's sign. */ -static void secp256k1_num_negate(secp256k1_num *r); - -#endif - -#endif /* SECP256K1_NUM_H */ diff --git a/src/num_gmp.h b/src/num_gmp.h deleted file mode 100644 index cc6c51a..0000000 --- a/src/num_gmp.h +++ /dev/null @@ -1,20 +0,0 @@ -/*********************************************************************** - * Copyright (c) 2013, 2014 Pieter Wuille * - * Distributed under the MIT software license, see the accompanying * - * file COPYING or https://www.opensource.org/licenses/mit-license.php.* - ***********************************************************************/ - -#ifndef SECP256K1_NUM_REPR_H -#define SECP256K1_NUM_REPR_H - -#include - -#define NUM_LIMBS ((256+GMP_NUMB_BITS-1)/GMP_NUMB_BITS) - -typedef struct { - mp_limb_t data[2*NUM_LIMBS]; - int neg; - int limbs; -} secp256k1_num; - -#endif /* SECP256K1_NUM_REPR_H */ diff --git a/src/num_gmp_impl.h b/src/num_gmp_impl.h deleted file mode 100644 index c07947d..0000000 --- a/src/num_gmp_impl.h +++ /dev/null @@ -1,288 +0,0 @@ -/*********************************************************************** - * Copyright (c) 2013, 2014 Pieter Wuille * - * Distributed under the MIT software license, see the accompanying * - * file COPYING or https://www.opensource.org/licenses/mit-license.php.* - ***********************************************************************/ - -#ifndef SECP256K1_NUM_REPR_IMPL_H -#define SECP256K1_NUM_REPR_IMPL_H - -#include -#include -#include - -#include "util.h" -#include "num.h" - -#ifdef VERIFY -static void secp256k1_num_sanity(const secp256k1_num *a) { - VERIFY_CHECK(a->limbs == 1 || (a->limbs > 1 && a->data[a->limbs-1] != 0)); -} -#else -#define secp256k1_num_sanity(a) do { } while(0) -#endif - -static void secp256k1_num_copy(secp256k1_num *r, const secp256k1_num *a) { - *r = *a; -} - -static void secp256k1_num_get_bin(unsigned char *r, unsigned int rlen, const secp256k1_num *a) { - unsigned char tmp[65]; - int len = 0; - int shift = 0; - if (a->limbs>1 || a->data[0] != 0) { - len = mpn_get_str(tmp, 256, (mp_limb_t*)a->data, a->limbs); - } - while (shift < len && tmp[shift] == 0) shift++; - VERIFY_CHECK(len-shift <= (int)rlen); - memset(r, 0, rlen - len + shift); - if (len > shift) { - memcpy(r + rlen - len + shift, tmp + shift, len - shift); - } - memset(tmp, 0, sizeof(tmp)); -} - -static void secp256k1_num_set_bin(secp256k1_num *r, const unsigned char *a, unsigned int alen) { - int len; - VERIFY_CHECK(alen > 0); - VERIFY_CHECK(alen <= 64); - len = mpn_set_str(r->data, a, alen, 256); - if (len == 0) { - r->data[0] = 0; - len = 1; - } - VERIFY_CHECK(len <= NUM_LIMBS*2); - r->limbs = len; - r->neg = 0; - while (r->limbs > 1 && r->data[r->limbs-1]==0) { - r->limbs--; - } -} - -static void secp256k1_num_add_abs(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b) { - mp_limb_t c = mpn_add(r->data, a->data, a->limbs, b->data, b->limbs); - r->limbs = a->limbs; - if (c != 0) { - VERIFY_CHECK(r->limbs < 2*NUM_LIMBS); - r->data[r->limbs++] = c; - } -} - -static void secp256k1_num_sub_abs(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b) { - mp_limb_t c = mpn_sub(r->data, a->data, a->limbs, b->data, b->limbs); - (void)c; - VERIFY_CHECK(c == 0); - r->limbs = a->limbs; - while (r->limbs > 1 && r->data[r->limbs-1]==0) { - r->limbs--; - } -} - -static void secp256k1_num_mod(secp256k1_num *r, const secp256k1_num *m) { - secp256k1_num_sanity(r); - secp256k1_num_sanity(m); - - if (r->limbs >= m->limbs) { - mp_limb_t t[2*NUM_LIMBS]; - mpn_tdiv_qr(t, r->data, 0, r->data, r->limbs, m->data, m->limbs); - memset(t, 0, sizeof(t)); - r->limbs = m->limbs; - while (r->limbs > 1 && r->data[r->limbs-1]==0) { - r->limbs--; - } - } - - if (r->neg && (r->limbs > 1 || r->data[0] != 0)) { - secp256k1_num_sub_abs(r, m, r); - r->neg = 0; - } -} - -static void secp256k1_num_mod_inverse(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *m) { - int i; - mp_limb_t g[NUM_LIMBS+1]; - mp_limb_t u[NUM_LIMBS+1]; - mp_limb_t v[NUM_LIMBS+1]; - mp_size_t sn; - mp_size_t gn; - secp256k1_num_sanity(a); - secp256k1_num_sanity(m); - - /** mpn_gcdext computes: (G,S) = gcdext(U,V), where - * * G = gcd(U,V) - * * G = U*S + V*T - * * U has equal or more limbs than V, and V has no padding - * If we set U to be (a padded version of) a, and V = m: - * G = a*S + m*T - * G = a*S mod m - * Assuming G=1: - * S = 1/a mod m - */ - VERIFY_CHECK(m->limbs <= NUM_LIMBS); - VERIFY_CHECK(m->data[m->limbs-1] != 0); - for (i = 0; i < m->limbs; i++) { - u[i] = (i < a->limbs) ? a->data[i] : 0; - v[i] = m->data[i]; - } - sn = NUM_LIMBS+1; - gn = mpn_gcdext(g, r->data, &sn, u, m->limbs, v, m->limbs); - (void)gn; - VERIFY_CHECK(gn == 1); - VERIFY_CHECK(g[0] == 1); - r->neg = a->neg ^ m->neg; - if (sn < 0) { - mpn_sub(r->data, m->data, m->limbs, r->data, -sn); - r->limbs = m->limbs; - while (r->limbs > 1 && r->data[r->limbs-1]==0) { - r->limbs--; - } - } else { - r->limbs = sn; - } - memset(g, 0, sizeof(g)); - memset(u, 0, sizeof(u)); - memset(v, 0, sizeof(v)); -} - -static int secp256k1_num_jacobi(const secp256k1_num *a, const secp256k1_num *b) { - int ret; - mpz_t ga, gb; - secp256k1_num_sanity(a); - secp256k1_num_sanity(b); - VERIFY_CHECK(!b->neg && (b->limbs > 0) && (b->data[0] & 1)); - - mpz_inits(ga, gb, NULL); - - mpz_import(gb, b->limbs, -1, sizeof(mp_limb_t), 0, 0, b->data); - mpz_import(ga, a->limbs, -1, sizeof(mp_limb_t), 0, 0, a->data); - if (a->neg) { - mpz_neg(ga, ga); - } - - ret = mpz_jacobi(ga, gb); - - mpz_clears(ga, gb, NULL); - - return ret; -} - -static int secp256k1_num_is_one(const secp256k1_num *a) { - return (a->limbs == 1 && a->data[0] == 1); -} - -static int secp256k1_num_is_zero(const secp256k1_num *a) { - return (a->limbs == 1 && a->data[0] == 0); -} - -static int secp256k1_num_is_neg(const secp256k1_num *a) { - return (a->limbs > 1 || a->data[0] != 0) && a->neg; -} - -static int secp256k1_num_cmp(const secp256k1_num *a, const secp256k1_num *b) { - if (a->limbs > b->limbs) { - return 1; - } - if (a->limbs < b->limbs) { - return -1; - } - return mpn_cmp(a->data, b->data, a->limbs); -} - -static int secp256k1_num_eq(const secp256k1_num *a, const secp256k1_num *b) { - if (a->limbs > b->limbs) { - return 0; - } - if (a->limbs < b->limbs) { - return 0; - } - if ((a->neg && !secp256k1_num_is_zero(a)) != (b->neg && !secp256k1_num_is_zero(b))) { - return 0; - } - return mpn_cmp(a->data, b->data, a->limbs) == 0; -} - -static void secp256k1_num_subadd(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b, int bneg) { - if (!(b->neg ^ bneg ^ a->neg)) { /* a and b have the same sign */ - r->neg = a->neg; - if (a->limbs >= b->limbs) { - secp256k1_num_add_abs(r, a, b); - } else { - secp256k1_num_add_abs(r, b, a); - } - } else { - if (secp256k1_num_cmp(a, b) > 0) { - r->neg = a->neg; - secp256k1_num_sub_abs(r, a, b); - } else { - r->neg = b->neg ^ bneg; - secp256k1_num_sub_abs(r, b, a); - } - } -} - -static void secp256k1_num_add(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b) { - secp256k1_num_sanity(a); - secp256k1_num_sanity(b); - secp256k1_num_subadd(r, a, b, 0); -} - -static void secp256k1_num_sub(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b) { - secp256k1_num_sanity(a); - secp256k1_num_sanity(b); - secp256k1_num_subadd(r, a, b, 1); -} - -static void secp256k1_num_mul(secp256k1_num *r, const secp256k1_num *a, const secp256k1_num *b) { - mp_limb_t tmp[2*NUM_LIMBS+1]; - secp256k1_num_sanity(a); - secp256k1_num_sanity(b); - - VERIFY_CHECK(a->limbs + b->limbs <= 2*NUM_LIMBS+1); - if ((a->limbs==1 && a->data[0]==0) || (b->limbs==1 && b->data[0]==0)) { - r->limbs = 1; - r->neg = 0; - r->data[0] = 0; - return; - } - if (a->limbs >= b->limbs) { - mpn_mul(tmp, a->data, a->limbs, b->data, b->limbs); - } else { - mpn_mul(tmp, b->data, b->limbs, a->data, a->limbs); - } - r->limbs = a->limbs + b->limbs; - if (r->limbs > 1 && tmp[r->limbs - 1]==0) { - r->limbs--; - } - VERIFY_CHECK(r->limbs <= 2*NUM_LIMBS); - mpn_copyi(r->data, tmp, r->limbs); - r->neg = a->neg ^ b->neg; - memset(tmp, 0, sizeof(tmp)); -} - -static void secp256k1_num_shift(secp256k1_num *r, int bits) { - if (bits % GMP_NUMB_BITS) { - /* Shift within limbs. */ - mpn_rshift(r->data, r->data, r->limbs, bits % GMP_NUMB_BITS); - } - if (bits >= GMP_NUMB_BITS) { - int i; - /* Shift full limbs. */ - for (i = 0; i < r->limbs; i++) { - int index = i + (bits / GMP_NUMB_BITS); - if (index < r->limbs && index < 2*NUM_LIMBS) { - r->data[i] = r->data[index]; - } else { - r->data[i] = 0; - } - } - } - while (r->limbs>1 && r->data[r->limbs-1]==0) { - r->limbs--; - } -} - -static void secp256k1_num_negate(secp256k1_num *r) { - r->neg ^= 1; -} - -#endif /* SECP256K1_NUM_REPR_IMPL_H */ diff --git a/src/num_impl.h b/src/num_impl.h deleted file mode 100644 index 880598e..0000000 --- a/src/num_impl.h +++ /dev/null @@ -1,24 +0,0 @@ -/*********************************************************************** - * Copyright (c) 2013, 2014 Pieter Wuille * - * Distributed under the MIT software license, see the accompanying * - * file COPYING or https://www.opensource.org/licenses/mit-license.php.* - ***********************************************************************/ - -#ifndef SECP256K1_NUM_IMPL_H -#define SECP256K1_NUM_IMPL_H - -#if defined HAVE_CONFIG_H -#include "libsecp256k1-config.h" -#endif - -#include "num.h" - -#if defined(USE_NUM_GMP) -#include "num_gmp_impl.h" -#elif defined(USE_NUM_NONE) -/* Nothing. */ -#else -#error "Please select num implementation" -#endif - -#endif /* SECP256K1_NUM_IMPL_H */ diff --git a/src/scalar.h b/src/scalar.h index 0b737f9..aaaa3d8 100644 --- a/src/scalar.h +++ b/src/scalar.h @@ -7,7 +7,6 @@ #ifndef SECP256K1_SCALAR_H #define SECP256K1_SCALAR_H -#include "num.h" #include "util.h" #if defined HAVE_CONFIG_H @@ -63,9 +62,6 @@ static void secp256k1_scalar_mul(secp256k1_scalar *r, const secp256k1_scalar *a, * the low bits that were shifted off */ static int secp256k1_scalar_shr_int(secp256k1_scalar *r, int n); -/** Compute the square of a scalar (modulo the group order). */ -static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a); - /** Compute the inverse of a scalar (modulo the group order). */ static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *a); @@ -91,14 +87,6 @@ static int secp256k1_scalar_is_high(const secp256k1_scalar *a); * Returns -1 if the number was negated, 1 otherwise */ static int secp256k1_scalar_cond_negate(secp256k1_scalar *a, int flag); -#ifndef USE_NUM_NONE -/** Convert a scalar to a number. */ -static void secp256k1_scalar_get_num(secp256k1_num *r, const secp256k1_scalar *a); - -/** Get the order of the group as a number. */ -static void secp256k1_scalar_order_get_num(secp256k1_num *r); -#endif - /** Compare two scalars. */ static int secp256k1_scalar_eq(const secp256k1_scalar *a, const secp256k1_scalar *b); diff --git a/src/scalar_4x64_impl.h b/src/scalar_4x64_impl.h index 3eaa041..a1def26 100644 --- a/src/scalar_4x64_impl.h +++ b/src/scalar_4x64_impl.h @@ -7,6 +7,8 @@ #ifndef SECP256K1_SCALAR_REPR_IMPL_H #define SECP256K1_SCALAR_REPR_IMPL_H +#include "modinv64_impl.h" + /* Limbs of the secp256k1 order. */ #define SECP256K1_N_0 ((uint64_t)0xBFD25E8CD0364141ULL) #define SECP256K1_N_1 ((uint64_t)0xBAAEDCE6AF48A03BULL) @@ -212,28 +214,6 @@ static int secp256k1_scalar_cond_negate(secp256k1_scalar *r, int flag) { VERIFY_CHECK(c1 >= th); \ } -/** Add 2*a*b to the number defined by (c0,c1,c2). c2 must never overflow. */ -#define muladd2(a,b) { \ - uint64_t tl, th, th2, tl2; \ - { \ - uint128_t t = (uint128_t)a * b; \ - th = t >> 64; /* at most 0xFFFFFFFFFFFFFFFE */ \ - tl = t; \ - } \ - th2 = th + th; /* at most 0xFFFFFFFFFFFFFFFE (in case th was 0x7FFFFFFFFFFFFFFF) */ \ - c2 += (th2 < th); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((th2 >= th) || (c2 != 0)); \ - tl2 = tl + tl; /* at most 0xFFFFFFFFFFFFFFFE (in case the lowest 63 bits of tl were 0x7FFFFFFFFFFFFFFF) */ \ - th2 += (tl2 < tl); /* at most 0xFFFFFFFFFFFFFFFF */ \ - c0 += tl2; /* overflow is handled on the next line */ \ - th2 += (c0 < tl2); /* second overflow is handled on the next line */ \ - c2 += (c0 < tl2) & (th2 == 0); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((c0 >= tl2) || (th2 != 0) || (c2 != 0)); \ - c1 += th2; /* overflow is handled on the next line */ \ - c2 += (c1 < th2); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((c1 >= th2) || (c2 != 0)); \ -} - /** Add a to the number defined by (c0,c1,c2). c2 must never overflow. */ #define sumadd(a) { \ unsigned int over; \ @@ -743,148 +723,10 @@ static void secp256k1_scalar_mul_512(uint64_t l[8], const secp256k1_scalar *a, c #endif } -static void secp256k1_scalar_sqr_512(uint64_t l[8], const secp256k1_scalar *a) { -#ifdef USE_ASM_X86_64 - __asm__ __volatile__( - /* Preload */ - "movq 0(%%rdi), %%r11\n" - "movq 8(%%rdi), %%r12\n" - "movq 16(%%rdi), %%r13\n" - "movq 24(%%rdi), %%r14\n" - /* (rax,rdx) = a0 * a0 */ - "movq %%r11, %%rax\n" - "mulq %%r11\n" - /* Extract l0 */ - "movq %%rax, 0(%%rsi)\n" - /* (r8,r9,r10) = (rdx,0) */ - "movq %%rdx, %%r8\n" - "xorq %%r9, %%r9\n" - "xorq %%r10, %%r10\n" - /* (r8,r9,r10) += 2 * a0 * a1 */ - "movq %%r11, %%rax\n" - "mulq %%r12\n" - "addq %%rax, %%r8\n" - "adcq %%rdx, %%r9\n" - "adcq $0, %%r10\n" - "addq %%rax, %%r8\n" - "adcq %%rdx, %%r9\n" - "adcq $0, %%r10\n" - /* Extract l1 */ - "movq %%r8, 8(%%rsi)\n" - "xorq %%r8, %%r8\n" - /* (r9,r10,r8) += 2 * a0 * a2 */ - "movq %%r11, %%rax\n" - "mulq %%r13\n" - "addq %%rax, %%r9\n" - "adcq %%rdx, %%r10\n" - "adcq $0, %%r8\n" - "addq %%rax, %%r9\n" - "adcq %%rdx, %%r10\n" - "adcq $0, %%r8\n" - /* (r9,r10,r8) += a1 * a1 */ - "movq %%r12, %%rax\n" - "mulq %%r12\n" - "addq %%rax, %%r9\n" - "adcq %%rdx, %%r10\n" - "adcq $0, %%r8\n" - /* Extract l2 */ - "movq %%r9, 16(%%rsi)\n" - "xorq %%r9, %%r9\n" - /* (r10,r8,r9) += 2 * a0 * a3 */ - "movq %%r11, %%rax\n" - "mulq %%r14\n" - "addq %%rax, %%r10\n" - "adcq %%rdx, %%r8\n" - "adcq $0, %%r9\n" - "addq %%rax, %%r10\n" - "adcq %%rdx, %%r8\n" - "adcq $0, %%r9\n" - /* (r10,r8,r9) += 2 * a1 * a2 */ - "movq %%r12, %%rax\n" - "mulq %%r13\n" - "addq %%rax, %%r10\n" - "adcq %%rdx, %%r8\n" - "adcq $0, %%r9\n" - "addq %%rax, %%r10\n" - "adcq %%rdx, %%r8\n" - "adcq $0, %%r9\n" - /* Extract l3 */ - "movq %%r10, 24(%%rsi)\n" - "xorq %%r10, %%r10\n" - /* (r8,r9,r10) += 2 * a1 * a3 */ - "movq %%r12, %%rax\n" - "mulq %%r14\n" - "addq %%rax, %%r8\n" - "adcq %%rdx, %%r9\n" - "adcq $0, %%r10\n" - "addq %%rax, %%r8\n" - "adcq %%rdx, %%r9\n" - "adcq $0, %%r10\n" - /* (r8,r9,r10) += a2 * a2 */ - "movq %%r13, %%rax\n" - "mulq %%r13\n" - "addq %%rax, %%r8\n" - "adcq %%rdx, %%r9\n" - "adcq $0, %%r10\n" - /* Extract l4 */ - "movq %%r8, 32(%%rsi)\n" - "xorq %%r8, %%r8\n" - /* (r9,r10,r8) += 2 * a2 * a3 */ - "movq %%r13, %%rax\n" - "mulq %%r14\n" - "addq %%rax, %%r9\n" - "adcq %%rdx, %%r10\n" - "adcq $0, %%r8\n" - "addq %%rax, %%r9\n" - "adcq %%rdx, %%r10\n" - "adcq $0, %%r8\n" - /* Extract l5 */ - "movq %%r9, 40(%%rsi)\n" - /* (r10,r8) += a3 * a3 */ - "movq %%r14, %%rax\n" - "mulq %%r14\n" - "addq %%rax, %%r10\n" - "adcq %%rdx, %%r8\n" - /* Extract l6 */ - "movq %%r10, 48(%%rsi)\n" - /* Extract l7 */ - "movq %%r8, 56(%%rsi)\n" - : - : "S"(l), "D"(a->d) - : "rax", "rdx", "r8", "r9", "r10", "r11", "r12", "r13", "r14", "cc", "memory"); -#else - /* 160 bit accumulator. */ - uint64_t c0 = 0, c1 = 0; - uint32_t c2 = 0; - - /* l[0..7] = a[0..3] * b[0..3]. */ - muladd_fast(a->d[0], a->d[0]); - extract_fast(l[0]); - muladd2(a->d[0], a->d[1]); - extract(l[1]); - muladd2(a->d[0], a->d[2]); - muladd(a->d[1], a->d[1]); - extract(l[2]); - muladd2(a->d[0], a->d[3]); - muladd2(a->d[1], a->d[2]); - extract(l[3]); - muladd2(a->d[1], a->d[3]); - muladd(a->d[2], a->d[2]); - extract(l[4]); - muladd2(a->d[2], a->d[3]); - extract(l[5]); - muladd_fast(a->d[3], a->d[3]); - extract_fast(l[6]); - VERIFY_CHECK(c1 == 0); - l[7] = c0; -#endif -} - #undef sumadd #undef sumadd_fast #undef muladd #undef muladd_fast -#undef muladd2 #undef extract #undef extract_fast @@ -906,12 +748,6 @@ static int secp256k1_scalar_shr_int(secp256k1_scalar *r, int n) { return ret; } -static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a) { - uint64_t l[8]; - secp256k1_scalar_sqr_512(l, a); - secp256k1_scalar_reduce_512(r, l); -} - static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) { r1->d[0] = k->d[0]; r1->d[1] = k->d[1]; @@ -955,4 +791,78 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se r->d[3] = (r->d[3] & mask0) | (a->d[3] & mask1); } +static void secp256k1_scalar_from_signed62(secp256k1_scalar *r, const secp256k1_modinv64_signed62 *a) { + const uint64_t a0 = a->v[0], a1 = a->v[1], a2 = a->v[2], a3 = a->v[3], a4 = a->v[4]; + + /* The output from secp256k1_modinv64{_var} should be normalized to range [0,modulus), and + * have limbs in [0,2^62). The modulus is < 2^256, so the top limb must be below 2^(256-62*4). + */ + VERIFY_CHECK(a0 >> 62 == 0); + VERIFY_CHECK(a1 >> 62 == 0); + VERIFY_CHECK(a2 >> 62 == 0); + VERIFY_CHECK(a3 >> 62 == 0); + VERIFY_CHECK(a4 >> 8 == 0); + + r->d[0] = a0 | a1 << 62; + r->d[1] = a1 >> 2 | a2 << 60; + r->d[2] = a2 >> 4 | a3 << 58; + r->d[3] = a3 >> 6 | a4 << 56; + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_check_overflow(r) == 0); +#endif +} + +static void secp256k1_scalar_to_signed62(secp256k1_modinv64_signed62 *r, const secp256k1_scalar *a) { + const uint64_t M62 = UINT64_MAX >> 2; + const uint64_t a0 = a->d[0], a1 = a->d[1], a2 = a->d[2], a3 = a->d[3]; + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_check_overflow(a) == 0); +#endif + + r->v[0] = a0 & M62; + r->v[1] = (a0 >> 62 | a1 << 2) & M62; + r->v[2] = (a1 >> 60 | a2 << 4) & M62; + r->v[3] = (a2 >> 58 | a3 << 6) & M62; + r->v[4] = a3 >> 56; +} + +static const secp256k1_modinv64_modinfo secp256k1_const_modinfo_scalar = { + {{0x3FD25E8CD0364141LL, 0x2ABB739ABD2280EELL, -0x15LL, 0, 256}}, + 0x34F20099AA774EC1LL +}; + +static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) { + secp256k1_modinv64_signed62 s; +#ifdef VERIFY + int zero_in = secp256k1_scalar_is_zero(x); +#endif + secp256k1_scalar_to_signed62(&s, x); + secp256k1_modinv64(&s, &secp256k1_const_modinfo_scalar); + secp256k1_scalar_from_signed62(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_is_zero(r) == zero_in); +#endif +} + +static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) { + secp256k1_modinv64_signed62 s; +#ifdef VERIFY + int zero_in = secp256k1_scalar_is_zero(x); +#endif + secp256k1_scalar_to_signed62(&s, x); + secp256k1_modinv64_var(&s, &secp256k1_const_modinfo_scalar); + secp256k1_scalar_from_signed62(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_is_zero(r) == zero_in); +#endif +} + +SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) { + return !(a->d[0] & 1); +} + #endif /* SECP256K1_SCALAR_REPR_IMPL_H */ diff --git a/src/scalar_8x32_impl.h b/src/scalar_8x32_impl.h index bf98e01..62c7ae7 100644 --- a/src/scalar_8x32_impl.h +++ b/src/scalar_8x32_impl.h @@ -7,6 +7,8 @@ #ifndef SECP256K1_SCALAR_REPR_IMPL_H #define SECP256K1_SCALAR_REPR_IMPL_H +#include "modinv32_impl.h" + /* Limbs of the secp256k1 order. */ #define SECP256K1_N_0 ((uint32_t)0xD0364141UL) #define SECP256K1_N_1 ((uint32_t)0xBFD25E8CUL) @@ -291,28 +293,6 @@ static int secp256k1_scalar_cond_negate(secp256k1_scalar *r, int flag) { VERIFY_CHECK(c1 >= th); \ } -/** Add 2*a*b to the number defined by (c0,c1,c2). c2 must never overflow. */ -#define muladd2(a,b) { \ - uint32_t tl, th, th2, tl2; \ - { \ - uint64_t t = (uint64_t)a * b; \ - th = t >> 32; /* at most 0xFFFFFFFE */ \ - tl = t; \ - } \ - th2 = th + th; /* at most 0xFFFFFFFE (in case th was 0x7FFFFFFF) */ \ - c2 += (th2 < th); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((th2 >= th) || (c2 != 0)); \ - tl2 = tl + tl; /* at most 0xFFFFFFFE (in case the lowest 63 bits of tl were 0x7FFFFFFF) */ \ - th2 += (tl2 < tl); /* at most 0xFFFFFFFF */ \ - c0 += tl2; /* overflow is handled on the next line */ \ - th2 += (c0 < tl2); /* second overflow is handled on the next line */ \ - c2 += (c0 < tl2) & (th2 == 0); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((c0 >= tl2) || (th2 != 0) || (c2 != 0)); \ - c1 += th2; /* overflow is handled on the next line */ \ - c2 += (c1 < th2); /* never overflows by contract (verified the next line) */ \ - VERIFY_CHECK((c1 >= th2) || (c2 != 0)); \ -} - /** Add a to the number defined by (c0,c1,c2). c2 must never overflow. */ #define sumadd(a) { \ unsigned int over; \ @@ -576,71 +556,10 @@ static void secp256k1_scalar_mul_512(uint32_t *l, const secp256k1_scalar *a, con l[15] = c0; } -static void secp256k1_scalar_sqr_512(uint32_t *l, const secp256k1_scalar *a) { - /* 96 bit accumulator. */ - uint32_t c0 = 0, c1 = 0, c2 = 0; - - /* l[0..15] = a[0..7]^2. */ - muladd_fast(a->d[0], a->d[0]); - extract_fast(l[0]); - muladd2(a->d[0], a->d[1]); - extract(l[1]); - muladd2(a->d[0], a->d[2]); - muladd(a->d[1], a->d[1]); - extract(l[2]); - muladd2(a->d[0], a->d[3]); - muladd2(a->d[1], a->d[2]); - extract(l[3]); - muladd2(a->d[0], a->d[4]); - muladd2(a->d[1], a->d[3]); - muladd(a->d[2], a->d[2]); - extract(l[4]); - muladd2(a->d[0], a->d[5]); - muladd2(a->d[1], a->d[4]); - muladd2(a->d[2], a->d[3]); - extract(l[5]); - muladd2(a->d[0], a->d[6]); - muladd2(a->d[1], a->d[5]); - muladd2(a->d[2], a->d[4]); - muladd(a->d[3], a->d[3]); - extract(l[6]); - muladd2(a->d[0], a->d[7]); - muladd2(a->d[1], a->d[6]); - muladd2(a->d[2], a->d[5]); - muladd2(a->d[3], a->d[4]); - extract(l[7]); - muladd2(a->d[1], a->d[7]); - muladd2(a->d[2], a->d[6]); - muladd2(a->d[3], a->d[5]); - muladd(a->d[4], a->d[4]); - extract(l[8]); - muladd2(a->d[2], a->d[7]); - muladd2(a->d[3], a->d[6]); - muladd2(a->d[4], a->d[5]); - extract(l[9]); - muladd2(a->d[3], a->d[7]); - muladd2(a->d[4], a->d[6]); - muladd(a->d[5], a->d[5]); - extract(l[10]); - muladd2(a->d[4], a->d[7]); - muladd2(a->d[5], a->d[6]); - extract(l[11]); - muladd2(a->d[5], a->d[7]); - muladd(a->d[6], a->d[6]); - extract(l[12]); - muladd2(a->d[6], a->d[7]); - extract(l[13]); - muladd_fast(a->d[7], a->d[7]); - extract_fast(l[14]); - VERIFY_CHECK(c1 == 0); - l[15] = c0; -} - #undef sumadd #undef sumadd_fast #undef muladd #undef muladd_fast -#undef muladd2 #undef extract #undef extract_fast @@ -666,12 +585,6 @@ static int secp256k1_scalar_shr_int(secp256k1_scalar *r, int n) { return ret; } -static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a) { - uint32_t l[16]; - secp256k1_scalar_sqr_512(l, a); - secp256k1_scalar_reduce_512(r, l); -} - static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) { r1->d[0] = k->d[0]; r1->d[1] = k->d[1]; @@ -731,4 +644,92 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se r->d[7] = (r->d[7] & mask0) | (a->d[7] & mask1); } +static void secp256k1_scalar_from_signed30(secp256k1_scalar *r, const secp256k1_modinv32_signed30 *a) { + const uint32_t a0 = a->v[0], a1 = a->v[1], a2 = a->v[2], a3 = a->v[3], a4 = a->v[4], + a5 = a->v[5], a6 = a->v[6], a7 = a->v[7], a8 = a->v[8]; + + /* The output from secp256k1_modinv32{_var} should be normalized to range [0,modulus), and + * have limbs in [0,2^30). The modulus is < 2^256, so the top limb must be below 2^(256-30*8). + */ + VERIFY_CHECK(a0 >> 30 == 0); + VERIFY_CHECK(a1 >> 30 == 0); + VERIFY_CHECK(a2 >> 30 == 0); + VERIFY_CHECK(a3 >> 30 == 0); + VERIFY_CHECK(a4 >> 30 == 0); + VERIFY_CHECK(a5 >> 30 == 0); + VERIFY_CHECK(a6 >> 30 == 0); + VERIFY_CHECK(a7 >> 30 == 0); + VERIFY_CHECK(a8 >> 16 == 0); + + r->d[0] = a0 | a1 << 30; + r->d[1] = a1 >> 2 | a2 << 28; + r->d[2] = a2 >> 4 | a3 << 26; + r->d[3] = a3 >> 6 | a4 << 24; + r->d[4] = a4 >> 8 | a5 << 22; + r->d[5] = a5 >> 10 | a6 << 20; + r->d[6] = a6 >> 12 | a7 << 18; + r->d[7] = a7 >> 14 | a8 << 16; + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_check_overflow(r) == 0); +#endif +} + +static void secp256k1_scalar_to_signed30(secp256k1_modinv32_signed30 *r, const secp256k1_scalar *a) { + const uint32_t M30 = UINT32_MAX >> 2; + const uint32_t a0 = a->d[0], a1 = a->d[1], a2 = a->d[2], a3 = a->d[3], + a4 = a->d[4], a5 = a->d[5], a6 = a->d[6], a7 = a->d[7]; + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_check_overflow(a) == 0); +#endif + + r->v[0] = a0 & M30; + r->v[1] = (a0 >> 30 | a1 << 2) & M30; + r->v[2] = (a1 >> 28 | a2 << 4) & M30; + r->v[3] = (a2 >> 26 | a3 << 6) & M30; + r->v[4] = (a3 >> 24 | a4 << 8) & M30; + r->v[5] = (a4 >> 22 | a5 << 10) & M30; + r->v[6] = (a5 >> 20 | a6 << 12) & M30; + r->v[7] = (a6 >> 18 | a7 << 14) & M30; + r->v[8] = a7 >> 16; +} + +static const secp256k1_modinv32_modinfo secp256k1_const_modinfo_scalar = { + {{0x10364141L, 0x3F497A33L, 0x348A03BBL, 0x2BB739ABL, -0x146L, 0, 0, 0, 65536}}, + 0x2A774EC1L +}; + +static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) { + secp256k1_modinv32_signed30 s; +#ifdef VERIFY + int zero_in = secp256k1_scalar_is_zero(x); +#endif + secp256k1_scalar_to_signed30(&s, x); + secp256k1_modinv32(&s, &secp256k1_const_modinfo_scalar); + secp256k1_scalar_from_signed30(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_is_zero(r) == zero_in); +#endif +} + +static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) { + secp256k1_modinv32_signed30 s; +#ifdef VERIFY + int zero_in = secp256k1_scalar_is_zero(x); +#endif + secp256k1_scalar_to_signed30(&s, x); + secp256k1_modinv32_var(&s, &secp256k1_const_modinfo_scalar); + secp256k1_scalar_from_signed30(r, &s); + +#ifdef VERIFY + VERIFY_CHECK(secp256k1_scalar_is_zero(r) == zero_in); +#endif +} + +SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) { + return !(a->d[0] & 1); +} + #endif /* SECP256K1_SCALAR_REPR_IMPL_H */ diff --git a/src/scalar_impl.h b/src/scalar_impl.h index 61c1fbd..e124474 100644 --- a/src/scalar_impl.h +++ b/src/scalar_impl.h @@ -31,231 +31,12 @@ static const secp256k1_scalar secp256k1_scalar_one = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1); static const secp256k1_scalar secp256k1_scalar_zero = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0); -#ifndef USE_NUM_NONE -static void secp256k1_scalar_get_num(secp256k1_num *r, const secp256k1_scalar *a) { - unsigned char c[32]; - secp256k1_scalar_get_b32(c, a); - secp256k1_num_set_bin(r, c, 32); -} - -/** secp256k1 curve order, see secp256k1_ecdsa_const_order_as_fe in ecdsa_impl.h */ -static void secp256k1_scalar_order_get_num(secp256k1_num *r) { -#if defined(EXHAUSTIVE_TEST_ORDER) - static const unsigned char order[32] = { - 0,0,0,0,0,0,0,0, - 0,0,0,0,0,0,0,0, - 0,0,0,0,0,0,0,0, - 0,0,0,0,0,0,0,EXHAUSTIVE_TEST_ORDER - }; -#else - static const unsigned char order[32] = { - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, - 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE, - 0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B, - 0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41 - }; -#endif - secp256k1_num_set_bin(r, order, 32); -} -#endif - static int secp256k1_scalar_set_b32_seckey(secp256k1_scalar *r, const unsigned char *bin) { int overflow; secp256k1_scalar_set_b32(r, bin, &overflow); return (!overflow) & (!secp256k1_scalar_is_zero(r)); } -static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) { -#if defined(EXHAUSTIVE_TEST_ORDER) - int i; - *r = 0; - for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++) - if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1) - *r = i; - /* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus - * have a composite group order; fix it in exhaustive_tests.c). */ - VERIFY_CHECK(*r != 0); -} -#else - secp256k1_scalar *t; - int i; - /* First compute xN as x ^ (2^N - 1) for some values of N, - * and uM as x ^ M for some values of M. */ - secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126; - secp256k1_scalar u2, u5, u9, u11, u13; - - secp256k1_scalar_sqr(&u2, x); - secp256k1_scalar_mul(&x2, &u2, x); - secp256k1_scalar_mul(&u5, &u2, &x2); - secp256k1_scalar_mul(&x3, &u5, &u2); - secp256k1_scalar_mul(&u9, &x3, &u2); - secp256k1_scalar_mul(&u11, &u9, &u2); - secp256k1_scalar_mul(&u13, &u11, &u2); - - secp256k1_scalar_sqr(&x6, &u13); - secp256k1_scalar_sqr(&x6, &x6); - secp256k1_scalar_mul(&x6, &x6, &u11); - - secp256k1_scalar_sqr(&x8, &x6); - secp256k1_scalar_sqr(&x8, &x8); - secp256k1_scalar_mul(&x8, &x8, &x2); - - secp256k1_scalar_sqr(&x14, &x8); - for (i = 0; i < 5; i++) { - secp256k1_scalar_sqr(&x14, &x14); - } - secp256k1_scalar_mul(&x14, &x14, &x6); - - secp256k1_scalar_sqr(&x28, &x14); - for (i = 0; i < 13; i++) { - secp256k1_scalar_sqr(&x28, &x28); - } - secp256k1_scalar_mul(&x28, &x28, &x14); - - secp256k1_scalar_sqr(&x56, &x28); - for (i = 0; i < 27; i++) { - secp256k1_scalar_sqr(&x56, &x56); - } - secp256k1_scalar_mul(&x56, &x56, &x28); - - secp256k1_scalar_sqr(&x112, &x56); - for (i = 0; i < 55; i++) { - secp256k1_scalar_sqr(&x112, &x112); - } - secp256k1_scalar_mul(&x112, &x112, &x56); - - secp256k1_scalar_sqr(&x126, &x112); - for (i = 0; i < 13; i++) { - secp256k1_scalar_sqr(&x126, &x126); - } - secp256k1_scalar_mul(&x126, &x126, &x14); - - /* Then accumulate the final result (t starts at x126). */ - t = &x126; - for (i = 0; i < 3; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u5); /* 101 */ - for (i = 0; i < 4; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 4; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u5); /* 101 */ - for (i = 0; i < 5; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u11); /* 1011 */ - for (i = 0; i < 4; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u11); /* 1011 */ - for (i = 0; i < 4; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 5; i++) { /* 00 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 6; i++) { /* 00 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u13); /* 1101 */ - for (i = 0; i < 4; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u5); /* 101 */ - for (i = 0; i < 3; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 5; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u9); /* 1001 */ - for (i = 0; i < 6; i++) { /* 000 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u5); /* 101 */ - for (i = 0; i < 10; i++) { /* 0000000 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 4; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x3); /* 111 */ - for (i = 0; i < 9; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x8); /* 11111111 */ - for (i = 0; i < 5; i++) { /* 0 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u9); /* 1001 */ - for (i = 0; i < 6; i++) { /* 00 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u11); /* 1011 */ - for (i = 0; i < 4; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u13); /* 1101 */ - for (i = 0; i < 5; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &x2); /* 11 */ - for (i = 0; i < 6; i++) { /* 00 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u13); /* 1101 */ - for (i = 0; i < 10; i++) { /* 000000 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u13); /* 1101 */ - for (i = 0; i < 4; i++) { - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, &u9); /* 1001 */ - for (i = 0; i < 6; i++) { /* 00000 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(t, t, x); /* 1 */ - for (i = 0; i < 8; i++) { /* 00 */ - secp256k1_scalar_sqr(t, t); - } - secp256k1_scalar_mul(r, t, &x6); /* 111111 */ -} - -SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) { - return !(a->d[0] & 1); -} -#endif - -static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) { -#if defined(USE_SCALAR_INV_BUILTIN) - secp256k1_scalar_inverse(r, x); -#elif defined(USE_SCALAR_INV_NUM) - unsigned char b[32]; - secp256k1_num n, m; - secp256k1_scalar t = *x; - secp256k1_scalar_get_b32(b, &t); - secp256k1_num_set_bin(&n, b, 32); - secp256k1_scalar_order_get_num(&m); - secp256k1_num_mod_inverse(&n, &n, &m); - secp256k1_num_get_bin(b, 32, &n); - secp256k1_scalar_set_b32(r, b, NULL); - /* Verify that the inverse was computed correctly, without GMP code. */ - secp256k1_scalar_mul(&t, &t, r); - CHECK(secp256k1_scalar_is_one(&t)); -#else -#error "Please select scalar inverse implementation" -#endif -} - /* These parameters are generated using sage/gen_exhaustive_groups.sage. */ #if defined(EXHAUSTIVE_TEST_ORDER) # if EXHAUSTIVE_TEST_ORDER == 13 diff --git a/src/scalar_low_impl.h b/src/scalar_low_impl.h index 98ffd15..7176f0b 100644 --- a/src/scalar_low_impl.h +++ b/src/scalar_low_impl.h @@ -104,10 +104,6 @@ static int secp256k1_scalar_shr_int(secp256k1_scalar *r, int n) { return ret; } -static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a) { - *r = (*a * *a) % EXHAUSTIVE_TEST_ORDER; -} - static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) { *r1 = *a; *r2 = 0; @@ -125,4 +121,19 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se *r = (*r & mask0) | (*a & mask1); } +static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) { + int i; + *r = 0; + for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++) + if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1) + *r = i; + /* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus + * have a composite group order; fix it in exhaustive_tests.c). */ + VERIFY_CHECK(*r != 0); +} + +static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) { + secp256k1_scalar_inverse(r, x); +} + #endif /* SECP256K1_SCALAR_REPR_IMPL_H */ diff --git a/src/secp256k1.c b/src/secp256k1.c index 4f56c27..aef3f99 100644 --- a/src/secp256k1.c +++ b/src/secp256k1.c @@ -9,7 +9,6 @@ #include "assumptions.h" #include "util.h" -#include "num_impl.h" #include "field_impl.h" #include "scalar_impl.h" #include "group_impl.h" diff --git a/src/tests.c b/src/tests.c index c2d5e28..ba645bb 100644 --- a/src/tests.c +++ b/src/tests.c @@ -18,6 +18,7 @@ #include "include/secp256k1.h" #include "include/secp256k1_preallocated.h" #include "testrand_impl.h" +#include "util.h" #ifdef ENABLE_OPENSSL_TESTS #include "openssl/bn.h" @@ -32,6 +33,11 @@ void ECDSA_SIG_get0(const ECDSA_SIG *sig, const BIGNUM **pr, const BIGNUM **ps) #include "contrib/lax_der_parsing.c" #include "contrib/lax_der_privatekey_parsing.c" +#include "modinv32_impl.h" +#ifdef SECP256K1_WIDEMUL_INT128 +#include "modinv64_impl.h" +#endif + static int count = 64; static secp256k1_context *ctx = NULL; @@ -416,6 +422,25 @@ void run_scratch_tests(void) { secp256k1_context_destroy(none); } +void run_ctz_tests(void) { + static const uint32_t b32[] = {1, 0xffffffff, 0x5e56968f, 0xe0d63129}; + static const uint64_t b64[] = {1, 0xffffffffffffffff, 0xbcd02462139b3fc3, 0x98b5f80c769693ef}; + int shift; + unsigned i; + for (i = 0; i < sizeof(b32) / sizeof(b32[0]); ++i) { + for (shift = 0; shift < 32; ++shift) { + CHECK(secp256k1_ctz32_var_debruijn(b32[i] << shift) == shift); + CHECK(secp256k1_ctz32_var(b32[i] << shift) == shift); + } + } + for (i = 0; i < sizeof(b64) / sizeof(b64[0]); ++i) { + for (shift = 0; shift < 64; ++shift) { + CHECK(secp256k1_ctz64_var_debruijn(b64[i] << shift) == shift); + CHECK(secp256k1_ctz64_var(b64[i] << shift) == shift); + } + } +} + /***** HASH TESTS *****/ void run_sha256_tests(void) { @@ -611,202 +636,448 @@ void run_rand_int(void) { } } -/***** NUM TESTS *****/ +/***** MODINV TESTS *****/ -#ifndef USE_NUM_NONE -void random_num_negate(secp256k1_num *num) { - if (secp256k1_testrand_bits(1)) { - secp256k1_num_negate(num); +/* Compute the modular inverse of (odd) x mod 2^64. */ +uint64_t modinv2p64(uint64_t x) { + /* If w = 1/x mod 2^(2^L), then w*(2 - w*x) = 1/x mod 2^(2^(L+1)). See + * Hacker's Delight second edition, Henry S. Warren, Jr., pages 245-247 for + * why. Start with L=0, for which it is true for every odd x that + * 1/x=1 mod 2. Iterating 6 times gives us 1/x mod 2^64. */ + int l; + uint64_t w = 1; + CHECK(x & 1); + for (l = 0; l < 6; ++l) w *= (2 - w*x); + return w; +} + +/* compute out = (a*b) mod m; if b=NULL, treat b=1. + * + * Out is a 512-bit number (represented as 32 uint16_t's in LE order). The other + * arguments are 256-bit numbers (represented as 16 uint16_t's in LE order). */ +void mulmod256(uint16_t* out, const uint16_t* a, const uint16_t* b, const uint16_t* m) { + uint16_t mul[32]; + uint64_t c = 0; + int i, j; + int m_bitlen = 0; + int mul_bitlen = 0; + + if (b != NULL) { + /* Compute the product of a and b, and put it in mul. */ + for (i = 0; i < 32; ++i) { + for (j = i <= 15 ? 0 : i - 15; j <= i && j <= 15; j++) { + c += (uint64_t)a[j] * b[i - j]; + } + mul[i] = c & 0xFFFF; + c >>= 16; + } + CHECK(c == 0); + + /* compute the highest set bit in mul */ + for (i = 511; i >= 0; --i) { + if ((mul[i >> 4] >> (i & 15)) & 1) { + mul_bitlen = i; + break; + } + } + } else { + /* if b==NULL, set mul=a. */ + memcpy(mul, a, 32); + memset(mul + 16, 0, 32); + /* compute the highest set bit in mul */ + for (i = 255; i >= 0; --i) { + if ((mul[i >> 4] >> (i & 15)) & 1) { + mul_bitlen = i; + break; + } + } } + + /* Compute the highest set bit in m. */ + for (i = 255; i >= 0; --i) { + if ((m[i >> 4] >> (i & 15)) & 1) { + m_bitlen = i; + break; + } + } + + /* Try do mul -= m<= 0; --i) { + uint16_t mul2[32]; + int64_t cs; + + /* Compute mul2 = mul - m<= 0 && bitpos < 256) { + sub |= ((m[bitpos >> 4] >> (bitpos & 15)) & 1) << p; + } + } + /* Add mul[j]-sub to accumulator, and shift bottom 16 bits out to mul2[j]. */ + cs += mul[j]; + cs -= sub; + mul2[j] = (cs & 0xFFFF); + cs >>= 16; + } + /* If remainder of subtraction is 0, set mul = mul2. */ + if (cs == 0) { + memcpy(mul, mul2, sizeof(mul)); + } + } + /* Sanity check: test that all limbs higher than m's highest are zero */ + for (i = (m_bitlen >> 4) + 1; i < 32; ++i) { + CHECK(mul[i] == 0); + } + memcpy(out, mul, 32); } -void random_num_order_test(secp256k1_num *num) { - secp256k1_scalar sc; - random_scalar_order_test(&sc); - secp256k1_scalar_get_num(num, &sc); -} - -void random_num_order(secp256k1_num *num) { - secp256k1_scalar sc; - random_scalar_order(&sc); - secp256k1_scalar_get_num(num, &sc); -} - -void test_num_negate(void) { - secp256k1_num n1; - secp256k1_num n2; - random_num_order_test(&n1); /* n1 = R */ - random_num_negate(&n1); - secp256k1_num_copy(&n2, &n1); /* n2 = R */ - secp256k1_num_sub(&n1, &n2, &n1); /* n1 = n2-n1 = 0 */ - CHECK(secp256k1_num_is_zero(&n1)); - secp256k1_num_copy(&n1, &n2); /* n1 = R */ - secp256k1_num_negate(&n1); /* n1 = -R */ - CHECK(!secp256k1_num_is_zero(&n1)); - secp256k1_num_add(&n1, &n2, &n1); /* n1 = n2+n1 = 0 */ - CHECK(secp256k1_num_is_zero(&n1)); - secp256k1_num_copy(&n1, &n2); /* n1 = R */ - secp256k1_num_negate(&n1); /* n1 = -R */ - CHECK(secp256k1_num_is_neg(&n1) != secp256k1_num_is_neg(&n2)); - secp256k1_num_negate(&n1); /* n1 = R */ - CHECK(secp256k1_num_eq(&n1, &n2)); -} - -void test_num_add_sub(void) { +/* Convert a 256-bit number represented as 16 uint16_t's to signed30 notation. */ +void uint16_to_signed30(secp256k1_modinv32_signed30* out, const uint16_t* in) { int i; - secp256k1_scalar s; - secp256k1_num n1; - secp256k1_num n2; - secp256k1_num n1p2, n2p1, n1m2, n2m1; - random_num_order_test(&n1); /* n1 = R1 */ - if (secp256k1_testrand_bits(1)) { - random_num_negate(&n1); - } - random_num_order_test(&n2); /* n2 = R2 */ - if (secp256k1_testrand_bits(1)) { - random_num_negate(&n2); - } - secp256k1_num_add(&n1p2, &n1, &n2); /* n1p2 = R1 + R2 */ - secp256k1_num_add(&n2p1, &n2, &n1); /* n2p1 = R2 + R1 */ - secp256k1_num_sub(&n1m2, &n1, &n2); /* n1m2 = R1 - R2 */ - secp256k1_num_sub(&n2m1, &n2, &n1); /* n2m1 = R2 - R1 */ - CHECK(secp256k1_num_eq(&n1p2, &n2p1)); - CHECK(!secp256k1_num_eq(&n1p2, &n1m2)); - secp256k1_num_negate(&n2m1); /* n2m1 = -R2 + R1 */ - CHECK(secp256k1_num_eq(&n2m1, &n1m2)); - CHECK(!secp256k1_num_eq(&n2m1, &n1)); - secp256k1_num_add(&n2m1, &n2m1, &n2); /* n2m1 = -R2 + R1 + R2 = R1 */ - CHECK(secp256k1_num_eq(&n2m1, &n1)); - CHECK(!secp256k1_num_eq(&n2p1, &n1)); - secp256k1_num_sub(&n2p1, &n2p1, &n2); /* n2p1 = R2 + R1 - R2 = R1 */ - CHECK(secp256k1_num_eq(&n2p1, &n1)); - - /* check is_one */ - secp256k1_scalar_set_int(&s, 1); - secp256k1_scalar_get_num(&n1, &s); - CHECK(secp256k1_num_is_one(&n1)); - /* check that 2^n + 1 is never 1 */ - secp256k1_scalar_get_num(&n2, &s); - for (i = 0; i < 250; ++i) { - secp256k1_num_add(&n1, &n1, &n1); /* n1 *= 2 */ - secp256k1_num_add(&n1p2, &n1, &n2); /* n1p2 = n1 + 1 */ - CHECK(!secp256k1_num_is_one(&n1p2)); + memset(out->v, 0, sizeof(out->v)); + for (i = 0; i < 256; ++i) { + out->v[i / 30] |= (int32_t)(((in[i >> 4]) >> (i & 15)) & 1) << (i % 30); } } -void test_num_mod(void) { +/* Convert a 256-bit number in signed30 notation to a representation as 16 uint16_t's. */ +void signed30_to_uint16(uint16_t* out, const secp256k1_modinv32_signed30* in) { int i; - secp256k1_scalar s; - secp256k1_num order, n; + memset(out, 0, 32); + for (i = 0; i < 256; ++i) { + out[i >> 4] |= (((in->v[i / 30]) >> (i % 30)) & 1) << (i & 15); + } +} - /* check that 0 mod anything is 0 */ - random_scalar_order_test(&s); - secp256k1_scalar_get_num(&order, &s); - secp256k1_scalar_set_int(&s, 0); - secp256k1_scalar_get_num(&n, &s); - secp256k1_num_mod(&n, &order); - CHECK(secp256k1_num_is_zero(&n)); +/* Randomly mutate the sign of limbs in signed30 representation, without changing the value. */ +void mutate_sign_signed30(secp256k1_modinv32_signed30* x) { + int i; + for (i = 0; i < 16; ++i) { + int pos = secp256k1_testrand_int(8); + if (x->v[pos] > 0 && x->v[pos + 1] <= 0x3fffffff) { + x->v[pos] -= 0x40000000; + x->v[pos + 1] += 1; + } else if (x->v[pos] < 0 && x->v[pos + 1] >= 0x3fffffff) { + x->v[pos] += 0x40000000; + x->v[pos + 1] -= 1; + } + } +} - /* check that anything mod 1 is 0 */ - secp256k1_scalar_set_int(&s, 1); - secp256k1_scalar_get_num(&order, &s); - secp256k1_scalar_get_num(&n, &s); - secp256k1_num_mod(&n, &order); - CHECK(secp256k1_num_is_zero(&n)); +/* Test secp256k1_modinv32{_var}, using inputs in 16-bit limb format, and returning inverse. */ +void test_modinv32_uint16(uint16_t* out, const uint16_t* in, const uint16_t* mod) { + uint16_t tmp[16]; + secp256k1_modinv32_signed30 x; + secp256k1_modinv32_modinfo m; + int i, vartime, nonzero; - /* check that increasing the number past 2^256 does not break this */ - random_scalar_order_test(&s); - secp256k1_scalar_get_num(&n, &s); - /* multiply by 2^8, which'll test this case with high probability */ + uint16_to_signed30(&x, in); + nonzero = (x.v[0] | x.v[1] | x.v[2] | x.v[3] | x.v[4] | x.v[5] | x.v[6] | x.v[7] | x.v[8]) != 0; + uint16_to_signed30(&m.modulus, mod); + mutate_sign_signed30(&m.modulus); + + /* compute 1/modulus mod 2^30 */ + m.modulus_inv30 = modinv2p64(m.modulus.v[0]) & 0x3fffffff; + CHECK(((m.modulus_inv30 * m.modulus.v[0]) & 0x3fffffff) == 1); + + for (vartime = 0; vartime < 2; ++vartime) { + /* compute inverse */ + (vartime ? secp256k1_modinv32_var : secp256k1_modinv32)(&x, &m); + + /* produce output */ + signed30_to_uint16(out, &x); + + /* check if the inverse times the input is 1 (mod m), unless x is 0. */ + mulmod256(tmp, out, in, mod); + CHECK(tmp[0] == nonzero); + for (i = 1; i < 16; ++i) CHECK(tmp[i] == 0); + + /* invert again */ + (vartime ? secp256k1_modinv32_var : secp256k1_modinv32)(&x, &m); + + /* check if the result is equal to the input */ + signed30_to_uint16(tmp, &x); + for (i = 0; i < 16; ++i) CHECK(tmp[i] == in[i]); + } +} + +#ifdef SECP256K1_WIDEMUL_INT128 +/* Convert a 256-bit number represented as 16 uint16_t's to signed62 notation. */ +void uint16_to_signed62(secp256k1_modinv64_signed62* out, const uint16_t* in) { + int i; + memset(out->v, 0, sizeof(out->v)); + for (i = 0; i < 256; ++i) { + out->v[i / 62] |= (int64_t)(((in[i >> 4]) >> (i & 15)) & 1) << (i % 62); + } +} + +/* Convert a 256-bit number in signed62 notation to a representation as 16 uint16_t's. */ +void signed62_to_uint16(uint16_t* out, const secp256k1_modinv64_signed62* in) { + int i; + memset(out, 0, 32); + for (i = 0; i < 256; ++i) { + out[i >> 4] |= (((in->v[i / 62]) >> (i % 62)) & 1) << (i & 15); + } +} + +/* Randomly mutate the sign of limbs in signed62 representation, without changing the value. */ +void mutate_sign_signed62(secp256k1_modinv64_signed62* x) { + static const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + int i; for (i = 0; i < 8; ++i) { - secp256k1_num_add(&n, &n, &n); + int pos = secp256k1_testrand_int(4); + if (x->v[pos] > 0 && x->v[pos + 1] <= M62) { + x->v[pos] -= (M62 + 1); + x->v[pos + 1] += 1; + } else if (x->v[pos] < 0 && x->v[pos + 1] >= -M62) { + x->v[pos] += (M62 + 1); + x->v[pos + 1] -= 1; + } } - secp256k1_num_mod(&n, &order); - CHECK(secp256k1_num_is_zero(&n)); } -void test_num_jacobi(void) { - secp256k1_scalar sqr; - secp256k1_scalar small; - secp256k1_scalar five; /* five is not a quadratic residue */ - secp256k1_num order, n; - int i; - /* squares mod 5 are 1, 4 */ - const int jacobi5[10] = { 0, 1, -1, -1, 1, 0, 1, -1, -1, 1 }; +/* Test secp256k1_modinv64{_var}, using inputs in 16-bit limb format, and returning inverse. */ +void test_modinv64_uint16(uint16_t* out, const uint16_t* in, const uint16_t* mod) { + static const int64_t M62 = (int64_t)(UINT64_MAX >> 2); + uint16_t tmp[16]; + secp256k1_modinv64_signed62 x; + secp256k1_modinv64_modinfo m; + int i, vartime, nonzero; - /* check some small values with 5 as the order */ - secp256k1_scalar_set_int(&five, 5); - secp256k1_scalar_get_num(&order, &five); - for (i = 0; i < 10; ++i) { - secp256k1_scalar_set_int(&small, i); - secp256k1_scalar_get_num(&n, &small); - CHECK(secp256k1_num_jacobi(&n, &order) == jacobi5[i]); - } + uint16_to_signed62(&x, in); + nonzero = (x.v[0] | x.v[1] | x.v[2] | x.v[3] | x.v[4]) != 0; + uint16_to_signed62(&m.modulus, mod); + mutate_sign_signed62(&m.modulus); - /** test large values with 5 as group order */ - secp256k1_scalar_get_num(&order, &five); - /* we first need a scalar which is not a multiple of 5 */ - do { - secp256k1_num fiven; - random_scalar_order_test(&sqr); - secp256k1_scalar_get_num(&fiven, &five); - secp256k1_scalar_get_num(&n, &sqr); - secp256k1_num_mod(&n, &fiven); - } while (secp256k1_num_is_zero(&n)); - /* next force it to be a residue. 2 is a nonresidue mod 5 so we can - * just multiply by two, i.e. add the number to itself */ - if (secp256k1_num_jacobi(&n, &order) == -1) { - secp256k1_num_add(&n, &n, &n); - } + /* compute 1/modulus mod 2^62 */ + m.modulus_inv62 = modinv2p64(m.modulus.v[0]) & M62; + CHECK(((m.modulus_inv62 * m.modulus.v[0]) & M62) == 1); - /* test residue */ - CHECK(secp256k1_num_jacobi(&n, &order) == 1); - /* test nonresidue */ - secp256k1_num_add(&n, &n, &n); - CHECK(secp256k1_num_jacobi(&n, &order) == -1); + for (vartime = 0; vartime < 2; ++vartime) { + /* compute inverse */ + (vartime ? secp256k1_modinv64_var : secp256k1_modinv64)(&x, &m); - /** test with secp group order as order */ - secp256k1_scalar_order_get_num(&order); - random_scalar_order_test(&sqr); - secp256k1_scalar_sqr(&sqr, &sqr); - /* test residue */ - secp256k1_scalar_get_num(&n, &sqr); - CHECK(secp256k1_num_jacobi(&n, &order) == 1); - /* test nonresidue */ - secp256k1_scalar_mul(&sqr, &sqr, &five); - secp256k1_scalar_get_num(&n, &sqr); - CHECK(secp256k1_num_jacobi(&n, &order) == -1); - /* test multiple of the order*/ - CHECK(secp256k1_num_jacobi(&order, &order) == 0); + /* produce output */ + signed62_to_uint16(out, &x); - /* check one less than the order */ - secp256k1_scalar_set_int(&small, 1); - secp256k1_scalar_get_num(&n, &small); - secp256k1_num_sub(&n, &order, &n); - CHECK(secp256k1_num_jacobi(&n, &order) == 1); /* sage confirms this is 1 */ -} + /* check if the inverse times the input is 1 (mod m), unless x is 0. */ + mulmod256(tmp, out, in, mod); + CHECK(tmp[0] == nonzero); + for (i = 1; i < 16; ++i) CHECK(tmp[i] == 0); -void run_num_smalltests(void) { - int i; - for (i = 0; i < 100*count; i++) { - test_num_negate(); - test_num_add_sub(); - test_num_mod(); - test_num_jacobi(); + /* invert again */ + (vartime ? secp256k1_modinv64_var : secp256k1_modinv64)(&x, &m); + + /* check if the result is equal to the input */ + signed62_to_uint16(tmp, &x); + for (i = 0; i < 16; ++i) CHECK(tmp[i] == in[i]); } } #endif +/* test if a and b are coprime */ +int coprime(const uint16_t* a, const uint16_t* b) { + uint16_t x[16], y[16], t[16]; + int i; + int iszero; + memcpy(x, a, 32); + memcpy(y, b, 32); + + /* simple gcd loop: while x!=0, (x,y)=(y%x,x) */ + while (1) { + iszero = 1; + for (i = 0; i < 16; ++i) { + if (x[i] != 0) { + iszero = 0; + break; + } + } + if (iszero) break; + mulmod256(t, y, NULL, x); + memcpy(y, x, 32); + memcpy(x, t, 32); + } + + /* return whether y=1 */ + if (y[0] != 1) return 0; + for (i = 1; i < 16; ++i) { + if (y[i] != 0) return 0; + } + return 1; +} + +void run_modinv_tests(void) { + /* Fixed test cases. Each tuple is (input, modulus, output), each as 16x16 bits in LE order. */ + static const uint16_t CASES[][3][16] = { + /* Test case known to need 713 divsteps */ + {{0x1513, 0x5389, 0x54e9, 0x2798, 0x1957, 0x66a0, 0x8057, 0x3477, + 0x7784, 0x1052, 0x326a, 0x9331, 0x6506, 0xa95c, 0x91f3, 0xfb5e}, + {0x2bdd, 0x8df4, 0xcc61, 0x481f, 0xdae5, 0x5ca7, 0xf43b, 0x7d54, + 0x13d6, 0x469b, 0x2294, 0x20f4, 0xb2a4, 0xa2d1, 0x3ff1, 0xfd4b}, + {0xffd8, 0xd9a0, 0x456e, 0x81bb, 0xbabd, 0x6cea, 0x6dbd, 0x73ab, + 0xbb94, 0x3d3c, 0xdf08, 0x31c4, 0x3e32, 0xc179, 0x2486, 0xb86b}}, + /* Test case known to need 589 divsteps, reaching delta=-140 and + delta=141. */ + {{0x3fb1, 0x903b, 0x4eb7, 0x4813, 0xd863, 0x26bf, 0xd89f, 0xa8a9, + 0x02fe, 0x57c6, 0x554a, 0x4eab, 0x165e, 0x3d61, 0xee1e, 0x456c}, + {0x9295, 0x823b, 0x5c1f, 0x5386, 0x48e0, 0x02ff, 0x4c2a, 0xa2da, + 0xe58f, 0x967c, 0xc97e, 0x3f5a, 0x69fb, 0x52d9, 0x0a86, 0xb4a3}, + {0x3d30, 0xb893, 0xa809, 0xa7a8, 0x26f5, 0x5b42, 0x55be, 0xf4d0, + 0x12c2, 0x7e6a, 0xe41a, 0x90c7, 0xebfa, 0xf920, 0x304e, 0x1419}}, + /* Test case known to need 650 divsteps, and doing 65 consecutive (f,g/2) steps. */ + {{0x8583, 0x5058, 0xbeae, 0xeb69, 0x48bc, 0x52bb, 0x6a9d, 0xcc94, + 0x2a21, 0x87d5, 0x5b0d, 0x42f6, 0x5b8a, 0x2214, 0xe9d6, 0xa040}, + {0x7531, 0x27cb, 0x7e53, 0xb739, 0x6a5f, 0x83f5, 0xa45c, 0xcb1d, + 0x8a87, 0x1c9c, 0x51d7, 0x851c, 0xb9d8, 0x1fbe, 0xc241, 0xd4a3}, + {0xcdb4, 0x275c, 0x7d22, 0xa906, 0x0173, 0xc054, 0x7fdf, 0x5005, + 0x7fb8, 0x9059, 0xdf51, 0x99df, 0x2654, 0x8f6e, 0x070f, 0xb347}}, + /* Test case with the group order as modulus, needing 635 divsteps. */ + {{0x95ed, 0x6c01, 0xd113, 0x5ff1, 0xd7d0, 0x29cc, 0x5817, 0x6120, + 0xca8e, 0xaad1, 0x25ae, 0x8e84, 0x9af6, 0x30bf, 0xf0ed, 0x1686}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x1631, 0xbf4a, 0x286a, 0x2716, 0x469f, 0x2ac8, 0x1312, 0xe9bc, + 0x04f4, 0x304b, 0x9931, 0x113b, 0xd932, 0xc8f4, 0x0d0d, 0x01a1}}, + /* Test case with the field size as modulus, needing 637 divsteps. */ + {{0x9ec3, 0x1919, 0xca84, 0x7c11, 0xf996, 0x06f3, 0x5408, 0x6688, + 0x1320, 0xdb8a, 0x632a, 0x0dcb, 0x8a84, 0x6bee, 0x9c95, 0xe34e}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x18e5, 0x19b6, 0xdf92, 0x1aaa, 0x09fb, 0x8a3f, 0x52b0, 0x8701, + 0xac0c, 0x2582, 0xda44, 0x9bcc, 0x6828, 0x1c53, 0xbd8f, 0xbd2c}}, + /* Test case with the field size as modulus, needing 935 divsteps with + broken eta handling. */ + {{0x1b37, 0xbdc3, 0x8bcd, 0x25e3, 0x1eae, 0x567d, 0x30b6, 0xf0d8, + 0x9277, 0x0cf8, 0x9c2e, 0xecd7, 0x631d, 0xe38f, 0xd4f8, 0x5c93}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x1622, 0xe05b, 0xe880, 0x7de9, 0x3e45, 0xb682, 0xee6c, 0x67ed, + 0xa179, 0x15db, 0x6b0d, 0xa656, 0x7ccb, 0x8ef7, 0xa2ff, 0xe279}}, + /* Test case with the group size as modulus, needing 981 divsteps with + broken eta handling. */ + {{0xfeb9, 0xb877, 0xee41, 0x7fa3, 0x87da, 0x94c4, 0x9d04, 0xc5ae, + 0x5708, 0x0994, 0xfc79, 0x0916, 0xbf32, 0x3ad8, 0xe11c, 0x5ca2}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x0f12, 0x075e, 0xce1c, 0x6f92, 0xc80f, 0xca92, 0x9a04, 0x6126, + 0x4b6c, 0x57d6, 0xca31, 0x97f3, 0x1f99, 0xf4fd, 0xda4d, 0x42ce}}, + /* Test case with the field size as modulus, input = 0. */ + {{0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}}, + /* Test case with the field size as modulus, input = 1. */ + {{0x0001, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x0001, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}}, + /* Test case with the field size as modulus, input = 2. */ + {{0x0002, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0xfe18, 0x7fff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0x7fff}}, + /* Test case with the field size as modulus, input = field - 1. */ + {{0xfc2e, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0xfc2f, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0xfc2e, 0xffff, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}}, + /* Test case with the group size as modulus, input = 0. */ + {{0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}}, + /* Test case with the group size as modulus, input = 1. */ + {{0x0001, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x0001, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}}, + /* Test case with the group size as modulus, input = 2. */ + {{0x0002, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, + 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x20a1, 0x681b, 0x2f46, 0xdfe9, 0x501d, 0x57a4, 0x6e73, 0x5d57, + 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0x7fff}}, + /* Test case with the group size as modulus, input = group - 1. */ + {{0x4140, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x4141, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}, + {0x4140, 0xd036, 0x5e8c, 0xbfd2, 0xa03b, 0xaf48, 0xdce6, 0xbaae, + 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff, 0xffff}} + }; + + int i, j, ok; + + /* Test known inputs/outputs */ + for (i = 0; (size_t)i < sizeof(CASES) / sizeof(CASES[0]); ++i) { + uint16_t out[16]; + test_modinv32_uint16(out, CASES[i][0], CASES[i][1]); + for (j = 0; j < 16; ++j) CHECK(out[j] == CASES[i][2][j]); +#ifdef SECP256K1_WIDEMUL_INT128 + test_modinv64_uint16(out, CASES[i][0], CASES[i][1]); + for (j = 0; j < 16; ++j) CHECK(out[j] == CASES[i][2][j]); +#endif + } + + for (i = 0; i < 100 * count; ++i) { + /* 256-bit numbers in 16-uint16_t's notation */ + static const uint16_t ZERO[16] = {0}; + uint16_t xd[16]; /* the number (in range [0,2^256)) to be inverted */ + uint16_t md[16]; /* the modulus (odd, in range [3,2^256)) */ + uint16_t id[16]; /* the inverse of xd mod md */ + + /* generate random xd and md, so that md is odd, md>1, xd 0)); - secp256k1_scalar_negate(&neg, &s); - secp256k1_num_sub(&negnum, &order, &snum); - secp256k1_num_mod(&negnum, &order); - /* Check that comparison with the half order is equal to testing for high scalar after negation. */ - CHECK(secp256k1_scalar_is_high(&neg) == (secp256k1_num_cmp(&negnum, &half_order) > 0)); - /* Negating should change the high property, unless the value was already zero. */ - CHECK((secp256k1_scalar_is_high(&s) == secp256k1_scalar_is_high(&neg)) == secp256k1_scalar_is_zero(&s)); - secp256k1_scalar_get_num(&negnum2, &neg); - /* Negating a scalar should be equal to (order - n) mod order on the number. */ - CHECK(secp256k1_num_eq(&negnum, &negnum2)); - secp256k1_scalar_add(&neg, &neg, &s); - /* Adding a number to its negation should result in zero. */ - CHECK(secp256k1_scalar_is_zero(&neg)); - secp256k1_scalar_negate(&neg, &neg); - /* Negating zero should still result in zero. */ - CHECK(secp256k1_scalar_is_zero(&neg)); - } - - { - /* Test secp256k1_scalar_mul_shift_var. */ - secp256k1_scalar r; - secp256k1_num one; - secp256k1_num rnum; - secp256k1_num rnum2; - unsigned char cone[1] = {0x01}; - unsigned int shift = 256 + secp256k1_testrand_int(257); - secp256k1_scalar_mul_shift_var(&r, &s1, &s2, shift); - secp256k1_num_mul(&rnum, &s1num, &s2num); - secp256k1_num_shift(&rnum, shift - 1); - secp256k1_num_set_bin(&one, cone, 1); - secp256k1_num_add(&rnum, &rnum, &one); - secp256k1_num_shift(&rnum, 1); - secp256k1_scalar_get_num(&rnum2, &r); - CHECK(secp256k1_num_eq(&rnum, &rnum2)); - } - { /* test secp256k1_scalar_shr_int */ secp256k1_scalar r; @@ -955,34 +1142,6 @@ void scalar_test(void) { CHECK(expected == low); } } -#endif - - { - /* Test that scalar inverses are equal to the inverse of their number modulo the order. */ - if (!secp256k1_scalar_is_zero(&s)) { - secp256k1_scalar inv; -#ifndef USE_NUM_NONE - secp256k1_num invnum; - secp256k1_num invnum2; -#endif - secp256k1_scalar_inverse(&inv, &s); -#ifndef USE_NUM_NONE - secp256k1_num_mod_inverse(&invnum, &snum, &order); - secp256k1_scalar_get_num(&invnum2, &inv); - CHECK(secp256k1_num_eq(&invnum, &invnum2)); -#endif - secp256k1_scalar_mul(&inv, &inv, &s); - /* Multiplying a scalar with its inverse must result in one. */ - CHECK(secp256k1_scalar_is_one(&inv)); - secp256k1_scalar_inverse(&inv, &inv); - /* Inverting one must result in one. */ - CHECK(secp256k1_scalar_is_one(&inv)); -#ifndef USE_NUM_NONE - secp256k1_scalar_get_num(&invnum, &inv); - CHECK(secp256k1_num_is_one(&invnum)); -#endif - } - } { /* Test commutativity of add. */ @@ -1054,14 +1213,6 @@ void scalar_test(void) { CHECK(secp256k1_scalar_eq(&r1, &r2)); } - { - /* Test square. */ - secp256k1_scalar r1, r2; - secp256k1_scalar_sqr(&r1, &s1); - secp256k1_scalar_mul(&r2, &s1, &s1); - CHECK(secp256k1_scalar_eq(&r1, &r2)); - } - { /* Test multiplicative identity. */ secp256k1_scalar r1, v1; @@ -1126,48 +1277,6 @@ void run_scalar_tests(void) { CHECK(secp256k1_scalar_is_zero(&o)); } -#ifndef USE_NUM_NONE - { - /* Test secp256k1_scalar_set_b32 boundary conditions */ - secp256k1_num order; - secp256k1_scalar scalar; - unsigned char bin[32]; - unsigned char bin_tmp[32]; - int overflow = 0; - /* 2^256-1 - order */ - static const secp256k1_scalar all_ones_minus_order = SECP256K1_SCALAR_CONST( - 0x00000000UL, 0x00000000UL, 0x00000000UL, 0x00000001UL, - 0x45512319UL, 0x50B75FC4UL, 0x402DA173UL, 0x2FC9BEBEUL - ); - - /* A scalar set to 0s should be 0. */ - memset(bin, 0, 32); - secp256k1_scalar_set_b32(&scalar, bin, &overflow); - CHECK(overflow == 0); - CHECK(secp256k1_scalar_is_zero(&scalar)); - - /* A scalar with value of the curve order should be 0. */ - secp256k1_scalar_order_get_num(&order); - secp256k1_num_get_bin(bin, 32, &order); - secp256k1_scalar_set_b32(&scalar, bin, &overflow); - CHECK(overflow == 1); - CHECK(secp256k1_scalar_is_zero(&scalar)); - - /* A scalar with value of the curve order minus one should not overflow. */ - bin[31] -= 1; - secp256k1_scalar_set_b32(&scalar, bin, &overflow); - CHECK(overflow == 0); - secp256k1_scalar_get_b32(bin_tmp, &scalar); - CHECK(secp256k1_memcmp_var(bin, bin_tmp, 32) == 0); - - /* A scalar set to all 1s should overflow. */ - memset(bin, 0xFF, 32); - secp256k1_scalar_set_b32(&scalar, bin, &overflow); - CHECK(overflow == 1); - CHECK(secp256k1_scalar_eq(&scalar, &all_ones_minus_order)); - } -#endif - { /* Does check_overflow check catch all ones? */ static const secp256k1_scalar overflowed = SECP256K1_SCALAR_CONST( @@ -1190,9 +1299,7 @@ void run_scalar_tests(void) { secp256k1_scalar one; secp256k1_scalar r1; secp256k1_scalar r2; -#if defined(USE_SCALAR_INV_NUM) secp256k1_scalar zzv; -#endif int overflow; unsigned char chal[33][2][32] = { {{0xff, 0xff, 0x03, 0x07, 0x00, 0x00, 0x00, 0x00, @@ -1742,10 +1849,8 @@ void run_scalar_tests(void) { if (!secp256k1_scalar_is_zero(&y)) { secp256k1_scalar_inverse(&zz, &y); CHECK(!secp256k1_scalar_check_overflow(&zz)); -#if defined(USE_SCALAR_INV_NUM) secp256k1_scalar_inverse_var(&zzv, &y); CHECK(secp256k1_scalar_eq(&zzv, &zz)); -#endif secp256k1_scalar_mul(&z, &z, &zz); CHECK(!secp256k1_scalar_check_overflow(&z)); CHECK(secp256k1_scalar_eq(&x, &z)); @@ -1753,12 +1858,6 @@ void run_scalar_tests(void) { CHECK(!secp256k1_scalar_check_overflow(&zz)); CHECK(secp256k1_scalar_eq(&one, &zz)); } - secp256k1_scalar_mul(&z, &x, &x); - CHECK(!secp256k1_scalar_check_overflow(&z)); - secp256k1_scalar_sqr(&zz, &x); - CHECK(!secp256k1_scalar_check_overflow(&zz)); - CHECK(secp256k1_scalar_eq(&zz, &z)); - CHECK(secp256k1_scalar_eq(&r2, &zz)); } } } @@ -1814,13 +1913,6 @@ int check_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b) { return secp256k1_fe_equal_var(&an, &bn); } -int check_fe_inverse(const secp256k1_fe *a, const secp256k1_fe *ai) { - secp256k1_fe x; - secp256k1_fe one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1); - secp256k1_fe_mul(&x, a, ai); - return check_fe_equal(&x, &one); -} - void run_field_convert(void) { static const unsigned char b32[32] = { 0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, @@ -1940,30 +2032,6 @@ void run_field_misc(void) { } } -void run_field_inv(void) { - secp256k1_fe x, xi, xii; - int i; - for (i = 0; i < 10*count; i++) { - random_fe_non_zero(&x); - secp256k1_fe_inv(&xi, &x); - CHECK(check_fe_inverse(&x, &xi)); - secp256k1_fe_inv(&xii, &xi); - CHECK(check_fe_equal(&x, &xii)); - } -} - -void run_field_inv_var(void) { - secp256k1_fe x, xi, xii; - int i; - for (i = 0; i < 10*count; i++) { - random_fe_non_zero(&x); - secp256k1_fe_inv_var(&xi, &x); - CHECK(check_fe_inverse(&x, &xi)); - secp256k1_fe_inv_var(&xii, &xi); - CHECK(check_fe_equal(&x, &xii)); - } -} - void run_sqr(void) { secp256k1_fe x, s; @@ -2028,6 +2096,169 @@ void run_sqrt(void) { } } +/***** FIELD/SCALAR INVERSE TESTS *****/ + +static const secp256k1_scalar scalar_minus_one = SECP256K1_SCALAR_CONST( + 0xFFFFFFFF, 0xFFFFFFFF, 0xFFFFFFFF, 0xFFFFFFFE, + 0xBAAEDCE6, 0xAF48A03B, 0xBFD25E8C, 0xD0364140 +); + +static const secp256k1_fe fe_minus_one = SECP256K1_FE_CONST( + 0xFFFFFFFF, 0xFFFFFFFF, 0xFFFFFFFF, 0xFFFFFFFF, + 0xFFFFFFFF, 0xFFFFFFFF, 0xFFFFFFFE, 0xFFFFFC2E +); + +/* These tests test the following identities: + * + * for x==0: 1/x == 0 + * for x!=0: x*(1/x) == 1 + * for x!=0 and x!=1: 1/(1/x - 1) + 1 == -1/(x-1) + */ + +void test_inverse_scalar(secp256k1_scalar* out, const secp256k1_scalar* x, int var) +{ + secp256k1_scalar l, r, t; + + (var ? secp256k1_scalar_inverse_var : secp256k1_scalar_inverse_var)(&l, x); /* l = 1/x */ + if (out) *out = l; + if (secp256k1_scalar_is_zero(x)) { + CHECK(secp256k1_scalar_is_zero(&l)); + return; + } + secp256k1_scalar_mul(&t, x, &l); /* t = x*(1/x) */ + CHECK(secp256k1_scalar_is_one(&t)); /* x*(1/x) == 1 */ + secp256k1_scalar_add(&r, x, &scalar_minus_one); /* r = x-1 */ + if (secp256k1_scalar_is_zero(&r)) return; + (var ? secp256k1_scalar_inverse_var : secp256k1_scalar_inverse_var)(&r, &r); /* r = 1/(x-1) */ + secp256k1_scalar_add(&l, &scalar_minus_one, &l); /* l = 1/x-1 */ + (var ? secp256k1_scalar_inverse_var : secp256k1_scalar_inverse_var)(&l, &l); /* l = 1/(1/x-1) */ + secp256k1_scalar_add(&l, &l, &secp256k1_scalar_one); /* l = 1/(1/x-1)+1 */ + secp256k1_scalar_add(&l, &r, &l); /* l = 1/(1/x-1)+1 + 1/(x-1) */ + CHECK(secp256k1_scalar_is_zero(&l)); /* l == 0 */ +} + +void test_inverse_field(secp256k1_fe* out, const secp256k1_fe* x, int var) +{ + secp256k1_fe l, r, t; + + (var ? secp256k1_fe_inv_var : secp256k1_fe_inv)(&l, x) ; /* l = 1/x */ + if (out) *out = l; + t = *x; /* t = x */ + if (secp256k1_fe_normalizes_to_zero_var(&t)) { + CHECK(secp256k1_fe_normalizes_to_zero(&l)); + return; + } + secp256k1_fe_mul(&t, x, &l); /* t = x*(1/x) */ + secp256k1_fe_add(&t, &fe_minus_one); /* t = x*(1/x)-1 */ + CHECK(secp256k1_fe_normalizes_to_zero(&t)); /* x*(1/x)-1 == 0 */ + r = *x; /* r = x */ + secp256k1_fe_add(&r, &fe_minus_one); /* r = x-1 */ + if (secp256k1_fe_normalizes_to_zero_var(&r)) return; + (var ? secp256k1_fe_inv_var : secp256k1_fe_inv)(&r, &r); /* r = 1/(x-1) */ + secp256k1_fe_add(&l, &fe_minus_one); /* l = 1/x-1 */ + (var ? secp256k1_fe_inv_var : secp256k1_fe_inv)(&l, &l); /* l = 1/(1/x-1) */ + secp256k1_fe_add(&l, &secp256k1_fe_one); /* l = 1/(1/x-1)+1 */ + secp256k1_fe_add(&l, &r); /* l = 1/(1/x-1)+1 + 1/(x-1) */ + CHECK(secp256k1_fe_normalizes_to_zero_var(&l)); /* l == 0 */ +} + +void run_inverse_tests(void) +{ + /* Fixed test cases for field inverses: pairs of (x, 1/x) mod p. */ + static const secp256k1_fe fe_cases[][2] = { + /* 0 */ + {SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0), + SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0)}, + /* 1 */ + {SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1), + SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1)}, + /* -1 */ + {SECP256K1_FE_CONST(0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0xfffffc2e), + SECP256K1_FE_CONST(0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0xfffffc2e)}, + /* 2 */ + {SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 2), + SECP256K1_FE_CONST(0x7fffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0x7ffffe18)}, + /* 2**128 */ + {SECP256K1_FE_CONST(0, 0, 0, 1, 0, 0, 0, 0), + SECP256K1_FE_CONST(0xbcb223fe, 0xdc24a059, 0xd838091d, 0xd2253530, 0xffffffff, 0xffffffff, 0xffffffff, 0x434dd931)}, + /* Input known to need 637 divsteps */ + {SECP256K1_FE_CONST(0xe34e9c95, 0x6bee8a84, 0x0dcb632a, 0xdb8a1320, 0x66885408, 0x06f3f996, 0x7c11ca84, 0x19199ec3), + SECP256K1_FE_CONST(0xbd2cbd8f, 0x1c536828, 0x9bccda44, 0x2582ac0c, 0x870152b0, 0x8a3f09fb, 0x1aaadf92, 0x19b618e5)} + }; + /* Fixed test cases for scalar inverses: pairs of (x, 1/x) mod n. */ + static const secp256k1_scalar scalar_cases[][2] = { + /* 0 */ + {SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0), + SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0)}, + /* 1 */ + {SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1), + SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1)}, + /* -1 */ + {SECP256K1_SCALAR_CONST(0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0xbaaedce6, 0xaf48a03b, 0xbfd25e8c, 0xd0364140), + SECP256K1_SCALAR_CONST(0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0xbaaedce6, 0xaf48a03b, 0xbfd25e8c, 0xd0364140)}, + /* 2 */ + {SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 2), + SECP256K1_SCALAR_CONST(0x7fffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0x5d576e73, 0x57a4501d, 0xdfe92f46, 0x681b20a1)}, + /* 2**128 */ + {SECP256K1_SCALAR_CONST(0, 0, 0, 1, 0, 0, 0, 0), + SECP256K1_SCALAR_CONST(0x50a51ac8, 0x34b9ec24, 0x4b0dff66, 0x5588b13e, 0x9984d5b3, 0xcf80ef0f, 0xd6a23766, 0xa3ee9f22)}, + /* Input known to need 635 divsteps */ + {SECP256K1_SCALAR_CONST(0xcb9f1d35, 0xdd4416c2, 0xcd71bf3f, 0x6365da66, 0x3c9b3376, 0x8feb7ae9, 0x32a5ef60, 0x19199ec3), + SECP256K1_SCALAR_CONST(0x1d7c7bba, 0xf1893d53, 0xb834bd09, 0x36b411dc, 0x42c2e42f, 0xec72c428, 0x5e189791, 0x8e9bc708)} + }; + int i, var, testrand; + unsigned char b32[32]; + secp256k1_fe x_fe; + secp256k1_scalar x_scalar; + memset(b32, 0, sizeof(b32)); + /* Test fixed test cases through test_inverse_{scalar,field}, both ways. */ + for (i = 0; (size_t)i < sizeof(fe_cases)/sizeof(fe_cases[0]); ++i) { + for (var = 0; var <= 1; ++var) { + test_inverse_field(&x_fe, &fe_cases[i][0], var); + check_fe_equal(&x_fe, &fe_cases[i][1]); + test_inverse_field(&x_fe, &fe_cases[i][1], var); + check_fe_equal(&x_fe, &fe_cases[i][0]); + } + } + for (i = 0; (size_t)i < sizeof(scalar_cases)/sizeof(scalar_cases[0]); ++i) { + for (var = 0; var <= 1; ++var) { + test_inverse_scalar(&x_scalar, &scalar_cases[i][0], var); + CHECK(secp256k1_scalar_eq(&x_scalar, &scalar_cases[i][1])); + test_inverse_scalar(&x_scalar, &scalar_cases[i][1], var); + CHECK(secp256k1_scalar_eq(&x_scalar, &scalar_cases[i][0])); + } + } + /* Test inputs 0..999 and their respective negations. */ + for (i = 0; i < 1000; ++i) { + b32[31] = i & 0xff; + b32[30] = (i >> 8) & 0xff; + secp256k1_scalar_set_b32(&x_scalar, b32, NULL); + secp256k1_fe_set_b32(&x_fe, b32); + for (var = 0; var <= 1; ++var) { + test_inverse_scalar(NULL, &x_scalar, var); + test_inverse_field(NULL, &x_fe, var); + } + secp256k1_scalar_negate(&x_scalar, &x_scalar); + secp256k1_fe_negate(&x_fe, &x_fe, 1); + for (var = 0; var <= 1; ++var) { + test_inverse_scalar(NULL, &x_scalar, var); + test_inverse_field(NULL, &x_fe, var); + } + } + /* test 128*count random inputs; half with testrand256_test, half with testrand256 */ + for (testrand = 0; testrand <= 1; ++testrand) { + for (i = 0; i < 64 * count; ++i) { + (testrand ? secp256k1_testrand256_test : secp256k1_testrand256)(b32); + secp256k1_scalar_set_b32(&x_scalar, b32, NULL); + secp256k1_fe_set_b32(&x_fe, b32); + for (var = 0; var <= 1; ++var) { + test_inverse_scalar(NULL, &x_scalar, var); + test_inverse_field(NULL, &x_fe, var); + } + } + } +} + /***** GROUP TESTS *****/ void ge_equals_ge(const secp256k1_ge *a, const secp256k1_ge *b) { @@ -2407,64 +2638,35 @@ void run_ec_combine(void) { void test_group_decompress(const secp256k1_fe* x) { /* The input itself, normalized. */ secp256k1_fe fex = *x; - secp256k1_fe fez; - /* Results of set_xquad_var, set_xo_var(..., 0), set_xo_var(..., 1). */ - secp256k1_ge ge_quad, ge_even, ge_odd; - secp256k1_gej gej_quad; + /* Results of set_xo_var(..., 0), set_xo_var(..., 1). */ + secp256k1_ge ge_even, ge_odd; /* Return values of the above calls. */ - int res_quad, res_even, res_odd; + int res_even, res_odd; secp256k1_fe_normalize_var(&fex); - res_quad = secp256k1_ge_set_xquad(&ge_quad, &fex); res_even = secp256k1_ge_set_xo_var(&ge_even, &fex, 0); res_odd = secp256k1_ge_set_xo_var(&ge_odd, &fex, 1); - CHECK(res_quad == res_even); - CHECK(res_quad == res_odd); + CHECK(res_even == res_odd); - if (res_quad) { - secp256k1_fe_normalize_var(&ge_quad.x); + if (res_even) { secp256k1_fe_normalize_var(&ge_odd.x); secp256k1_fe_normalize_var(&ge_even.x); - secp256k1_fe_normalize_var(&ge_quad.y); secp256k1_fe_normalize_var(&ge_odd.y); secp256k1_fe_normalize_var(&ge_even.y); /* No infinity allowed. */ - CHECK(!ge_quad.infinity); CHECK(!ge_even.infinity); CHECK(!ge_odd.infinity); /* Check that the x coordinates check out. */ - CHECK(secp256k1_fe_equal_var(&ge_quad.x, x)); CHECK(secp256k1_fe_equal_var(&ge_even.x, x)); CHECK(secp256k1_fe_equal_var(&ge_odd.x, x)); - /* Check that the Y coordinate result in ge_quad is a square. */ - CHECK(secp256k1_fe_is_quad_var(&ge_quad.y)); - /* Check odd/even Y in ge_odd, ge_even. */ CHECK(secp256k1_fe_is_odd(&ge_odd.y)); CHECK(!secp256k1_fe_is_odd(&ge_even.y)); - - /* Check secp256k1_gej_has_quad_y_var. */ - secp256k1_gej_set_ge(&gej_quad, &ge_quad); - CHECK(secp256k1_gej_has_quad_y_var(&gej_quad)); - do { - random_fe_test(&fez); - } while (secp256k1_fe_is_zero(&fez)); - secp256k1_gej_rescale(&gej_quad, &fez); - CHECK(secp256k1_gej_has_quad_y_var(&gej_quad)); - secp256k1_gej_neg(&gej_quad, &gej_quad); - CHECK(!secp256k1_gej_has_quad_y_var(&gej_quad)); - do { - random_fe_test(&fez); - } while (secp256k1_fe_is_zero(&fez)); - secp256k1_gej_rescale(&gej_quad, &fez); - CHECK(!secp256k1_gej_has_quad_y_var(&gej_quad)); - secp256k1_gej_neg(&gej_quad, &gej_quad); - CHECK(secp256k1_gej_has_quad_y_var(&gej_quad)); } } @@ -5606,21 +5808,18 @@ int main(int argc, char **argv) { run_rand_bits(); run_rand_int(); + run_ctz_tests(); + run_modinv_tests(); + run_inverse_tests(); + run_sha256_tests(); run_hmac_sha256_tests(); run_rfc6979_hmac_sha256_tests(); -#ifndef USE_NUM_NONE - /* num tests */ - run_num_smalltests(); -#endif - /* scalar tests */ run_scalar_tests(); /* field tests */ - run_field_inv(); - run_field_inv_var(); run_field_misc(); run_field_convert(); run_sqr(); diff --git a/src/util.h b/src/util.h index b7e457c..f788468 100644 --- a/src/util.h +++ b/src/util.h @@ -276,4 +276,69 @@ SECP256K1_GNUC_EXT typedef __int128 int128_t; # endif #endif +#ifndef __has_builtin +#define __has_builtin(x) 0 +#endif + +/* Determine the number of trailing zero bits in a (non-zero) 32-bit x. + * This function is only intended to be used as fallback for + * secp256k1_ctz32_var, but permits it to be tested separately. */ +static SECP256K1_INLINE int secp256k1_ctz32_var_debruijn(uint32_t x) { + static const uint8_t debruijn[32] = { + 0x00, 0x01, 0x02, 0x18, 0x03, 0x13, 0x06, 0x19, 0x16, 0x04, 0x14, 0x0A, + 0x10, 0x07, 0x0C, 0x1A, 0x1F, 0x17, 0x12, 0x05, 0x15, 0x09, 0x0F, 0x0B, + 0x1E, 0x11, 0x08, 0x0E, 0x1D, 0x0D, 0x1C, 0x1B + }; + return debruijn[((x & -x) * 0x04D7651F) >> 27]; +} + +/* Determine the number of trailing zero bits in a (non-zero) 64-bit x. + * This function is only intended to be used as fallback for + * secp256k1_ctz64_var, but permits it to be tested separately. */ +static SECP256K1_INLINE int secp256k1_ctz64_var_debruijn(uint64_t x) { + static const uint8_t debruijn[64] = { + 0, 1, 2, 53, 3, 7, 54, 27, 4, 38, 41, 8, 34, 55, 48, 28, + 62, 5, 39, 46, 44, 42, 22, 9, 24, 35, 59, 56, 49, 18, 29, 11, + 63, 52, 6, 26, 37, 40, 33, 47, 61, 45, 43, 21, 23, 58, 17, 10, + 51, 25, 36, 32, 60, 20, 57, 16, 50, 31, 19, 15, 30, 14, 13, 12 + }; + return debruijn[((x & -x) * 0x022FDD63CC95386D) >> 58]; +} + +/* Determine the number of trailing zero bits in a (non-zero) 32-bit x. */ +static SECP256K1_INLINE int secp256k1_ctz32_var(uint32_t x) { + VERIFY_CHECK(x != 0); +#if (__has_builtin(__builtin_ctz) || SECP256K1_GNUC_PREREQ(3,4)) + /* If the unsigned type is sufficient to represent the largest uint32_t, consider __builtin_ctz. */ + if (((unsigned)UINT32_MAX) == UINT32_MAX) { + return __builtin_ctz(x); + } +#endif +#if (__has_builtin(__builtin_ctzl) || SECP256K1_GNUC_PREREQ(3,4)) + /* Otherwise consider __builtin_ctzl (the unsigned long type is always at least 32 bits). */ + return __builtin_ctzl(x); +#else + /* If no suitable CTZ builtin is available, use a (variable time) software emulation. */ + return secp256k1_ctz32_var_debruijn(x); +#endif +} + +/* Determine the number of trailing zero bits in a (non-zero) 64-bit x. */ +static SECP256K1_INLINE int secp256k1_ctz64_var(uint64_t x) { + VERIFY_CHECK(x != 0); +#if (__has_builtin(__builtin_ctzl) || SECP256K1_GNUC_PREREQ(3,4)) + /* If the unsigned long type is sufficient to represent the largest uint64_t, consider __builtin_ctzl. */ + if (((unsigned long)UINT64_MAX) == UINT64_MAX) { + return __builtin_ctzl(x); + } +#endif +#if (__has_builtin(__builtin_ctzll) || SECP256K1_GNUC_PREREQ(3,4)) + /* Otherwise consider __builtin_ctzll (the unsigned long long type is always at least 64 bits). */ + return __builtin_ctzll(x); +#else + /* If no suitable CTZ builtin is available, use a (variable time) software emulation. */ + return secp256k1_ctz64_var_debruijn(x); +#endif +} + #endif /* SECP256K1_UTIL_H */