Add more curves for testing: Curve25519, BLS12-377, BN446, FKM-447, BLS12-461, BN462

2020-03-21 13:05:58 +01:00 · 2020-03-21 13:05:58 +01:00 · 1855d14497
parent 9e78cd5d6d
commit 1855d14497
11 changed files with 297 additions and 90 deletions
--- a/README.md
+++ b/README.md
@ -35,6 +35,36 @@ The library focuses on following properties:

 in this order

+## Curves supported
+
+At the moment the following curves are supported, adding a new curve only requires adding the prime modulus
+and its bitsize in [constantine/config/curves.nim]().
+
+The following curves are configured:
+
+### ECDH / ECDSA curves
+
+- NIST P-224
+- Curve25519
+- NIST P-256 / Secp256r1
+- Secp256k1 (Bitcoin, Ethereum 1)
+
+### Pairing-Friendly curves
+
+Families:
+- BN: Barreto-Naerig
+- BLS: Barreto-Lynn-Scott
+- FKM: Fotiadis-Konstantinou-Martindale
+
+Curves:
+- BN254 (Zero-Knowledge Proofs, Snarks, Starks, Zcash, Ethereum 1)
+- BLS12-377 (Zexe)
+- BLS12-381 (Algorand, Chia Networks, Dfinity, Ethereum 2, Filecoin, Zcash Sapling)
+- BN446
+- FKM12-447
+- BLS12-461
+- BN462
+
 ## Security

 Hardening an implementation against all existing and upcoming attack vectors is an extremely complex task.
@ -82,6 +112,50 @@ The previous implementation was 15x slower and one of the key optimizations
 was changing the elliptic curve cryptography backend.
 It had a direct implication on hardware cost and/or cloud computing resources required.

+## Measuring performance
+
+To measure the performance of Constantine
+
+```bash
+git clone https://github.com/mratsim/constantine
+nimble bench_fp_clang
+nimble bench_fp2_clang
+```
+
+As mentioned in the [Compiler caveats](#compiler-caveats) section, GCC is up to 2x slower than Clang due to mishandling of carries and register usage.
+
+On my machine, for selected benchmarks on the prime field for popular pairing-friendly curves.
+
+```
+⚠️ Measurements are approximate and use the CPU nominal clock: Turbo-Boost and overclocking will skew them.
+==========================================================================================================
+
+All benchmarks are using constant-time implementations to protect against side-channel attacks.
+
+Compiled with Clang
+Running on Intel(R) Core(TM) i9-9980XE CPU @ 3.00GHz (overclocked all-core Turbo @4.1GHz)
+
+--------------------------------------------------------------------------------
+Addition        Fp[BN254]               0 ns         0 cycles
+Substraction    Fp[BN254]               0 ns         0 cycles
+Negation        Fp[BN254]               0 ns         0 cycles
+Multiplication  Fp[BN254]              21 ns        65 cycles
+Squaring        Fp[BN254]              18 ns        55 cycles
+Inversion       Fp[BN254]            6266 ns     18799 cycles
+--------------------------------------------------------------------------------
+Addition        Fp[BLS12_381]           0 ns         0 cycles
+Substraction    Fp[BLS12_381]           0 ns         0 cycles
+Negation        Fp[BLS12_381]           0 ns         0 cycles
+Multiplication  Fp[BLS12_381]          45 ns       136 cycles
+Squaring        Fp[BLS12_381]          39 ns       118 cycles
+Inversion       Fp[BLS12_381]       15683 ns     47050 cycles
+--------------------------------------------------------------------------------
+
+Notes:
+  GCC is significantly slower than Clang on multiprecision arithmetic.
+  The simplest operations might be optimized away by the compiler.
+```
+
 ### Compiler caveats

 Unfortunately compilers and in particular GCC are not very good at optimizing big integers and/or cryptographic code even when using intrinsics like `addcarry_u64`.
--- a/benchmarks/bench_fp.nim
+++ b/benchmarks/bench_fp.nim
@ -27,9 +27,15 @@ const InvIters = 1000
 const AvailableCurves = [
  P224,
  BN254,
+  Curve25519,
  P256,
  Secp256k1,
-  BLS12_381
+  BLS12_377,
+  BLS12_381,
+  BN446,
+  FKM12_447,
+  BLS12_461,
+  BN462
 ]

 proc main() =
--- a/benchmarks/bench_fp2.nim
+++ b/benchmarks/bench_fp2.nim
@ -26,8 +26,14 @@ import
 const Iters = 1_000_000
 const InvIters = 1000
 const AvailableCurves = [
+  # Pairing-Friendly curves
  BN254,
-  BLS12_381
+  BLS12_377,
+  BLS12_381,
+  BN446,
+  FKM12_447,
+  BLS12_461,
+  BN462
 ]

 proc main() =
--- a/benchmarks/bench_fp6.nim
+++ b/benchmarks/bench_fp6.nim
@ -26,8 +26,14 @@ import
 const Iters = 1_000_000
 const InvIters = 1000
 const AvailableCurves = [
+  # Pairing-Friendly curves
  BN254,
-  BLS12_381
+  BLS12_377,
+  BLS12_381,
+  BN446,
+  FKM12_447,
+  BLS12_461,
+  BN462
 ]

 proc main() =
--- a/constantine/config/curves.nim
+++ b/constantine/config/curves.nim
@ -39,54 +39,93 @@ import
 # - type Curve* = enum
 # - proc Mod*(curve: static Curve): auto
 #   which returns the field modulus of the curve
-when not defined(testingCurves):
-  declareCurves:
-    curve P224: # NIST P-224
-      bitsize: 224
-      modulus: "0xffffffff_ffffffff_ffffffff_ffffffff_00000000_00000000_00000001"
-    curve BN254:
-      bitsize: 254
-      modulus: "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"
-      # Equation: Y^2 = X^3 + 3
-    curve P256: # secp256r1 / NIST P-256
-      bitsize: 256
-      modulus: "0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff"
-    curve Secp256k1: # Bitcoin curve
-      bitsize: 256
-      modulus: "0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F"
-    curve BLS12_381:
-      bitsize: 381
-      modulus: "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
-      # Equation: y^2 = x^3 + 4
-else:
-  # Fake curve for testing field arithmetic
-  declareCurves:
-    curve Fake101:
-      bitsize: 7
-      modulus: "0x65" # 101 in hex
-    curve Mersenne61:
-      bitsize: 61
-      modulus: "0x1fffffffffffffff" # 2^61 - 1
-    curve Mersenne127:
-      bitsize: 127
-      modulus: "0x7fffffffffffffffffffffffffffffff" # 2^127 - 1
-    curve P224: # NIST P-224
-      bitsize: 224
-      modulus: "0xffffffff_ffffffff_ffffffff_ffffffff_00000000_00000000_00000001"
-    curve BN254: # Zero-Knowledge proofs curve (SNARKS, STARKS)
-      bitsize: 254
-      modulus: "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"
-      # Equation: Y^2 = X^3 + 3
-    curve P256: # secp256r1 / NIST P-256
-      bitsize: 256
-      modulus: "0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff"
-    curve Secp256k1: # Bitcoin curve
-      bitsize: 256
-      modulus: "0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F"
-    curve BLS12_381:
-      bitsize: 381
-      modulus: "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
-      # Equation: y^2 = x^3 + 4
+
+declareCurves:
+  # -----------------------------------------------------------------------------
+  # Curves added when passed "-d:testingCurves"
+  curve Fake101:
+    testingCurve: true
+    bitsize: 7
+    modulus: "0x65" # 101 in hex
+  curve Mersenne61:
+    testingCurve: true
+    bitsize: 61
+    modulus: "0x1fffffffffffffff" # 2^61 - 1
+  curve Mersenne127:
+    testingCurve: true
+    bitsize: 127
+    modulus: "0x7fffffffffffffffffffffffffffffff" # 2^127 - 1
+  # -----------------------------------------------------------------------------
+  curve P224: # NIST P-224
+    bitsize: 224
+    modulus: "0xffffffff_ffffffff_ffffffff_ffffffff_00000000_00000000_00000001"
+  curve BN254: # Zero-Knowledge proofs curve (SNARKS, STARKS)
+    bitsize: 254
+    modulus: "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"
+    # Equation: Y^2 = X^3 + 3
+  curve Curve25519: # Bernstein curve
+    bitsize: 255
+    modulus: "0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed"
+  curve P256: # secp256r1 / NIST P-256
+    bitsize: 256
+    modulus: "0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff"
+  curve Secp256k1: # Bitcoin curve
+    bitsize: 256
+    modulus: "0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F"
+  curve BLS12_377:
+    # Zexe curve
+    # (p41) https://eprint.iacr.org/2018/962.pdf
+    # https://github.com/ethereum/EIPs/blob/41dea9615/EIPS/eip-2539.md
+    bitsize: 377
+    modulus: "0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001"
+  curve BLS12_381:
+    bitsize: 381
+    modulus: "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
+    # Equation: y^2 = x^3 + 4
+  curve BN446:
+    bitsize: 446
+    modulus: "0x2400000000000000002400000002d00000000d800000021c0000001800000000870000000b0400000057c00000015c000000132000000067"
+    # u = 2^110 + 2^36 + 1
+  curve FKM12_447: # Fotiadis-Konstantinou-Martindale
+    bitsize: 447
+    modulus: "0x4ce300001338c00001c08180000f20cfffffe5a8bffffd08a000000f228000007e8ffffffaddfffffffdc00000009efffffffca000000007"
+    # TNFS Resistant Families of Pairing-Friendly Elliptic Curves
+    # Georgios Fotiadis and Elisavet Konstantinou, 2018
+    # https://eprint.iacr.org/2018/1017
+    #
+    # Family 17 choice b of
+    # Optimal TNFS-secure pairings on elliptic curves with composite embedding degree
+    # Georgios Fotiadis and Chloe Martindale, 2019
+    # https://eprint.iacr.org/2019/555
+    #
+    # A short-list of pairing-friendly curves resistant toSpecial TNFS at the 128-bit security level
+    # Aurore Guillevic
+    # https://hal.inria.fr/hal-02396352v2/document
+    #
+    # p(x) = 1728x^6 + 2160x^5 + 1548x^4 + 756x^3 + 240x^2 + 54x + 7
+    # t(x) = −6x² + 1,  r(x) = 36x^4 + 36x^3 + 18x^2 + 6x + 1.
+    # Choice (b):u=−2^72 − 2^71 − 2^36
+    #
+    # Note the paper mentions 446-bit but it's 447
+  curve BLS12_461:
+    # Updating Key Size Estimations for Pairings
+    # Barbulescu, R. and S. Duquesne, 2018
+    # https://hal.archives-ouvertes.fr/hal-01534101/file/main.pdf
+    bitsize: 461
+    modulus: "0x15555545554d5a555a55d69414935fbd6f1e32d8bacca47b14848b42a8dffa5c1cc00f26aa91557f00400020000555554aaaaaac0000aaaaaaab"
+    # u = −2^77 + 2^50 + 2^33
+    # p = (u - 1)^2 (u^4 - u^2 + 1)/3 + u
+  curve BN462:
+    # Pairing-Friendly Curves
+    # IETF Draft
+    # https://tools.ietf.org/id/draft-irtf-cfrg-pairing-friendly-curves-02.html
+
+    # Updating Key Size Estimations for Pairings
+    # Barbulescu, R. and S. Duquesne, 2018
+    # https://hal.archives-ouvertes.fr/hal-01534101/file/main.pdf
+    bitsize: 462
+    modulus: "0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908f41c8020ffffffffff6ff66fc6ff687f640000000002401b00840138013"
+    # u = 2^114 + 2^101 − 2^14 − 1

 # ############################################################
 #
@ -253,6 +292,10 @@ macro genMontyMagics(T: typed): untyped =

 genMontyMagics(Curve)

+func getCurveBitSize*(C: static Curve): static int =
+  ## Returns the number of bits taken by the curve modulus
+  result = static(CurveBitSize[C])
+
 macro canUseNoCarryMontyMul*(C: static Curve): untyped =
  ## Returns true if the Modulus is compatible with a fast
  ## Montgomery multiplication that avoids many carries
--- a/constantine/config/curves_parser.nim
+++ b/constantine/config/curves_parser.nim
@ -71,38 +71,45 @@ macro declareCurves*(curves: untyped): untyped =

    let curve = curveDesc[1]

-    let sizeSection = curveDesc[2][0]
+    var offset = 0
+    var testCurve = false
+    if curveDesc[2][0][0].eqident"testingCurve":
+      offset = 1
+      testCurve = curveDesc[2][0][1].boolVal
+
+    let sizeSection = curveDesc[2][offset]
    doAssert sizeSection[0].eqIdent"bitsize"
    sizeSection[1].expectKind(nnkStmtList)
    let bitSize = sizeSection[1][0]

-    let modSection = curveDesc[2][1]
+    let modSection = curveDesc[2][offset+1]
    doAssert modSection[0].eqIdent"modulus"
    modSection[1].expectKind(nnkStmtList)
    let modulus = modSection[1][0]

-    Curves.add curve
-    # "BN254: 254" for array construction
-    CurveBitSize.add nnkExprColonExpr.newTree(
-      curve, bitSize
-    )
+    if not testCurve or defined(testingCurves):
+      Curves.add curve
+      # "BN254: 254" for array construction
+      CurveBitSize.add nnkExprColonExpr.newTree(
+        curve, bitSize
+      )

-    # const BN254_Modulus = Fp[BN254](value: fromHex(BigInt[254], "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"))
-    let modulusID = ident($curve & "_Modulus")
-    curveModStmts.add newConstStmt(
-      modulusID,
-      nnkObjConstr.newTree(
-        nnkBracketExpr.newTree(Fp, curve),
-        nnkExprColonExpr.newTree(
-          ident"mres",
-          newCall(
-            bindSym"fromHex",
-            nnkBracketExpr.newTree(bindSym"BigInt", bitSize),
-            modulus
+      # const BN254_Modulus = Fp[BN254](value: fromHex(BigInt[254], "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"))
+      let modulusID = ident($curve & "_Modulus")
+      curveModStmts.add newConstStmt(
+        modulusID,
+        nnkObjConstr.newTree(
+          nnkBracketExpr.newTree(Fp, curve),
+          nnkExprColonExpr.newTree(
+            ident"mres",
+            newCall(
+              bindSym"fromHex",
+              nnkBracketExpr.newTree(bindSym"BigInt", bitSize),
+              modulus
+            )
          )
        )
      )
-    )

  # end for ---------------------------------------------------

--- a/constantine/pairing/README.md
+++ b/constantine/pairing/README.md
@ -15,6 +15,10 @@
  Ben Lynn, 2007\
  https://crypto.stanford.edu/pbc/thesis.pdf

+- An Analysis of Affine Coordinates for Pairing Computation\
+  Kristin Lauter, Peter L. Montgomery, and Michael Naehrig, 2010\
+  https://eprint.iacr.org/2010/363.pdf
+
 - Pairings for beginners\
  Craig Costello, 2012 (?)\
  http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
@ -30,11 +34,14 @@
  https://scholarworks.rit.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=10083&context=theses
  https://github.com/rajeevakarv/FiniteFieldComputations

-
 - A taxonomy of pairings, their security, their complexity\
  Razvan Barbulescu, Nadia El Mrabet, and Loubna Ghammam, 2019\
  https://hal.archives-ouvertes.fr/hal-02129868/file/2019-485.pdf

+- A short-list of pairing-friendly curves resistantto Special TNFS at the 128-bit security level\
+  Aurore Guillevic\
+  https://eprint.iacr.org/2019/1371.pdf
+
 ### Presentations

 - Introduction to pairings\
--- a/constantine/tower_field_extensions/README.md
+++ b/constantine/tower_field_extensions/README.md
@ -79,6 +79,10 @@ From Ben Edgington, https://hackmd.io/@benjaminion/bls12-381
  Naomi Benger and Michael Scott, 2009\
  https://eprint.iacr.org/2009/556

+- Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions\
+  Robert Granger and Michael Scott, 2009\
+  https://eprint.iacr.org/2009/565.pdf
+
 - High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves\
  Jean-Luc Beuchat and Jorge Enrique González Díaz and Shigeo Mitsunari and Eiji Okamoto and Francisco Rodríguez-Henríquez and Tadanori Teruya, 2010\
  https://eprint.iacr.org/2010/354
--- a/tests/test_finite_fields_vs_gmp.nim
+++ b/tests/test_finite_fields_vs_gmp.nim
@ -19,11 +19,17 @@ import

 var RNG {.compileTime.} = initRand(1234)
 const CurveParams = [
-  P224: (224, "0xffffffffffffffffffffffffffffffff000000000000000000000001"),
-  BN254: (254, "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47"),
-  P256: (256, "0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff"),
-  Secp256k1: (256, "0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F"),
-  BLS12_381: (381, "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab")
+  P224,
+  BN254,
+  Curve25519,
+  P256,
+  Secp256k1,
+  BLS12_377,
+  BLS12_381,
+  BN446,
+  FKM12_447,
+  BLS12_461,
+  BN462
 ]

 const AvailableCurves = [P224, BN254, P256, Secp256k1, BLS12_381]
@ -49,13 +55,13 @@ proc binary_prologue[C: static Curve, N: static int](
        a, b, p: var mpz_t,
        aTest, bTest: var Fp[C],
        aBuf, bBuf: var array[N, byte]) =
-  const bits = CurveParams[C][0]
+  const bits = C.getCurveBitSize()

  # Generate random value in the range 0 ..< 2^(bits-1)
  mpz_urandomb(a, gmpRng, uint bits)
  mpz_urandomb(b, gmpRng, uint bits)
  # Set modulus to curve modulus
-  let err = mpz_set_str(p, CurveParams[C][1], 0)
+  let err = mpz_set_str(p, Curve(C).Mod.mres.toHex(), 0)
  doAssert err == 0, "Error on prime for curve " & $Curve(C)

  #########################################################
@ -116,7 +122,7 @@ proc addTests(gmpRng: var gmp_randstate_t, a, b, p, r: var mpz_t, C: static Curv
  # echo "Testing: random modular addition on ", $C

  const
-    bits = CurveParams[C][0]
+    bits = C.getCurveBitSize()
    bufLen = (bits + 7) div 8
  var
    aTest, bTest{.noInit.}: Fp[C]
@ -139,7 +145,7 @@ proc subTests(gmpRng: var gmp_randstate_t, a, b, p, r: var mpz_t, C: static Curv
  # echo "Testing: random modular substraction on ", $C

  const
-    bits = CurveParams[C][0]
+    bits = C.getCurveBitSize()
    bufLen = (bits + 7) div 8
  var
    aTest, bTest{.noInit.}: Fp[C]
@ -162,7 +168,7 @@ proc mulTests(gmpRng: var gmp_randstate_t, a, b, p, r: var mpz_t, C: static Curv
  # echo "Testing: random modular multiplication on ", $C

  const
-    bits = CurveParams[C][0]
+    bits = C.getCurveBitSize()
    bufLen = (bits + 7) div 8
  var
    aTest, bTest{.noInit.}: Fp[C]
@ -182,7 +188,7 @@ proc invTests(gmpRng: var gmp_randstate_t, a, b, p, r: var mpz_t, C: static Curv
  # echo "Testing: random modular inversion on ", $C

  const
-    bits = CurveParams[C][0]
+    bits = C.getCurveBitSize()
    bufLen = (bits + 7) div 8
  var
    aTest, bTest{.noInit.}: Fp[C]
--- a/tests/test_fp2.nim
+++ b/tests/test_fp2.nim
@ -76,9 +76,14 @@ suite "𝔽p2 = 𝔽p[𝑖] (irreducible polynomial x²+1)":
        testInstance()

    test(BN254)
-    test(BLS12_381)
    test(P256)
    test(Secp256k1)
+    test(BLS12_377)
+    test(BLS12_381)
+    test(BN446)
+    test(FKM12_447)
+    test(BLS12_461)
+    test(BN462)

  test "Multiplication by 0 and 1":
    template test(C: static Curve, body: untyped) =
@ -192,9 +197,14 @@ suite "𝔽p2 = 𝔽p[𝑖] (irreducible polynomial x²+1)":
          bool(r0 == r4)

    abelianGroup(BN254)
-    abelianGroup(BLS12_381)
-    abelianGroup(Secp256k1)
    abelianGroup(P256)
+    abelianGroup(Secp256k1)
+    abelianGroup(BLS12_377)
+    abelianGroup(BLS12_381)
+    abelianGroup(BN446)
+    abelianGroup(FKM12_447)
+    abelianGroup(BLS12_461)
+    abelianGroup(BN462)

  test "𝔽p2 = 𝔽p[𝑖] multiplication is associative and commutative":
    proc commutativeRing(curve: static Curve) =
@ -239,9 +249,14 @@ suite "𝔽p2 = 𝔽p[𝑖] (irreducible polynomial x²+1)":
          bool(r0 == r4)

    commutativeRing(BN254)
-    commutativeRing(BLS12_381)
-    commutativeRing(Secp256k1)
    commutativeRing(P256)
+    commutativeRing(Secp256k1)
+    commutativeRing(BLS12_377)
+    commutativeRing(BLS12_381)
+    commutativeRing(BN446)
+    commutativeRing(FKM12_447)
+    commutativeRing(BLS12_461)
+    commutativeRing(BN462)

  test "𝔽p2 = 𝔽p[𝑖] extension field multiplicative inverse":
    proc mulInvOne(curve: static Curve) =
@ -259,6 +274,11 @@ suite "𝔽p2 = 𝔽p[𝑖] (irreducible polynomial x²+1)":
        check: bool(r == one)

    mulInvOne(BN254)
-    mulInvOne(BLS12_381)
-    mulInvOne(Secp256k1)
    mulInvOne(P256)
+    mulInvOne(Secp256k1)
+    mulInvOne(BLS12_377)
+    mulInvOne(BLS12_381)
+    mulInvOne(BN446)
+    mulInvOne(FKM12_447)
+    mulInvOne(BLS12_461)
+    mulInvOne(BN462)
--- a/tests/test_fp6.nim
+++ b/tests/test_fp6.nim
@ -51,7 +51,14 @@ suite "𝔽p6 = 𝔽p2[∛(1+𝑖)] (irreducible polynomial x³ - (1+𝑖))":
        testInstance()

    test(BN254)
+    test(P256)
+    test(Secp256k1)
+    test(BLS12_377)
    test(BLS12_381)
+    test(BN446)
+    test(FKM12_447)
+    test(BLS12_461)
+    test(BN462)

  test "Squaring 2 returns 4":
    template test(C: static Curve) =
@ -76,7 +83,14 @@ suite "𝔽p6 = 𝔽p2[∛(1+𝑖)] (irreducible polynomial x³ - (1+𝑖))":
        testInstance()

    test(BN254)
+    test(P256)
+    test(Secp256k1)
+    test(BLS12_377)
    test(BLS12_381)
+    test(BN446)
+    test(FKM12_447)
+    test(BLS12_461)
+    test(BN462)

  test "Squaring 3 returns 9":
    template test(C: static Curve) =
@ -103,7 +117,14 @@ suite "𝔽p6 = 𝔽p2[∛(1+𝑖)] (irreducible polynomial x³ - (1+𝑖))":
        testInstance()

    test(BN254)
+    test(P256)
+    test(Secp256k1)
+    test(BLS12_377)
    test(BLS12_381)
+    test(BN446)
+    test(FKM12_447)
+    test(BLS12_461)
+    test(BN462)

  test "Squaring -3 returns 9":
    template test(C: static Curve) =
@ -130,4 +151,11 @@ suite "𝔽p6 = 𝔽p2[∛(1+𝑖)] (irreducible polynomial x³ - (1+𝑖))":
        testInstance()

    test(BN254)
+    test(P256)
+    test(Secp256k1)
+    test(BLS12_377)
    test(BLS12_381)
+    test(BN446)
+    test(FKM12_447)
+    test(BLS12_461)
+    test(BN462)