Use Montgomery representation by default for Finite Field
- Fix montyMagic, modular inversion mode 2^2k was missing an iteration - Fix test for buffer size in BigInt serialization - Add UINT/Hex serialization for finite fields - Montgomery conversion and redc
This commit is contained in:
Mamy André-Ratsimbazafy
parent
f418e08746
commit
301cf20195
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@ -21,3 +21,4 @@ task test, "Run all tests":
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test "", "tests/test_bigints.nim"
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test "", "tests/test_bigints_multimod.nim"
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test "", "tests/test_bigints_vs_gmp.nim"
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test "", "tests/test_finite_fields.nim"
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@ -18,57 +18,51 @@ import
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#
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# ############################################################
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func montyMagic(M: static BigInt): static Word {.inline.} =
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# Note: The declareCurve macro builds an exported montyMagic*(C: static Curve) on top of the one below
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func montyMagic(M: BigInt): BaseType =
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## Returns the Montgomery domain magic constant for the input modulus:
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## -1/M[0] mod LimbSize
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## M[0] is the least significant limb of M
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## M must be odd and greater than 2.
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# We use BaseType for return value because static distinct type
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# confuses Nim semchecks [UPSTREAM BUG]
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# We don't enforce compile-time evaluation here
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# because static BigInt[bits] also causes semcheck troubles [UPSTREAM BUG]
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# Test vectors: https://www.researchgate.net/publication/4107322_Montgomery_modular_multiplication_architecture_for_public_key_cryptosystems
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# on p354
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# Reference C impl: http://www.hackersdelight.org/hdcodetxt/mont64.c.txt
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# Explanation: https://eprint.iacr.org/2017/1057.pdf
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# ######################################################################
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# Implementation of modular multiplicative inverse
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# Assuming 2 positive integers a and m the modulo
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#
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# We are looking for z that solves `az ≡ 1 mod m`
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#
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# References:
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# - Knuth, The Art of Computer Programming, Vol2 p342
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# - Menezes, Handbook of Applied Cryptography (HAC), p610
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# http://cacr.uwaterloo.ca/hac/about/chap14.pdf
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# Starting from the extended GCD formula (Bezout identity),
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# `ax + by = gcd(x,y)` with input x,y and outputs a, b, gcd
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# We assume a and m are coprimes, i.e. gcd is 1, otherwise no inverse
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# `ax + my = 1` <=> `ax + my ≡ 1 mod m` <=> `ax ≡ 1 mod m`
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# Modular inverse algorithm:
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# Explanation p11 "Dumas iterations" based on Newton-Raphson:
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# - Cetin Kaya Koc (2017), https://eprint.iacr.org/2017/411
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# - Jean-Guillaume Dumas (2012), https://arxiv.org/pdf/1209.6626v2.pdf
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# - Colin Plumb (1994), http://groups.google.com/groups?selm=1994Apr6.093116.27805%40mnemosyne.cs.du.edu
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# Other sources:
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# - https://crypto.stackexchange.com/questions/47493/how-to-determine-the-multiplicative-inverse-modulo-64-or-other-power-of-two
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# - https://mumble.net/~campbell/2015/01/21/inverse-mod-power-of-two
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# - http://marc-b-reynolds.github.io/math/2017/09/18/ModInverse.html
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# For Montgomery magic number, we are in a special case
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# where a = M and m = 2^LimbSize.
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# For a and m to be coprimes, a must be odd.
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# `m` (2^LimbSize) being a power of 2 greatly simplifies computation:
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# - https://crypto.stackexchange.com/questions/47493/how-to-determine-the-multiplicative-inverse-modulo-64-or-other-power-of-two
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# - http://groups.google.com/groups?selm=1994Apr6.093116.27805%40mnemosyne.cs.du.edu
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# - https://mumble.net/~campbell/2015/01/21/inverse-mod-power-of-two
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# - https://eprint.iacr.org/2017/411
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# We have the following relation
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# ax ≡ 1 (mod 2^k) <=> ax(2 - ax) ≡ 1 (mod 2^(2k))
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#
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# To get -1/M0 mod LimbSize
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# we can either negate the resulting x of `ax(2 - ax) ≡ 1 (mod 2^(2k))`
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# or do ax(2 + ax) ≡ 1 (mod 2^(2k))
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#
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# To get the the modular inverse of 2^k' with arbitrary k' (like k=63 in our case)
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# we can do modInv(a, 2^64) mod 2^63 as mentionned in Koc paper.
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const
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M0 = M.limbs[0]
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k = log2(WordBitSize)
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let
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M0 = BaseType(M.limbs[0])
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k = log2(WordPhysBitSize)
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result = M0 # Start from an inverse of M0 modulo 2, M0 is odd and it's own inverse
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for _ in 0 ..< k:
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result *= 2 + M * result # x' = x(2 + ax) (`+` to avoid negating at the end)
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for _ in 0 ..< k: # at each iteration we get the inverse mod(2^2k)
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result *= 2 + M0 * result # x' = x(2 + ax) (`+` to avoid negating at the end)
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# Our actual word size is 2^63 not 2^64
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result = result and BaseType(MaxWord)
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# ############################################################
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#
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@ -139,7 +139,7 @@ macro declareCurves*(curves: untyped): untyped =
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let C = ident"C"
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let matchingBigInt = ident"matchingBigInt"
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result.add newProc(
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name = matchingBigInt,
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name = nnkPostFix.newTree(ident"*", matchingBigInt),
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params = [ident"untyped", newIdentDefs(C, nnkStaticTy.newTree(Curve))],
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body = nnkBracketExpr.newTree(bindSym"BigInt", nnkBracketExpr.newTree(cbs, C)),
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procType = nnkTemplateDef
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@ -158,16 +158,13 @@ macro declareCurves*(curves: untyped): untyped =
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nnkGenericParams.newTree(newIdentDefs(
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C, nnkStaticTy.newTree(Curve), newEmptyNode()
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)),
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# TODO: where should I put the nnkCommentStmt?
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nnkObjectTy.newTree(
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newEmptyNode(),
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newEmptyNode(),
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nnkRecList.newTree(
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nnkCommentStmt.newTree "All operations on a field are modulo P",
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nnkCommentStmt.newTree "P being the prime modulus of the Curve C",
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nnkCommentStmt.newTree "Internally, data is stored in Montgomery n-residue form",
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nnkCommentStmt.newTree "with the magic constant chosen for convenient division (a power of 2 depending on P bitsize)"
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newIdentDefs(
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nnkPostfix.newTree(ident"*", ident"value"),
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nnkPostfix.newTree(ident"*", ident"nres"),
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newCall(matchingBigInt, C)
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)
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)
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@ -203,7 +200,7 @@ macro declareCurves*(curves: untyped): untyped =
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# proc MontyMagic(curve: static Curve): static Word
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result.add newProc(
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name = nnkPostfix.newTree(ident"*", ident"MontyMagic"),
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name = nnkPostfix.newTree(ident"*", ident"montyMagic"),
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params = [
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ident"auto",
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newIdentDefs(
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@ -213,10 +210,14 @@ macro declareCurves*(curves: untyped): untyped =
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],
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body = newCall(
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ident"montyMagic",
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newCall(ident"Mod", ident"curve")
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# curve.Mod().nres
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nnkDotExpr.newTree(
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newCall(ident"Mod", ident"curve"),
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ident"nres"
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)
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),
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procType = nnkFuncDef,
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pragmas = nnkPragma.newTree(ident"compileTime")
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procType = nnkFuncDef
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)
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# echo result.toStrLit()
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@ -181,13 +181,14 @@ func serializeRawUint*(
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dstEndianness: static Endianness) =
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## Serialize a bigint into its canonical big-endian or little endian
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## representation.
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## A destination buffer of size "BigInt.bits div 8" at minimum is needed.
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## A destination buffer of size "(BigInt.bits + 7) div 8" at minimum is needed,
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## i.e. bits -> byte conversion rounded up
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##
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## If the buffer is bigger, output will be zero-padded left for big-endian
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## or zero-padded right for little-endian.
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## I.e least significant bit is aligned to buffer boundary
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assert dst.len >= static(BigInt.bits div 8), "BigInt -> Raw int conversion: destination buffer is too small"
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assert dst.len >= (BigInt.bits + 7) div 8, "BigInt -> Raw int conversion: destination buffer is too small"
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when BigInt.bits == 0:
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zeroMem(dst, dst.len)
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@ -339,16 +340,7 @@ func toHex*(big: BigInt, order: static Endianness = bigEndian): string =
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## Note. Leading zeros are not removed.
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## Result is prefixed with 0x
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##
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## This is a raw memory dump. Output will be padded with 0
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## if the big int does not use the full memory allocated for it.
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##
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## Regardless of the machine endianness the output will be big-endian hex.
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##
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## For example a BigInt representing 10 will be
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## - 0x0a for BigInt[8]
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## - 0x000a for BigInt[16]
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## - 0x00000000_0000000a for BigInt[64]
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## (underscore added for docuentation readability only)
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## Output will be padded with 0s to maintain constant-time.
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##
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## CT:
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## - no leaks
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@ -8,7 +8,8 @@
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import
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./io_bigints,
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../math/finite_fields
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../config/curves,
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../math/[bigints_checked, finite_fields]
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# ############################################################
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#
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@ -16,8 +17,34 @@ import
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#
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# ############################################################
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func fromUint*(dst: var Fp,
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func fromUint*(dst: var Fq,
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src: SomeUnsignedInt) =
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## Parse a regular unsigned integer
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## and store it into a BigInt of size `bits`
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dst.value.fromRawUint(cast[array[sizeof(src), byte]](src), cpuEndian)
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dst.nres.fromRawUint(cast[array[sizeof(src), byte]](src), cpuEndian)
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dst.nres.unsafeMontgomeryResidue(Fq.C.Mod.nres)
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func serializeRawUint*(dst: var openarray[byte],
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src: Fq,
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dstEndianness: static Endianness) =
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## Serialize a finite field element to its canonical big-endian or little-endian
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## representation
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## With `bits` the number of bits of the field modulus
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## a buffer of size "(bits + 7) div 8" at minimum is needed
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## i.e. bits -> byte conversion rounded up
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##
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## If the buffer is bigger, output will be zero-padded left for big-endian
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## or zero-padded right for little-endian.
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## I.e least significant bit is aligned to buffer boundary
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serializeRawUint(dst, src.toBig(), dstEndianness)
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func toHex*(f: Fq, order: static Endianness = bigEndian): string =
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## Stringify a finite field element to hex.
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## Note. Leading zeros are not removed.
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## Result is prefixed with 0x
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##
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## Output will be padded with 0s to maintain constant-time.
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##
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## CT:
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## - no leaks
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result = f.toBig().toHex()
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@ -110,3 +110,24 @@ func reduce*[aBits, mBits](r: var BigInt[mBits], a: BigInt[aBits], M: BigInt[mBi
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# but we don't want to inline it as it would increase codesize, better have Nim
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# pass a pointer+length to a fixed session of the BSS.
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reduce(r.view, a.view, M.view)
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func unsafeMontgomeryResidue*[mBits](nres: var BigInt[mBits], N: BigInt[mBits]) =
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## Convert a BigInt from its natural representation
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## to the Montgomery n-residue form
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##
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## `nres` is modified in-place
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##
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## Caller must take care of properly switching between
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## the natural and montgomery domain.
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## Nesting Montgomery form is possible by applying this function twice.
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montgomeryResidue(nres.view, N.view)
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func unsafeRedc*[mBits](nres: var BigInt[mBits], N: BigInt[mBits], montyMagic: static BaseType) =
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## Convert a BigInt from its Montgomery n-residue form
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## to the natural representation
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##
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## `nres` is modified in-place
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##
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## Caller must take care of properly switching between
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## the natural and montgomery domain.
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redc(nres.view, N.view, Word(montyMagic))
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@ -181,6 +181,12 @@ template checkValidModulus(m: BigIntViewConst) =
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debug:
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assert not m[^1].isZero.bool, "Internal Error: the modulus must use all declared bits"
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template checkOddModulus(m: BigIntViewConst) =
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## CHeck that the modulus is odd
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## and valid for use in the Montgomery n-residue representation
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debug:
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assert bool(BaseType(m[0]) and 1), "Internal Error: the modulus must be odd to use the Montgomery representation."
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debug:
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func `$`*(a: BigIntViewAny): string =
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let len = a.numLimbs()
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@ -375,3 +381,56 @@ func reduce*(r: BigIntViewMut, a: BigIntViewAny, M: BigIntViewConst) =
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# Now shift-left the copied words while adding the new word modulo M
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for i in countdown(aOffset, 0):
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r.shlAddMod(a[i], M)
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func montgomeryResidue*(a: BigIntViewMut, N: BigIntViewConst) =
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## Transform a bigint ``a`` from it's natural representation (mod N)
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## to a the Montgomery n-residue representation
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## i.e. Does "a * (2^LimbSize)^W (mod N), where W is the number
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## of words needed to represent n in base 2^LimbSize
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##
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## `a`: The source BigInt in the natural representation. `a` in [0, N) range
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## `N`: The field modulus. N must be odd.
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##
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## Important: `a` is overwritten
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# Reference: https://eprint.iacr.org/2017/1057.pdf
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checkValidModulus(N)
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checkOddModulus(N)
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checkMatchingBitlengths(a, N)
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let nLen = N.numLimbs()
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for i in countdown(nLen, 1):
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a.shlAddMod(Zero, N)
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func redc*(a: BigIntViewMut, N: BigIntViewConst, montyMagic: Word) =
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## Transform a bigint ``a`` from it's Montgomery N-residue representation (mod N)
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## to the regular natural representation (mod N)
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##
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## i.e. Does "a * ((2^LimbSize)^W)^-1 (mod N), where W is the number
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## of words needed to represent n in base 2^LimbSize
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##
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## This is called a Montgomery Reduction
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## The Montgomery Magic Constant is µ = -1/N mod N
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## is used internally and can be precomputed with montyMagic(Curve)
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# References:
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# - https://eprint.iacr.org/2017/1057.pdf (Montgomery)
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# page: Radix-r interleaved multiplication algorithm
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# - https://en.wikipedia.org/wiki/Montgomery_modular_multiplication#Montgomery_arithmetic_on_multiprecision_(variable-radix)_integers
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# - http://langevin.univ-tln.fr/cours/MLC/extra/montgomery.pdf
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# Montgomery original paper
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#
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checkValidModulus(N)
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checkOddModulus(N)
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checkMatchingBitlengths(a, N)
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let nLen = N.numLimbs()
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for i in 0 ..< nLen:
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let z0 = Word(BaseType(a[0]) * BaseType(montyMagic)) and MaxWord
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var carry = DoubleWord(0)
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for j in 0 ..< nLen:
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let z = DoubleWord(a[i]) + unsafeExtPrecMul(z0, N[i]) + carry
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carry = z shr WordBitSize
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if j != 0:
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a[j] = Word(z) and MaxWord
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a[^1] = Word(carry)
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@ -42,16 +42,21 @@ debug:
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# No exceptions allowed
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{.push raises: [].}
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func toMonty*[C: static Curve](a: Fq[C]): Montgomery[C] =
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## Convert a big integer over Fq to its montgomery representation
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## over Fq.
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## i.e. Does "a * (2^LimbSize)^W (mod p), where W is the number
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## of words needed to represent p in base 2^LimbSize
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# ############################################################
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#
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# Conversion
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#
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# ############################################################
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result = a
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for i in static(countdown(C.Mod.limbs.high, 1)):
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shiftAdd(result, 0)
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func fromBig*(T: type Fq, src: BigInt): T =
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## Convert a BigInt to its Montgomery form
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result.nres = src
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result.nres.unsafeMontgomeryResidue(Fq.C.Mod)
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func toBig*[C: static Curve](src: Fq[C]): auto =
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## Convert a finite-field element to a BigInt in natral representation
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result = src.nres
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result.unsafeRedC(C.Mod.nres, montyMagic(C))
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# ############################################################
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#
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@ -66,7 +71,7 @@ template add(a: var Fq, b: Fq, ctl: CTBool[Word]): CTBool[Word] =
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##
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## a and b MAY be the same buffer
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## a and b MUST have the same announced bitlength (i.e. `bits` static parameters)
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add(a.value, b.value, ctl)
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add(a.nres, b.nres, ctl)
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template sub(a: var Fq, b: Fq, ctl: CTBool[Word]): CTBool[Word] =
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## Constant-time big integer in-place optional substraction
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@ -75,7 +80,7 @@ template sub(a: var Fq, b: Fq, ctl: CTBool[Word]): CTBool[Word] =
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##
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## a and b MAY be the same buffer
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## a and b MUST have the same announced bitlength (i.e. `bits` static parameters)
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sub(a.value, b.value, ctl)
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sub(a.nres, b.nres, ctl)
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# ############################################################
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#
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@ -83,9 +88,6 @@ template sub(a: var Fq, b: Fq, ctl: CTBool[Word]): CTBool[Word] =
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#
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# ############################################################
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# No exceptions allowed
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{.push raises: [].}
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func `+=`*(a: var Fq, b: Fq) =
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## Addition over Fq
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var ctl = add(a, b, CtTrue)
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@ -11,6 +11,8 @@ import unittest, random,
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../constantine/io/io_fields,
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../constantine/config/curves
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import ../constantine/io/io_bigints
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static: doAssert defined(testingCurves), "This modules requires the -d:testingCurves compile option"
|
||||
|
||||
proc main() =
|
||||
|
|
@ -24,7 +26,15 @@ proc main() =
|
|||
z.fromUint(90'u32)
|
||||
|
||||
x += y
|
||||
check: bool(z == x)
|
||||
|
||||
var x_bytes: array[8, byte]
|
||||
x_bytes.serializeRawUint(x, cpuEndian)
|
||||
|
||||
check:
|
||||
# Check equality in the Montgomery domain
|
||||
bool(z == x)
|
||||
# Check equality when converting back to natural domain
|
||||
90'u64 == cast[uint64](x_bytes)
|
||||
|
||||
block:
|
||||
var x, y, z: Fq[Fake101]
|
||||
|
|
@ -44,7 +54,15 @@ proc main() =
|
|||
z.fromUint(1'u32)
|
||||
|
||||
x += y
|
||||
check: bool(z == x)
|
||||
|
||||
var x_bytes: array[8, byte]
|
||||
x_bytes.serializeRawUint(x, cpuEndian)
|
||||
|
||||
check:
|
||||
# Check equality in the Montgomery domain
|
||||
bool(z == x)
|
||||
# Check equality when converting back to natural domain
|
||||
1'u64 == cast[uint64](x_bytes)
|
||||
|
||||
test "Substraction mod 101":
|
||||
block:
|
||||
|
|
@ -55,7 +73,15 @@ proc main() =
|
|||
z.fromUint(70'u32)
|
||||
|
||||
x -= y
|
||||
check: bool(z == x)
|
||||
|
||||
var x_bytes: array[8, byte]
|
||||
x_bytes.serializeRawUint(x, cpuEndian)
|
||||
|
||||
check:
|
||||
# Check equality in the Montgomery domain
|
||||
bool(z == x)
|
||||
# Check equality when converting back to natural domain
|
||||
70'u64 == cast[uint64](x_bytes)
|
||||
|
||||
block:
|
||||
var x, y, z: Fq[Fake101]
|
||||
|
|
@ -65,7 +91,15 @@ proc main() =
|
|||
z.fromUint(0'u32)
|
||||
|
||||
x -= y
|
||||
check: bool(z == x)
|
||||
|
||||
var x_bytes: array[8, byte]
|
||||
x_bytes.serializeRawUint(x, cpuEndian)
|
||||
|
||||
check:
|
||||
# Check equality in the Montgomery domain
|
||||
bool(z == x)
|
||||
# Check equality when converting back to natural domain
|
||||
0'u64 == cast[uint64](x_bytes)
|
||||
|
||||
block:
|
||||
var x, y, z: Fq[Fake101]
|
||||
|
|
@ -75,6 +109,14 @@ proc main() =
|
|||
z.fromUint(100'u32)
|
||||
|
||||
x -= y
|
||||
check: bool(z == x)
|
||||
|
||||
var x_bytes: array[8, byte]
|
||||
x_bytes.serializeRawUint(x, cpuEndian)
|
||||
|
||||
check:
|
||||
# Check equality in the Montgomery domain
|
||||
bool(z == x)
|
||||
# Check equality when converting back to natural domain
|
||||
100'u64 == cast[uint64](x_bytes)
|
||||
|
||||
main()
|
||||
|
|
|
|||
Loading…
Reference in New Issue