diff options
Diffstat (limited to 'src/crypto/elliptic')
| -rw-r--r-- | src/crypto/elliptic/elliptic.go | 373 | ||||
| -rw-r--r-- | src/crypto/elliptic/elliptic_test.go | 458 | ||||
| -rw-r--r-- | src/crypto/elliptic/p224.go | 765 | ||||
| -rw-r--r-- | src/crypto/elliptic/p224_test.go | 47 | ||||
| -rw-r--r-- | src/crypto/elliptic/p256.go | 1186 |
5 files changed, 2829 insertions, 0 deletions
diff --git a/src/crypto/elliptic/elliptic.go b/src/crypto/elliptic/elliptic.go new file mode 100644 index 000000000..ba673f80c --- /dev/null +++ b/src/crypto/elliptic/elliptic.go @@ -0,0 +1,373 @@ +// Copyright 2010 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +// Package elliptic implements several standard elliptic curves over prime +// fields. +package elliptic + +// This package operates, internally, on Jacobian coordinates. For a given +// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1) +// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole +// calculation can be performed within the transform (as in ScalarMult and +// ScalarBaseMult). But even for Add and Double, it's faster to apply and +// reverse the transform than to operate in affine coordinates. + +import ( + "io" + "math/big" + "sync" +) + +// A Curve represents a short-form Weierstrass curve with a=-3. +// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html +type Curve interface { + // Params returns the parameters for the curve. + Params() *CurveParams + // IsOnCurve returns true if the given (x,y) lies on the curve. + IsOnCurve(x, y *big.Int) bool + // Add returns the sum of (x1,y1) and (x2,y2) + Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) + // Double returns 2*(x,y) + Double(x1, y1 *big.Int) (x, y *big.Int) + // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. + ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) + // ScalarBaseMult returns k*G, where G is the base point of the group + // and k is an integer in big-endian form. + ScalarBaseMult(k []byte) (x, y *big.Int) +} + +// CurveParams contains the parameters of an elliptic curve and also provides +// a generic, non-constant time implementation of Curve. +type CurveParams struct { + P *big.Int // the order of the underlying field + N *big.Int // the order of the base point + B *big.Int // the constant of the curve equation + Gx, Gy *big.Int // (x,y) of the base point + BitSize int // the size of the underlying field +} + +func (curve *CurveParams) Params() *CurveParams { + return curve +} + +func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool { + // y² = x³ - 3x + b + y2 := new(big.Int).Mul(y, y) + y2.Mod(y2, curve.P) + + x3 := new(big.Int).Mul(x, x) + x3.Mul(x3, x) + + threeX := new(big.Int).Lsh(x, 1) + threeX.Add(threeX, x) + + x3.Sub(x3, threeX) + x3.Add(x3, curve.B) + x3.Mod(x3, curve.P) + + return x3.Cmp(y2) == 0 +} + +// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and +// y are zero, it assumes that they represent the point at infinity because (0, +// 0) is not on the any of the curves handled here. +func zForAffine(x, y *big.Int) *big.Int { + z := new(big.Int) + if x.Sign() != 0 || y.Sign() != 0 { + z.SetInt64(1) + } + return z +} + +// affineFromJacobian reverses the Jacobian transform. See the comment at the +// top of the file. If the point is ∞ it returns 0, 0. +func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) { + if z.Sign() == 0 { + return new(big.Int), new(big.Int) + } + + zinv := new(big.Int).ModInverse(z, curve.P) + zinvsq := new(big.Int).Mul(zinv, zinv) + + xOut = new(big.Int).Mul(x, zinvsq) + xOut.Mod(xOut, curve.P) + zinvsq.Mul(zinvsq, zinv) + yOut = new(big.Int).Mul(y, zinvsq) + yOut.Mod(yOut, curve.P) + return +} + +func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) { + z1 := zForAffine(x1, y1) + z2 := zForAffine(x2, y2) + return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2)) +} + +// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and +// (x2, y2, z2) and returns their sum, also in Jacobian form. +func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) { + // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl + x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int) + if z1.Sign() == 0 { + x3.Set(x2) + y3.Set(y2) + z3.Set(z2) + return x3, y3, z3 + } + if z2.Sign() == 0 { + x3.Set(x1) + y3.Set(y1) + z3.Set(z1) + return x3, y3, z3 + } + + z1z1 := new(big.Int).Mul(z1, z1) + z1z1.Mod(z1z1, curve.P) + z2z2 := new(big.Int).Mul(z2, z2) + z2z2.Mod(z2z2, curve.P) + + u1 := new(big.Int).Mul(x1, z2z2) + u1.Mod(u1, curve.P) + u2 := new(big.Int).Mul(x2, z1z1) + u2.Mod(u2, curve.P) + h := new(big.Int).Sub(u2, u1) + xEqual := h.Sign() == 0 + if h.Sign() == -1 { + h.Add(h, curve.P) + } + i := new(big.Int).Lsh(h, 1) + i.Mul(i, i) + j := new(big.Int).Mul(h, i) + + s1 := new(big.Int).Mul(y1, z2) + s1.Mul(s1, z2z2) + s1.Mod(s1, curve.P) + s2 := new(big.Int).Mul(y2, z1) + s2.Mul(s2, z1z1) + s2.Mod(s2, curve.P) + r := new(big.Int).Sub(s2, s1) + if r.Sign() == -1 { + r.Add(r, curve.P) + } + yEqual := r.Sign() == 0 + if xEqual && yEqual { + return curve.doubleJacobian(x1, y1, z1) + } + r.Lsh(r, 1) + v := new(big.Int).Mul(u1, i) + + x3.Set(r) + x3.Mul(x3, x3) + x3.Sub(x3, j) + x3.Sub(x3, v) + x3.Sub(x3, v) + x3.Mod(x3, curve.P) + + y3.Set(r) + v.Sub(v, x3) + y3.Mul(y3, v) + s1.Mul(s1, j) + s1.Lsh(s1, 1) + y3.Sub(y3, s1) + y3.Mod(y3, curve.P) + + z3.Add(z1, z2) + z3.Mul(z3, z3) + z3.Sub(z3, z1z1) + z3.Sub(z3, z2z2) + z3.Mul(z3, h) + z3.Mod(z3, curve.P) + + return x3, y3, z3 +} + +func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) { + z1 := zForAffine(x1, y1) + return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1)) +} + +// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and +// returns its double, also in Jacobian form. +func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) { + // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b + delta := new(big.Int).Mul(z, z) + delta.Mod(delta, curve.P) + gamma := new(big.Int).Mul(y, y) + gamma.Mod(gamma, curve.P) + alpha := new(big.Int).Sub(x, delta) + if alpha.Sign() == -1 { + alpha.Add(alpha, curve.P) + } + alpha2 := new(big.Int).Add(x, delta) + alpha.Mul(alpha, alpha2) + alpha2.Set(alpha) + alpha.Lsh(alpha, 1) + alpha.Add(alpha, alpha2) + + beta := alpha2.Mul(x, gamma) + + x3 := new(big.Int).Mul(alpha, alpha) + beta8 := new(big.Int).Lsh(beta, 3) + x3.Sub(x3, beta8) + for x3.Sign() == -1 { + x3.Add(x3, curve.P) + } + x3.Mod(x3, curve.P) + + z3 := new(big.Int).Add(y, z) + z3.Mul(z3, z3) + z3.Sub(z3, gamma) + if z3.Sign() == -1 { + z3.Add(z3, curve.P) + } + z3.Sub(z3, delta) + if z3.Sign() == -1 { + z3.Add(z3, curve.P) + } + z3.Mod(z3, curve.P) + + beta.Lsh(beta, 2) + beta.Sub(beta, x3) + if beta.Sign() == -1 { + beta.Add(beta, curve.P) + } + y3 := alpha.Mul(alpha, beta) + + gamma.Mul(gamma, gamma) + gamma.Lsh(gamma, 3) + gamma.Mod(gamma, curve.P) + + y3.Sub(y3, gamma) + if y3.Sign() == -1 { + y3.Add(y3, curve.P) + } + y3.Mod(y3, curve.P) + + return x3, y3, z3 +} + +func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) { + Bz := new(big.Int).SetInt64(1) + x, y, z := new(big.Int), new(big.Int), new(big.Int) + + for _, byte := range k { + for bitNum := 0; bitNum < 8; bitNum++ { + x, y, z = curve.doubleJacobian(x, y, z) + if byte&0x80 == 0x80 { + x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z) + } + byte <<= 1 + } + } + + return curve.affineFromJacobian(x, y, z) +} + +func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) { + return curve.ScalarMult(curve.Gx, curve.Gy, k) +} + +var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f} + +// GenerateKey returns a public/private key pair. The private key is +// generated using the given reader, which must return random data. +func GenerateKey(curve Curve, rand io.Reader) (priv []byte, x, y *big.Int, err error) { + bitSize := curve.Params().BitSize + byteLen := (bitSize + 7) >> 3 + priv = make([]byte, byteLen) + + for x == nil { + _, err = io.ReadFull(rand, priv) + if err != nil { + return + } + // We have to mask off any excess bits in the case that the size of the + // underlying field is not a whole number of bytes. + priv[0] &= mask[bitSize%8] + // This is because, in tests, rand will return all zeros and we don't + // want to get the point at infinity and loop forever. + priv[1] ^= 0x42 + x, y = curve.ScalarBaseMult(priv) + } + return +} + +// Marshal converts a point into the form specified in section 4.3.6 of ANSI X9.62. +func Marshal(curve Curve, x, y *big.Int) []byte { + byteLen := (curve.Params().BitSize + 7) >> 3 + + ret := make([]byte, 1+2*byteLen) + ret[0] = 4 // uncompressed point + + xBytes := x.Bytes() + copy(ret[1+byteLen-len(xBytes):], xBytes) + yBytes := y.Bytes() + copy(ret[1+2*byteLen-len(yBytes):], yBytes) + return ret +} + +// Unmarshal converts a point, serialized by Marshal, into an x, y pair. On error, x = nil. +func Unmarshal(curve Curve, data []byte) (x, y *big.Int) { + byteLen := (curve.Params().BitSize + 7) >> 3 + if len(data) != 1+2*byteLen { + return + } + if data[0] != 4 { // uncompressed form + return + } + x = new(big.Int).SetBytes(data[1 : 1+byteLen]) + y = new(big.Int).SetBytes(data[1+byteLen:]) + return +} + +var initonce sync.Once +var p384 *CurveParams +var p521 *CurveParams + +func initAll() { + initP224() + initP256() + initP384() + initP521() +} + +func initP384() { + // See FIPS 186-3, section D.2.4 + p384 = new(CurveParams) + p384.P, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319", 10) + p384.N, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643", 10) + p384.B, _ = new(big.Int).SetString("b3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", 16) + p384.Gx, _ = new(big.Int).SetString("aa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", 16) + p384.Gy, _ = new(big.Int).SetString("3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f", 16) + p384.BitSize = 384 +} + +func initP521() { + // See FIPS 186-3, section D.2.5 + p521 = new(CurveParams) + p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10) + p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10) + p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16) + p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16) + p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16) + p521.BitSize = 521 +} + +// P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3) +func P256() Curve { + initonce.Do(initAll) + return p256 +} + +// P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4) +func P384() Curve { + initonce.Do(initAll) + return p384 +} + +// P521 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5) +func P521() Curve { + initonce.Do(initAll) + return p521 +} diff --git a/src/crypto/elliptic/elliptic_test.go b/src/crypto/elliptic/elliptic_test.go new file mode 100644 index 000000000..4dc27c92b --- /dev/null +++ b/src/crypto/elliptic/elliptic_test.go @@ -0,0 +1,458 @@ +// Copyright 2010 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package elliptic + +import ( + "crypto/rand" + "encoding/hex" + "fmt" + "math/big" + "testing" +) + +func TestOnCurve(t *testing.T) { + p224 := P224() + if !p224.IsOnCurve(p224.Params().Gx, p224.Params().Gy) { + t.Errorf("FAIL") + } +} + +type baseMultTest struct { + k string + x, y string +} + +var p224BaseMultTests = []baseMultTest{ + { + "1", + "b70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", + "bd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34", + }, + { + "2", + "706a46dc76dcb76798e60e6d89474788d16dc18032d268fd1a704fa6", + "1c2b76a7bc25e7702a704fa986892849fca629487acf3709d2e4e8bb", + }, + { + "3", + "df1b1d66a551d0d31eff822558b9d2cc75c2180279fe0d08fd896d04", + "a3f7f03cadd0be444c0aa56830130ddf77d317344e1af3591981a925", + }, + { + "4", + "ae99feebb5d26945b54892092a8aee02912930fa41cd114e40447301", + "482580a0ec5bc47e88bc8c378632cd196cb3fa058a7114eb03054c9", + }, + { + "5", + "31c49ae75bce7807cdff22055d94ee9021fedbb5ab51c57526f011aa", + "27e8bff1745635ec5ba0c9f1c2ede15414c6507d29ffe37e790a079b", + }, + { + "6", + "1f2483f82572251fca975fea40db821df8ad82a3c002ee6c57112408", + "89faf0ccb750d99b553c574fad7ecfb0438586eb3952af5b4b153c7e", + }, + { + "7", + "db2f6be630e246a5cf7d99b85194b123d487e2d466b94b24a03c3e28", + "f3a30085497f2f611ee2517b163ef8c53b715d18bb4e4808d02b963", + }, + { + "8", + "858e6f9cc6c12c31f5df124aa77767b05c8bc021bd683d2b55571550", + "46dcd3ea5c43898c5c5fc4fdac7db39c2f02ebee4e3541d1e78047a", + }, + { + "9", + "2fdcccfee720a77ef6cb3bfbb447f9383117e3daa4a07e36ed15f78d", + "371732e4f41bf4f7883035e6a79fcedc0e196eb07b48171697517463", + }, + { + "10", + "aea9e17a306517eb89152aa7096d2c381ec813c51aa880e7bee2c0fd", + "39bb30eab337e0a521b6cba1abe4b2b3a3e524c14a3fe3eb116b655f", + }, + { + "11", + "ef53b6294aca431f0f3c22dc82eb9050324f1d88d377e716448e507c", + "20b510004092e96636cfb7e32efded8265c266dfb754fa6d6491a6da", + }, + { + "12", + "6e31ee1dc137f81b056752e4deab1443a481033e9b4c93a3044f4f7a", + "207dddf0385bfdeab6e9acda8da06b3bbef224a93ab1e9e036109d13", + }, + { + "13", + "34e8e17a430e43289793c383fac9774247b40e9ebd3366981fcfaeca", + "252819f71c7fb7fbcb159be337d37d3336d7feb963724fdfb0ecb767", + }, + { + "14", + "a53640c83dc208603ded83e4ecf758f24c357d7cf48088b2ce01e9fa", + "d5814cd724199c4a5b974a43685fbf5b8bac69459c9469bc8f23ccaf", + }, + { + "15", + "baa4d8635511a7d288aebeedd12ce529ff102c91f97f867e21916bf9", + "979a5f4759f80f4fb4ec2e34f5566d595680a11735e7b61046127989", + }, + { + "16", + "b6ec4fe1777382404ef679997ba8d1cc5cd8e85349259f590c4c66d", + "3399d464345906b11b00e363ef429221f2ec720d2f665d7dead5b482", + }, + { + "17", + "b8357c3a6ceef288310e17b8bfeff9200846ca8c1942497c484403bc", + "ff149efa6606a6bd20ef7d1b06bd92f6904639dce5174db6cc554a26", + }, + { + "18", + "c9ff61b040874c0568479216824a15eab1a838a797d189746226e4cc", + "ea98d60e5ffc9b8fcf999fab1df7e7ef7084f20ddb61bb045a6ce002", + }, + { + "19", + "a1e81c04f30ce201c7c9ace785ed44cc33b455a022f2acdbc6cae83c", + "dcf1f6c3db09c70acc25391d492fe25b4a180babd6cea356c04719cd", + }, + { + "20", + "fcc7f2b45df1cd5a3c0c0731ca47a8af75cfb0347e8354eefe782455", + "d5d7110274cba7cdee90e1a8b0d394c376a5573db6be0bf2747f530", + }, + { + "112233445566778899", + "61f077c6f62ed802dad7c2f38f5c67f2cc453601e61bd076bb46179e", + "2272f9e9f5933e70388ee652513443b5e289dd135dcc0d0299b225e4", + }, + { + "112233445566778899112233445566778899", + "29895f0af496bfc62b6ef8d8a65c88c613949b03668aab4f0429e35", + "3ea6e53f9a841f2019ec24bde1a75677aa9b5902e61081c01064de93", + }, + { + "6950511619965839450988900688150712778015737983940691968051900319680", + "ab689930bcae4a4aa5f5cb085e823e8ae30fd365eb1da4aba9cf0379", + "3345a121bbd233548af0d210654eb40bab788a03666419be6fbd34e7", + }, + { + "13479972933410060327035789020509431695094902435494295338570602119423", + "bdb6a8817c1f89da1c2f3dd8e97feb4494f2ed302a4ce2bc7f5f4025", + "4c7020d57c00411889462d77a5438bb4e97d177700bf7243a07f1680", + }, + { + "13479971751745682581351455311314208093898607229429740618390390702079", + "d58b61aa41c32dd5eba462647dba75c5d67c83606c0af2bd928446a9", + "d24ba6a837be0460dd107ae77725696d211446c5609b4595976b16bd", + }, + { + "13479972931865328106486971546324465392952975980343228160962702868479", + "dc9fa77978a005510980e929a1485f63716df695d7a0c18bb518df03", + "ede2b016f2ddffc2a8c015b134928275ce09e5661b7ab14ce0d1d403", + }, + { + "11795773708834916026404142434151065506931607341523388140225443265536", + "499d8b2829cfb879c901f7d85d357045edab55028824d0f05ba279ba", + "bf929537b06e4015919639d94f57838fa33fc3d952598dcdbb44d638", + }, + { + "784254593043826236572847595991346435467177662189391577090", + "8246c999137186632c5f9eddf3b1b0e1764c5e8bd0e0d8a554b9cb77", + "e80ed8660bc1cb17ac7d845be40a7a022d3306f116ae9f81fea65947", + }, + { + 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"f2a28eefd8b345832116f1e574f2c6b2c895aa8c24941f40d8b80ad1", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368042", + "a1e81c04f30ce201c7c9ace785ed44cc33b455a022f2acdbc6cae83c", + "230e093c24f638f533dac6e2b6d01da3b5e7f45429315ca93fb8e634", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368043", + "c9ff61b040874c0568479216824a15eab1a838a797d189746226e4cc", + "156729f1a003647030666054e208180f8f7b0df2249e44fba5931fff", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368044", + "b8357c3a6ceef288310e17b8bfeff9200846ca8c1942497c484403bc", + "eb610599f95942df1082e4f9426d086fb9c6231ae8b24933aab5db", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368045", + "b6ec4fe1777382404ef679997ba8d1cc5cd8e85349259f590c4c66d", + "cc662b9bcba6f94ee4ff1c9c10bd6ddd0d138df2d099a282152a4b7f", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368046", + 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"26959946667150639794667015087019625940457807714424391721682722368051", + "aea9e17a306517eb89152aa7096d2c381ec813c51aa880e7bee2c0fd", + "c644cf154cc81f5ade49345e541b4d4b5c1adb3eb5c01c14ee949aa2", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368052", + "2fdcccfee720a77ef6cb3bfbb447f9383117e3daa4a07e36ed15f78d", + "c8e8cd1b0be40b0877cfca1958603122f1e6914f84b7e8e968ae8b9e", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368053", + "858e6f9cc6c12c31f5df124aa77767b05c8bc021bd683d2b55571550", + "fb9232c15a3bc7673a3a03b0253824c53d0fd1411b1cabe2e187fb87", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368054", + "db2f6be630e246a5cf7d99b85194b123d487e2d466b94b24a03c3e28", + "f0c5cff7ab680d09ee11dae84e9c1072ac48ea2e744b1b7f72fd469e", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368055", + "1f2483f82572251fca975fea40db821df8ad82a3c002ee6c57112408", + "76050f3348af2664aac3a8b05281304ebc7a7914c6ad50a4b4eac383", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368056", + "31c49ae75bce7807cdff22055d94ee9021fedbb5ab51c57526f011aa", + "d817400e8ba9ca13a45f360e3d121eaaeb39af82d6001c8186f5f866", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368057", + "ae99feebb5d26945b54892092a8aee02912930fa41cd114e40447301", + "fb7da7f5f13a43b81774373c879cd32d6934c05fa758eeb14fcfab38", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368058", + "df1b1d66a551d0d31eff822558b9d2cc75c2180279fe0d08fd896d04", + "5c080fc3522f41bbb3f55a97cfecf21f882ce8cbb1e50ca6e67e56dc", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368059", + "706a46dc76dcb76798e60e6d89474788d16dc18032d268fd1a704fa6", + "e3d4895843da188fd58fb0567976d7b50359d6b78530c8f62d1b1746", + }, + { + "26959946667150639794667015087019625940457807714424391721682722368060", + "b70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", + "42c89c774a08dc04b3dd201932bc8a5ea5f8b89bbb2a7e667aff81cd", + }, +} + +func TestBaseMult(t *testing.T) { + p224 := P224() + for i, e := range p224BaseMultTests { + k, ok := new(big.Int).SetString(e.k, 10) + if !ok { + t.Errorf("%d: bad value for k: %s", i, e.k) + } + x, y := p224.ScalarBaseMult(k.Bytes()) + if fmt.Sprintf("%x", x) != e.x || fmt.Sprintf("%x", y) != e.y { + t.Errorf("%d: bad output for k=%s: got (%x, %x), want (%s, %s)", i, e.k, x, y, e.x, e.y) + } + if testing.Short() && i > 5 { + break + } + } +} + +func TestGenericBaseMult(t *testing.T) { + // We use the P224 CurveParams directly in order to test the generic implementation. + p224 := P224().Params() + for i, e := range p224BaseMultTests { + k, ok := new(big.Int).SetString(e.k, 10) + if !ok { + t.Errorf("%d: bad value for k: %s", i, e.k) + } + x, y := p224.ScalarBaseMult(k.Bytes()) + if fmt.Sprintf("%x", x) != e.x || fmt.Sprintf("%x", y) != e.y { + t.Errorf("%d: bad output for k=%s: got (%x, %x), want (%s, %s)", i, e.k, x, y, e.x, e.y) + } + if testing.Short() && i > 5 { + break + } + } +} + +func TestP256BaseMult(t *testing.T) { + p256 := P256() + p256Generic := p256.Params() + + scalars := make([]*big.Int, 0, len(p224BaseMultTests)+1) + for _, e := range p224BaseMultTests { + k, _ := new(big.Int).SetString(e.k, 10) + scalars = append(scalars, k) + } + k := new(big.Int).SetInt64(1) + k.Lsh(k, 500) + scalars = append(scalars, k) + + for i, k := range scalars { + x, y := p256.ScalarBaseMult(k.Bytes()) + x2, y2 := p256Generic.ScalarBaseMult(k.Bytes()) + if x.Cmp(x2) != 0 || y.Cmp(y2) != 0 { + t.Errorf("#%d: got (%x, %x), want (%x, %x)", i, x, y, x2, y2) + } + + if testing.Short() && i > 5 { + break + } + } +} + +func TestP256Mult(t *testing.T) { + p256 := P256() + p256Generic := p256.Params() + + for i, e := range p224BaseMultTests { + x, _ := new(big.Int).SetString(e.x, 16) + y, _ := new(big.Int).SetString(e.y, 16) + k, _ := new(big.Int).SetString(e.k, 10) + + xx, yy := p256.ScalarMult(x, y, k.Bytes()) + xx2, yy2 := p256Generic.ScalarMult(x, y, k.Bytes()) + if xx.Cmp(xx2) != 0 || yy.Cmp(yy2) != 0 { + t.Errorf("#%d: got (%x, %x), want (%x, %x)", i, xx, yy, xx2, yy2) + } + if testing.Short() && i > 5 { + break + } + } +} + +func TestInfinity(t *testing.T) { + tests := []struct { + name string + curve Curve + }{ + {"p224", P224()}, + {"p256", P256()}, + } + + for _, test := range tests { + curve := test.curve + x, y := curve.ScalarBaseMult(nil) + if x.Sign() != 0 || y.Sign() != 0 { + t.Errorf("%s: x^0 != ∞", test.name) + } + x.SetInt64(0) + y.SetInt64(0) + + x2, y2 := curve.Double(x, y) + if x2.Sign() != 0 || y2.Sign() != 0 { + t.Errorf("%s: 2∞ != ∞", test.name) + } + + baseX := curve.Params().Gx + baseY := curve.Params().Gy + + x3, y3 := curve.Add(baseX, baseY, x, y) + if x3.Cmp(baseX) != 0 || y3.Cmp(baseY) != 0 { + t.Errorf("%s: x+∞ != x", test.name) + } + + x4, y4 := curve.Add(x, y, baseX, baseY) + if x4.Cmp(baseX) != 0 || y4.Cmp(baseY) != 0 { + t.Errorf("%s: ∞+x != x", test.name) + } + } +} + +func BenchmarkBaseMult(b *testing.B) { + b.ResetTimer() + p224 := P224() + e := p224BaseMultTests[25] + k, _ := new(big.Int).SetString(e.k, 10) + b.StartTimer() + for i := 0; i < b.N; i++ { + p224.ScalarBaseMult(k.Bytes()) + } +} + +func BenchmarkBaseMultP256(b *testing.B) { + b.ResetTimer() + p256 := P256() + e := p224BaseMultTests[25] + k, _ := new(big.Int).SetString(e.k, 10) + b.StartTimer() + for i := 0; i < b.N; i++ { + p256.ScalarBaseMult(k.Bytes()) + } +} + +func TestMarshal(t *testing.T) { + p224 := P224() + _, x, y, err := GenerateKey(p224, rand.Reader) + if err != nil { + t.Error(err) + return + } + serialized := Marshal(p224, x, y) + xx, yy := Unmarshal(p224, serialized) + if xx == nil { + t.Error("failed to unmarshal") + return + } + if xx.Cmp(x) != 0 || yy.Cmp(y) != 0 { + t.Error("unmarshal returned different values") + return + } +} + +func TestP224Overflow(t *testing.T) { + // This tests for a specific bug in the P224 implementation. + p224 := P224() + pointData, _ := hex.DecodeString("049B535B45FB0A2072398A6831834624C7E32CCFD5A4B933BCEAF77F1DD945E08BBE5178F5EDF5E733388F196D2A631D2E075BB16CBFEEA15B") + x, y := Unmarshal(p224, pointData) + if !p224.IsOnCurve(x, y) { + t.Error("P224 failed to validate a correct point") + } +} diff --git a/src/crypto/elliptic/p224.go b/src/crypto/elliptic/p224.go new file mode 100644 index 000000000..1f7ff3f9d --- /dev/null +++ b/src/crypto/elliptic/p224.go @@ -0,0 +1,765 @@ +// Copyright 2012 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package elliptic + +// This is a constant-time, 32-bit implementation of P224. See FIPS 186-3, +// section D.2.2. +// +// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. + +import ( + "math/big" +) + +var p224 p224Curve + +type p224Curve struct { + *CurveParams + gx, gy, b p224FieldElement +} + +func initP224() { + // See FIPS 186-3, section D.2.2 + p224.CurveParams = new(CurveParams) + p224.P, _ = new(big.Int).SetString("26959946667150639794667015087019630673557916260026308143510066298881", 10) + p224.N, _ = new(big.Int).SetString("26959946667150639794667015087019625940457807714424391721682722368061", 10) + p224.B, _ = new(big.Int).SetString("b4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4", 16) + p224.Gx, _ = new(big.Int).SetString("b70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", 16) + p224.Gy, _ = new(big.Int).SetString("bd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34", 16) + p224.BitSize = 224 + + p224FromBig(&p224.gx, p224.Gx) + p224FromBig(&p224.gy, p224.Gy) + p224FromBig(&p224.b, p224.B) +} + +// P224 returns a Curve which implements P-224 (see FIPS 186-3, section D.2.2) +func P224() Curve { + initonce.Do(initAll) + return p224 +} + +func (curve p224Curve) Params() *CurveParams { + return curve.CurveParams +} + +func (curve p224Curve) IsOnCurve(bigX, bigY *big.Int) bool { + var x, y p224FieldElement + p224FromBig(&x, bigX) + p224FromBig(&y, bigY) + + // y² = x³ - 3x + b + var tmp p224LargeFieldElement + var x3 p224FieldElement + p224Square(&x3, &x, &tmp) + p224Mul(&x3, &x3, &x, &tmp) + + for i := 0; i < 8; i++ { + x[i] *= 3 + } + p224Sub(&x3, &x3, &x) + p224Reduce(&x3) + p224Add(&x3, &x3, &curve.b) + p224Contract(&x3, &x3) + + p224Square(&y, &y, &tmp) + p224Contract(&y, &y) + + for i := 0; i < 8; i++ { + if y[i] != x3[i] { + return false + } + } + return true +} + +func (p224Curve) Add(bigX1, bigY1, bigX2, bigY2 *big.Int) (x, y *big.Int) { + var x1, y1, z1, x2, y2, z2, x3, y3, z3 p224FieldElement + + p224FromBig(&x1, bigX1) + p224FromBig(&y1, bigY1) + if bigX1.Sign() != 0 || bigY1.Sign() != 0 { + z1[0] = 1 + } + p224FromBig(&x2, bigX2) + p224FromBig(&y2, bigY2) + if bigX2.Sign() != 0 || bigY2.Sign() != 0 { + z2[0] = 1 + } + + p224AddJacobian(&x3, &y3, &z3, &x1, &y1, &z1, &x2, &y2, &z2) + return p224ToAffine(&x3, &y3, &z3) +} + +func (p224Curve) Double(bigX1, bigY1 *big.Int) (x, y *big.Int) { + var x1, y1, z1, x2, y2, z2 p224FieldElement + + p224FromBig(&x1, bigX1) + p224FromBig(&y1, bigY1) + z1[0] = 1 + + p224DoubleJacobian(&x2, &y2, &z2, &x1, &y1, &z1) + return p224ToAffine(&x2, &y2, &z2) +} + +func (p224Curve) ScalarMult(bigX1, bigY1 *big.Int, scalar []byte) (x, y *big.Int) { + var x1, y1, z1, x2, y2, z2 p224FieldElement + + p224FromBig(&x1, bigX1) + p224FromBig(&y1, bigY1) + z1[0] = 1 + + p224ScalarMult(&x2, &y2, &z2, &x1, &y1, &z1, scalar) + return p224ToAffine(&x2, &y2, &z2) +} + +func (curve p224Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) { + var z1, x2, y2, z2 p224FieldElement + + z1[0] = 1 + p224ScalarMult(&x2, &y2, &z2, &curve.gx, &curve.gy, &z1, scalar) + return p224ToAffine(&x2, &y2, &z2) +} + +// Field element functions. +// +// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1. +// +// Field elements are represented by a FieldElement, which is a typedef to an +// array of 8 uint32's. The value of a FieldElement, a, is: +// a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7] +// +// Using 28-bit limbs means that there's only 4 bits of headroom, which is less +// than we would really like. But it has the useful feature that we hit 2**224 +// exactly, making the reflections during a reduce much nicer. +type p224FieldElement [8]uint32 + +// p224P is the order of the field, represented as a p224FieldElement. +var p224P = [8]uint32{1, 0, 0, 0xffff000, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff} + +// p224IsZero returns 1 if a == 0 mod p and 0 otherwise. +// +// a[i] < 2**29 +func p224IsZero(a *p224FieldElement) uint32 { + // Since a p224FieldElement contains 224 bits there are two possible + // representations of 0: 0 and p. + var minimal p224FieldElement + p224Contract(&minimal, a) + + var isZero, isP uint32 + for i, v := range minimal { + isZero |= v + isP |= v - p224P[i] + } + + // If either isZero or isP is 0, then we should return 1. + isZero |= isZero >> 16 + isZero |= isZero >> 8 + isZero |= isZero >> 4 + isZero |= isZero >> 2 + isZero |= isZero >> 1 + + isP |= isP >> 16 + isP |= isP >> 8 + isP |= isP >> 4 + isP |= isP >> 2 + isP |= isP >> 1 + + // For isZero and isP, the LSB is 0 iff all the bits are zero. + result := isZero & isP + result = (^result) & 1 + + return result +} + +// p224Add computes *out = a+b +// +// a[i] + b[i] < 2**32 +func p224Add(out, a, b *p224FieldElement) { + for i := 0; i < 8; i++ { + out[i] = a[i] + b[i] + } +} + +const two31p3 = 1<<31 + 1<<3 +const two31m3 = 1<<31 - 1<<3 +const two31m15m3 = 1<<31 - 1<<15 - 1<<3 + +// p224ZeroModP31 is 0 mod p where bit 31 is set in all limbs so that we can +// subtract smaller amounts without underflow. See the section "Subtraction" in +// [1] for reasoning. +var p224ZeroModP31 = []uint32{two31p3, two31m3, two31m3, two31m15m3, two31m3, two31m3, two31m3, two31m3} + +// p224Sub computes *out = a-b +// +// a[i], b[i] < 2**30 +// out[i] < 2**32 +func p224Sub(out, a, b *p224FieldElement) { + for i := 0; i < 8; i++ { + out[i] = a[i] + p224ZeroModP31[i] - b[i] + } +} + +// LargeFieldElement also represents an element of the field. The limbs are +// still spaced 28-bits apart and in little-endian order. So the limbs are at +// 0, 28, 56, ..., 392 bits, each 64-bits wide. +type p224LargeFieldElement [15]uint64 + +const two63p35 = 1<<63 + 1<<35 +const two63m35 = 1<<63 - 1<<35 +const two63m35m19 = 1<<63 - 1<<35 - 1<<19 + +// p224ZeroModP63 is 0 mod p where bit 63 is set in all limbs. See the section +// "Subtraction" in [1] for why. +var p224ZeroModP63 = [8]uint64{two63p35, two63m35, two63m35, two63m35, two63m35m19, two63m35, two63m35, two63m35} + +const bottom12Bits = 0xfff +const bottom28Bits = 0xfffffff + +// p224Mul computes *out = a*b +// +// a[i] < 2**29, b[i] < 2**30 (or vice versa) +// out[i] < 2**29 +func p224Mul(out, a, b *p224FieldElement, tmp *p224LargeFieldElement) { + for i := 0; i < 15; i++ { + tmp[i] = 0 + } + + for i := 0; i < 8; i++ { + for j := 0; j < 8; j++ { + tmp[i+j] += uint64(a[i]) * uint64(b[j]) + } + } + + p224ReduceLarge(out, tmp) +} + +// Square computes *out = a*a +// +// a[i] < 2**29 +// out[i] < 2**29 +func p224Square(out, a *p224FieldElement, tmp *p224LargeFieldElement) { + for i := 0; i < 15; i++ { + tmp[i] = 0 + } + + for i := 0; i < 8; i++ { + for j := 0; j <= i; j++ { + r := uint64(a[i]) * uint64(a[j]) + if i == j { + tmp[i+j] += r + } else { + tmp[i+j] += r << 1 + } + } + } + + p224ReduceLarge(out, tmp) +} + +// ReduceLarge converts a p224LargeFieldElement to a p224FieldElement. +// +// in[i] < 2**62 +func p224ReduceLarge(out *p224FieldElement, in *p224LargeFieldElement) { + for i := 0; i < 8; i++ { + in[i] += p224ZeroModP63[i] + } + + // Eliminate the coefficients at 2**224 and greater. + for i := 14; i >= 8; i-- { + in[i-8] -= in[i] + in[i-5] += (in[i] & 0xffff) << 12 + in[i-4] += in[i] >> 16 + } + in[8] = 0 + // in[0..8] < 2**64 + + // As the values become small enough, we start to store them in |out| + // and use 32-bit operations. + for i := 1; i < 8; i++ { + in[i+1] += in[i] >> 28 + out[i] = uint32(in[i] & bottom28Bits) + } + in[0] -= in[8] + out[3] += uint32(in[8]&0xffff) << 12 + out[4] += uint32(in[8] >> 16) + // in[0] < 2**64 + // out[3] < 2**29 + // out[4] < 2**29 + // out[1,2,5..7] < 2**28 + + out[0] = uint32(in[0] & bottom28Bits) + out[1] += uint32((in[0] >> 28) & bottom28Bits) + out[2] += uint32(in[0] >> 56) + // out[0] < 2**28 + // out[1..4] < 2**29 + // out[5..7] < 2**28 +} + +// Reduce reduces the coefficients of a to smaller bounds. +// +// On entry: a[i] < 2**31 + 2**30 +// On exit: a[i] < 2**29 +func p224Reduce(a *p224FieldElement) { + for i := 0; i < 7; i++ { + a[i+1] += a[i] >> 28 + a[i] &= bottom28Bits + } + top := a[7] >> 28 + a[7] &= bottom28Bits + + // top < 2**4 + mask := top + mask |= mask >> 2 + mask |= mask >> 1 + mask <<= 31 + mask = uint32(int32(mask) >> 31) + // Mask is all ones if top != 0, all zero otherwise + + a[0] -= top + a[3] += top << 12 + + // We may have just made a[0] negative but, if we did, then we must + // have added something to a[3], this it's > 2**12. Therefore we can + // carry down to a[0]. + a[3] -= 1 & mask + a[2] += mask & (1<<28 - 1) + a[1] += mask & (1<<28 - 1) + a[0] += mask & (1 << 28) +} + +// p224Invert calculates *out = in**-1 by computing in**(2**224 - 2**96 - 1), +// i.e. Fermat's little theorem. +func p224Invert(out, in *p224FieldElement) { + var f1, f2, f3, f4 p224FieldElement + var c p224LargeFieldElement + + p224Square(&f1, in, &c) // 2 + p224Mul(&f1, &f1, in, &c) // 2**2 - 1 + p224Square(&f1, &f1, &c) // 2**3 - 2 + p224Mul(&f1, &f1, in, &c) // 2**3 - 1 + p224Square(&f2, &f1, &c) // 2**4 - 2 + p224Square(&f2, &f2, &c) // 2**5 - 4 + p224Square(&f2, &f2, &c) // 2**6 - 8 + p224Mul(&f1, &f1, &f2, &c) // 2**6 - 1 + p224Square(&f2, &f1, &c) // 2**7 - 2 + for i := 0; i < 5; i++ { // 2**12 - 2**6 + p224Square(&f2, &f2, &c) + } + p224Mul(&f2, &f2, &f1, &c) // 2**12 - 1 + p224Square(&f3, &f2, &c) // 2**13 - 2 + for i := 0; i < 11; i++ { // 2**24 - 2**12 + p224Square(&f3, &f3, &c) + } + p224Mul(&f2, &f3, &f2, &c) // 2**24 - 1 + p224Square(&f3, &f2, &c) // 2**25 - 2 + for i := 0; i < 23; i++ { // 2**48 - 2**24 + p224Square(&f3, &f3, &c) + } + p224Mul(&f3, &f3, &f2, &c) // 2**48 - 1 + p224Square(&f4, &f3, &c) // 2**49 - 2 + for i := 0; i < 47; i++ { // 2**96 - 2**48 + p224Square(&f4, &f4, &c) + } + p224Mul(&f3, &f3, &f4, &c) // 2**96 - 1 + p224Square(&f4, &f3, &c) // 2**97 - 2 + for i := 0; i < 23; i++ { // 2**120 - 2**24 + p224Square(&f4, &f4, &c) + } + p224Mul(&f2, &f4, &f2, &c) // 2**120 - 1 + for i := 0; i < 6; i++ { // 2**126 - 2**6 + p224Square(&f2, &f2, &c) + } + p224Mul(&f1, &f1, &f2, &c) // 2**126 - 1 + p224Square(&f1, &f1, &c) // 2**127 - 2 + p224Mul(&f1, &f1, in, &c) // 2**127 - 1 + for i := 0; i < 97; i++ { // 2**224 - 2**97 + p224Square(&f1, &f1, &c) + } + p224Mul(out, &f1, &f3, &c) // 2**224 - 2**96 - 1 +} + +// p224Contract converts a FieldElement to its unique, minimal form. +// +// On entry, in[i] < 2**29 +// On exit, in[i] < 2**28 +func p224Contract(out, in *p224FieldElement) { + copy(out[:], in[:]) + + for i := 0; i < 7; i++ { + out[i+1] += out[i] >> 28 + out[i] &= bottom28Bits + } + top := out[7] >> 28 + out[7] &= bottom28Bits + + out[0] -= top + out[3] += top << 12 + + // We may just have made out[i] negative. So we carry down. If we made + // out[0] negative then we know that out[3] is sufficiently positive + // because we just added to it. + for i := 0; i < 3; i++ { + mask := uint32(int32(out[i]) >> 31) + out[i] += (1 << 28) & mask + out[i+1] -= 1 & mask + } + + // We might have pushed out[3] over 2**28 so we perform another, partial, + // carry chain. + for i := 3; i < 7; i++ { + out[i+1] += out[i] >> 28 + out[i] &= bottom28Bits + } + top = out[7] >> 28 + out[7] &= bottom28Bits + + // Eliminate top while maintaining the same value mod p. + out[0] -= top + out[3] += top << 12 + + // There are two cases to consider for out[3]: + // 1) The first time that we eliminated top, we didn't push out[3] over + // 2**28. In this case, the partial carry chain didn't change any values + // and top is zero. + // 2) We did push out[3] over 2**28 the first time that we eliminated top. + // The first value of top was in [0..16), therefore, prior to eliminating + // the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after + // overflowing and being reduced by the second carry chain, out[3] <= + // 0xf000. Thus it cannot have overflowed when we eliminated top for the + // second time. + + // Again, we may just have made out[0] negative, so do the same carry down. + // As before, if we made out[0] negative then we know that out[3] is + // sufficiently positive. + for i := 0; i < 3; i++ { + mask := uint32(int32(out[i]) >> 31) + out[i] += (1 << 28) & mask + out[i+1] -= 1 & mask + } + + // Now we see if the value is >= p and, if so, subtract p. + + // First we build a mask from the top four limbs, which must all be + // equal to bottom28Bits if the whole value is >= p. If top4AllOnes + // ends up with any zero bits in the bottom 28 bits, then this wasn't + // true. + top4AllOnes := uint32(0xffffffff) + for i := 4; i < 8; i++ { + top4AllOnes &= out[i] + } + top4AllOnes |= 0xf0000000 + // Now we replicate any zero bits to all the bits in top4AllOnes. + top4AllOnes &= top4AllOnes >> 16 + top4AllOnes &= top4AllOnes >> 8 + top4AllOnes &= top4AllOnes >> 4 + top4AllOnes &= top4AllOnes >> 2 + top4AllOnes &= top4AllOnes >> 1 + top4AllOnes = uint32(int32(top4AllOnes<<31) >> 31) + + // Now we test whether the bottom three limbs are non-zero. + bottom3NonZero := out[0] | out[1] | out[2] + bottom3NonZero |= bottom3NonZero >> 16 + bottom3NonZero |= bottom3NonZero >> 8 + bottom3NonZero |= bottom3NonZero >> 4 + bottom3NonZero |= bottom3NonZero >> 2 + bottom3NonZero |= bottom3NonZero >> 1 + bottom3NonZero = uint32(int32(bottom3NonZero<<31) >> 31) + + // Everything depends on the value of out[3]. + // If it's > 0xffff000 and top4AllOnes != 0 then the whole value is >= p + // If it's = 0xffff000 and top4AllOnes != 0 and bottom3NonZero != 0, + // then the whole value is >= p + // If it's < 0xffff000, then the whole value is < p + n := out[3] - 0xffff000 + out3Equal := n + out3Equal |= out3Equal >> 16 + out3Equal |= out3Equal >> 8 + out3Equal |= out3Equal >> 4 + out3Equal |= out3Equal >> 2 + out3Equal |= out3Equal >> 1 + out3Equal = ^uint32(int32(out3Equal<<31) >> 31) + + // If out[3] > 0xffff000 then n's MSB will be zero. + out3GT := ^uint32(int32(n) >> 31) + + mask := top4AllOnes & ((out3Equal & bottom3NonZero) | out3GT) + out[0] -= 1 & mask + out[3] -= 0xffff000 & mask + out[4] -= 0xfffffff & mask + out[5] -= 0xfffffff & mask + out[6] -= 0xfffffff & mask + out[7] -= 0xfffffff & mask +} + +// Group element functions. +// +// These functions deal with group elements. The group is an elliptic curve +// group with a = -3 defined in FIPS 186-3, section D.2.2. + +// p224AddJacobian computes *out = a+b where a != b. +func p224AddJacobian(x3, y3, z3, x1, y1, z1, x2, y2, z2 *p224FieldElement) { + // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-p224Add-2007-bl + var z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v p224FieldElement + var c p224LargeFieldElement + + z1IsZero := p224IsZero(z1) + z2IsZero := p224IsZero(z2) + + // Z1Z1 = Z1² + p224Square(&z1z1, z1, &c) + // Z2Z2 = Z2² + p224Square(&z2z2, z2, &c) + // U1 = X1*Z2Z2 + p224Mul(&u1, x1, &z2z2, &c) + // U2 = X2*Z1Z1 + p224Mul(&u2, x2, &z1z1, &c) + // S1 = Y1*Z2*Z2Z2 + p224Mul(&s1, z2, &z2z2, &c) + p224Mul(&s1, y1, &s1, &c) + // S2 = Y2*Z1*Z1Z1 + p224Mul(&s2, z1, &z1z1, &c) + p224Mul(&s2, y2, &s2, &c) + // H = U2-U1 + p224Sub(&h, &u2, &u1) + p224Reduce(&h) + xEqual := p224IsZero(&h) + // I = (2*H)² + for j := 0; j < 8; j++ { + i[j] = h[j] << 1 + } + p224Reduce(&i) + p224Square(&i, &i, &c) + // J = H*I + p224Mul(&j, &h, &i, &c) + // r = 2*(S2-S1) + p224Sub(&r, &s2, &s1) + p224Reduce(&r) + yEqual := p224IsZero(&r) + if xEqual == 1 && yEqual == 1 && z1IsZero == 0 && z2IsZero == 0 { + p224DoubleJacobian(x3, y3, z3, x1, y1, z1) + return + } + for i := 0; i < 8; i++ { + r[i] <<= 1 + } + p224Reduce(&r) + // V = U1*I + p224Mul(&v, &u1, &i, &c) + // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H + p224Add(&z1z1, &z1z1, &z2z2) + p224Add(&z2z2, z1, z2) + p224Reduce(&z2z2) + p224Square(&z2z2, &z2z2, &c) + p224Sub(z3, &z2z2, &z1z1) + p224Reduce(z3) + p224Mul(z3, z3, &h, &c) + // X3 = r²-J-2*V + for i := 0; i < 8; i++ { + z1z1[i] = v[i] << 1 + } + p224Add(&z1z1, &j, &z1z1) + p224Reduce(&z1z1) + p224Square(x3, &r, &c) + p224Sub(x3, x3, &z1z1) + p224Reduce(x3) + // Y3 = r*(V-X3)-2*S1*J + for i := 0; i < 8; i++ { + s1[i] <<= 1 + } + p224Mul(&s1, &s1, &j, &c) + p224Sub(&z1z1, &v, x3) + p224Reduce(&z1z1) + p224Mul(&z1z1, &z1z1, &r, &c) + p224Sub(y3, &z1z1, &s1) + p224Reduce(y3) + + p224CopyConditional(x3, x2, z1IsZero) + p224CopyConditional(x3, x1, z2IsZero) + p224CopyConditional(y3, y2, z1IsZero) + p224CopyConditional(y3, y1, z2IsZero) + p224CopyConditional(z3, z2, z1IsZero) + p224CopyConditional(z3, z1, z2IsZero) +} + +// p224DoubleJacobian computes *out = a+a. +func p224DoubleJacobian(x3, y3, z3, x1, y1, z1 *p224FieldElement) { + var delta, gamma, beta, alpha, t p224FieldElement + var c p224LargeFieldElement + + p224Square(&delta, z1, &c) + p224Square(&gamma, y1, &c) + p224Mul(&beta, x1, &gamma, &c) + + // alpha = 3*(X1-delta)*(X1+delta) + p224Add(&t, x1, &delta) + for i := 0; i < 8; i++ { + t[i] += t[i] << 1 + } + p224Reduce(&t) + p224Sub(&alpha, x1, &delta) + p224Reduce(&alpha) + p224Mul(&alpha, &alpha, &t, &c) + + // Z3 = (Y1+Z1)²-gamma-delta + p224Add(z3, y1, z1) + p224Reduce(z3) + p224Square(z3, z3, &c) + p224Sub(z3, z3, &gamma) + p224Reduce(z3) + p224Sub(z3, z3, &delta) + p224Reduce(z3) + + // X3 = alpha²-8*beta + for i := 0; i < 8; i++ { + delta[i] = beta[i] << 3 + } + p224Reduce(&delta) + p224Square(x3, &alpha, &c) + p224Sub(x3, x3, &delta) + p224Reduce(x3) + + // Y3 = alpha*(4*beta-X3)-8*gamma² + for i := 0; i < 8; i++ { + beta[i] <<= 2 + } + p224Sub(&beta, &beta, x3) + p224Reduce(&beta) + p224Square(&gamma, &gamma, &c) + for i := 0; i < 8; i++ { + gamma[i] <<= 3 + } + p224Reduce(&gamma) + p224Mul(y3, &alpha, &beta, &c) + p224Sub(y3, y3, &gamma) + p224Reduce(y3) +} + +// p224CopyConditional sets *out = *in iff the least-significant-bit of control +// is true, and it runs in constant time. +func p224CopyConditional(out, in *p224FieldElement, control uint32) { + control <<= 31 + control = uint32(int32(control) >> 31) + + for i := 0; i < 8; i++ { + out[i] ^= (out[i] ^ in[i]) & control + } +} + +func p224ScalarMult(outX, outY, outZ, inX, inY, inZ *p224FieldElement, scalar []byte) { + var xx, yy, zz p224FieldElement + for i := 0; i < 8; i++ { + outX[i] = 0 + outY[i] = 0 + outZ[i] = 0 + } + + for _, byte := range scalar { + for bitNum := uint(0); bitNum < 8; bitNum++ { + p224DoubleJacobian(outX, outY, outZ, outX, outY, outZ) + bit := uint32((byte >> (7 - bitNum)) & 1) + p224AddJacobian(&xx, &yy, &zz, inX, inY, inZ, outX, outY, outZ) + p224CopyConditional(outX, &xx, bit) + p224CopyConditional(outY, &yy, bit) + p224CopyConditional(outZ, &zz, bit) + } + } +} + +// p224ToAffine converts from Jacobian to affine form. +func p224ToAffine(x, y, z *p224FieldElement) (*big.Int, *big.Int) { + var zinv, zinvsq, outx, outy p224FieldElement + var tmp p224LargeFieldElement + + if isPointAtInfinity := p224IsZero(z); isPointAtInfinity == 1 { + return new(big.Int), new(big.Int) + } + + p224Invert(&zinv, z) + p224Square(&zinvsq, &zinv, &tmp) + p224Mul(x, x, &zinvsq, &tmp) + p224Mul(&zinvsq, &zinvsq, &zinv, &tmp) + p224Mul(y, y, &zinvsq, &tmp) + + p224Contract(&outx, x) + p224Contract(&outy, y) + return p224ToBig(&outx), p224ToBig(&outy) +} + +// get28BitsFromEnd returns the least-significant 28 bits from buf>>shift, +// where buf is interpreted as a big-endian number. +func get28BitsFromEnd(buf []byte, shift uint) (uint32, []byte) { + var ret uint32 + + for i := uint(0); i < 4; i++ { + var b byte + if l := len(buf); l > 0 { + b = buf[l-1] + // We don't remove the byte if we're about to return and we're not + // reading all of it. + if i != 3 || shift == 4 { + buf = buf[:l-1] + } + } + ret |= uint32(b) << (8 * i) >> shift + } + ret &= bottom28Bits + return ret, buf +} + +// p224FromBig sets *out = *in. +func p224FromBig(out *p224FieldElement, in *big.Int) { + bytes := in.Bytes() + out[0], bytes = get28BitsFromEnd(bytes, 0) + out[1], bytes = get28BitsFromEnd(bytes, 4) + out[2], bytes = get28BitsFromEnd(bytes, 0) + out[3], bytes = get28BitsFromEnd(bytes, 4) + out[4], bytes = get28BitsFromEnd(bytes, 0) + out[5], bytes = get28BitsFromEnd(bytes, 4) + out[6], bytes = get28BitsFromEnd(bytes, 0) + out[7], bytes = get28BitsFromEnd(bytes, 4) +} + +// p224ToBig returns in as a big.Int. +func p224ToBig(in *p224FieldElement) *big.Int { + var buf [28]byte + buf[27] = byte(in[0]) + buf[26] = byte(in[0] >> 8) + buf[25] = byte(in[0] >> 16) + buf[24] = byte(((in[0] >> 24) & 0x0f) | (in[1]<<4)&0xf0) + + buf[23] = byte(in[1] >> 4) + buf[22] = byte(in[1] >> 12) + buf[21] = byte(in[1] >> 20) + + buf[20] = byte(in[2]) + buf[19] = byte(in[2] >> 8) + buf[18] = byte(in[2] >> 16) + buf[17] = byte(((in[2] >> 24) & 0x0f) | (in[3]<<4)&0xf0) + + buf[16] = byte(in[3] >> 4) + buf[15] = byte(in[3] >> 12) + buf[14] = byte(in[3] >> 20) + + buf[13] = byte(in[4]) + buf[12] = byte(in[4] >> 8) + buf[11] = byte(in[4] >> 16) + buf[10] = byte(((in[4] >> 24) & 0x0f) | (in[5]<<4)&0xf0) + + buf[9] = byte(in[5] >> 4) + buf[8] = byte(in[5] >> 12) + buf[7] = byte(in[5] >> 20) + + buf[6] = byte(in[6]) + buf[5] = byte(in[6] >> 8) + buf[4] = byte(in[6] >> 16) + buf[3] = byte(((in[6] >> 24) & 0x0f) | (in[7]<<4)&0xf0) + + buf[2] = byte(in[7] >> 4) + buf[1] = byte(in[7] >> 12) + buf[0] = byte(in[7] >> 20) + + return new(big.Int).SetBytes(buf[:]) +} diff --git a/src/crypto/elliptic/p224_test.go b/src/crypto/elliptic/p224_test.go new file mode 100644 index 000000000..4b26d1610 --- /dev/null +++ b/src/crypto/elliptic/p224_test.go @@ -0,0 +1,47 @@ +// Copyright 2012 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package elliptic + +import ( + "math/big" + "testing" +) + +var toFromBigTests = []string{ + "0", + "1", + "23", + "b70e0cb46bb4bf7f321390b94a03c1d356c01122343280d6105c1d21", + "706a46d476dcb76798e6046d89474788d164c18032d268fd10704fa6", +} + +func p224AlternativeToBig(in *p224FieldElement) *big.Int { + ret := new(big.Int) + tmp := new(big.Int) + + for i := uint(0); i < 8; i++ { + tmp.SetInt64(int64(in[i])) + tmp.Lsh(tmp, 28*i) + ret.Add(ret, tmp) + } + ret.Mod(ret, p224.P) + return ret +} + +func TestToFromBig(t *testing.T) { + for i, test := range toFromBigTests { + n, _ := new(big.Int).SetString(test, 16) + var x p224FieldElement + p224FromBig(&x, n) + m := p224ToBig(&x) + if n.Cmp(m) != 0 { + t.Errorf("#%d: %x != %x", i, n, m) + } + q := p224AlternativeToBig(&x) + if n.Cmp(q) != 0 { + t.Errorf("#%d: %x != %x (alternative)", i, n, m) + } + } +} diff --git a/src/crypto/elliptic/p256.go b/src/crypto/elliptic/p256.go new file mode 100644 index 000000000..82be51e62 --- /dev/null +++ b/src/crypto/elliptic/p256.go @@ -0,0 +1,1186 @@ +// Copyright 2013 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package elliptic + +// This file contains a constant-time, 32-bit implementation of P256. + +import ( + "math/big" +) + +type p256Curve struct { + *CurveParams +} + +var ( + p256 p256Curve + // RInverse contains 1/R mod p - the inverse of the Montgomery constant + // (2**257). + p256RInverse *big.Int +) + +func initP256() { + // See FIPS 186-3, section D.2.3 + p256.CurveParams = new(CurveParams) + p256.P, _ = new(big.Int).SetString("115792089210356248762697446949407573530086143415290314195533631308867097853951", 10) + p256.N, _ = new(big.Int).SetString("115792089210356248762697446949407573529996955224135760342422259061068512044369", 10) + p256.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", 16) + p256.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", 16) + p256.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5", 16) + p256.BitSize = 256 + + p256RInverse, _ = new(big.Int).SetString("7fffffff00000001fffffffe8000000100000000ffffffff0000000180000000", 16) +} + +func (curve p256Curve) Params() *CurveParams { + return curve.CurveParams +} + +// p256GetScalar endian-swaps the big-endian scalar value from in and writes it +// to out. If the scalar is equal or greater than the order of the group, it's +// reduced modulo that order. +func p256GetScalar(out *[32]byte, in []byte) { + n := new(big.Int).SetBytes(in) + var scalarBytes []byte + + if n.Cmp(p256.N) >= 0 { + n.Mod(n, p256.N) + scalarBytes = n.Bytes() + } else { + scalarBytes = in + } + + for i, v := range scalarBytes { + out[len(scalarBytes)-(1+i)] = v + } +} + +func (p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) { + var scalarReversed [32]byte + p256GetScalar(&scalarReversed, scalar) + + var x1, y1, z1 [p256Limbs]uint32 + p256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed) + return p256ToAffine(&x1, &y1, &z1) +} + +func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) { + var scalarReversed [32]byte + p256GetScalar(&scalarReversed, scalar) + + var px, py, x1, y1, z1 [p256Limbs]uint32 + p256FromBig(&px, bigX) + p256FromBig(&py, bigY) + p256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed) + return p256ToAffine(&x1, &y1, &z1) +} + +// Field elements are represented as nine, unsigned 32-bit words. +// +// The value of an field element is: +// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228) +// +// That is, each limb is alternately 29 or 28-bits wide in little-endian +// order. +// +// This means that a field element hits 2**257, rather than 2**256 as we would +// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes +// problems when multiplying as terms end up one bit short of a limb which +// would require much bit-shifting to correct. +// +// Finally, the values stored in a field element are in Montgomery form. So the +// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is +// 2**257. + +const ( + p256Limbs = 9 + bottom29Bits = 0x1fffffff +) + +var ( + // p256One is the number 1 as a field element. + p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0} + p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0} + // p256P is the prime modulus as a field element. + p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff} + // p2562P is the twice prime modulus as a field element. + p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff} +) + +// p256Precomputed contains precomputed values to aid the calculation of scalar +// multiples of the base point, G. It's actually two, equal length, tables +// concatenated. +// +// The first table contains (x,y) field element pairs for 16 multiples of the +// base point, G. +// +// Index | Index (binary) | Value +// 0 | 0000 | 0G (all zeros, omitted) +// 1 | 0001 | G +// 2 | 0010 | 2**64G +// 3 | 0011 | 2**64G + G +// 4 | 0100 | 2**128G +// 5 | 0101 | 2**128G + G +// 6 | 0110 | 2**128G + 2**64G +// 7 | 0111 | 2**128G + 2**64G + G +// 8 | 1000 | 2**192G +// 9 | 1001 | 2**192G + G +// 10 | 1010 | 2**192G + 2**64G +// 11 | 1011 | 2**192G + 2**64G + G +// 12 | 1100 | 2**192G + 2**128G +// 13 | 1101 | 2**192G + 2**128G + G +// 14 | 1110 | 2**192G + 2**128G + 2**64G +// 15 | 1111 | 2**192G + 2**128G + 2**64G + G +// +// The second table follows the same style, but the terms are 2**32G, +// 2**96G, 2**160G, 2**224G. +// +// This is ~2KB of data. +var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{ + 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee, + 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3, + 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c, + 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22, + 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050, + 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b, + 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa, + 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2, + 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609, + 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581, + 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca, + 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33, + 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6, + 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd, + 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0, + 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881, + 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a, + 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26, + 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b, + 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023, + 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133, + 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa, + 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29, + 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc, + 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8, + 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59, + 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39, + 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689, + 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa, + 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3, + 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1, + 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f, + 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72, + 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d, + 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b, + 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a, + 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a, + 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f, + 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb, + 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc, + 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9, + 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce, + 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2, + 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca, + 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229, + 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57, + 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c, + 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa, + 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651, + 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec, + 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7, + 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c, + 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927, + 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298, + 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8, + 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2, + 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d, + 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4, + 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8, + 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78, +} + +// Field element operations: + +// nonZeroToAllOnes returns: +// 0xffffffff for 0 < x <= 2**31 +// 0 for x == 0 or x > 2**31. +func nonZeroToAllOnes(x uint32) uint32 { + return ((x - 1) >> 31) - 1 +} + +// p256ReduceCarry adds a multiple of p in order to cancel |carry|, +// which is a term at 2**257. +// +// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28. +// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29. +func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) { + carry_mask := nonZeroToAllOnes(carry) + + inout[0] += carry << 1 + inout[3] += 0x10000000 & carry_mask + // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the + // previous line therefore this doesn't underflow. + inout[3] -= carry << 11 + inout[4] += (0x20000000 - 1) & carry_mask + inout[5] += (0x10000000 - 1) & carry_mask + inout[6] += (0x20000000 - 1) & carry_mask + inout[6] -= carry << 22 + // This may underflow if carry is non-zero but, if so, we'll fix it in the + // next line. + inout[7] -= 1 & carry_mask + inout[7] += carry << 25 +} + +// p256Sum sets out = in+in2. +// +// On entry, in[i]+in2[i] must not overflow a 32-bit word. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 +func p256Sum(out, in, in2 *[p256Limbs]uint32) { + carry := uint32(0) + for i := 0; ; i++ { + out[i] = in[i] + in2[i] + out[i] += carry + carry = out[i] >> 29 + out[i] &= bottom29Bits + + i++ + if i == p256Limbs { + break + } + + out[i] = in[i] + in2[i] + out[i] += carry + carry = out[i] >> 28 + out[i] &= bottom28Bits + } + + p256ReduceCarry(out, carry) +} + +const ( + two30m2 = 1<<30 - 1<<2 + two30p13m2 = 1<<30 + 1<<13 - 1<<2 + two31m2 = 1<<31 - 1<<2 + two31p24m2 = 1<<31 + 1<<24 - 1<<2 + two30m27m2 = 1<<30 - 1<<27 - 1<<2 +) + +// p256Zero31 is 0 mod p. +var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2} + +// p256Diff sets out = in-in2. +// +// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and +// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Diff(out, in, in2 *[p256Limbs]uint32) { + var carry uint32 + + for i := 0; ; i++ { + out[i] = in[i] - in2[i] + out[i] += p256Zero31[i] + out[i] += carry + carry = out[i] >> 29 + out[i] &= bottom29Bits + + i++ + if i == p256Limbs { + break + } + + out[i] = in[i] - in2[i] + out[i] += p256Zero31[i] + out[i] += carry + carry = out[i] >> 28 + out[i] &= bottom28Bits + } + + p256ReduceCarry(out, carry) +} + +// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with +// the same 29,28,... bit positions as an field element. +// +// The values in field elements are in Montgomery form: x*R mod p where R = +// 2**257. Since we just multiplied two Montgomery values together, the result +// is x*y*R*R mod p. We wish to divide by R in order for the result also to be +// in Montgomery form. +// +// On entry: tmp[i] < 2**64 +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29 +func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) { + // The following table may be helpful when reading this code: + // + // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10... + // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29 + // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285 + // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285 + var tmp2 [18]uint32 + var carry, x, xMask uint32 + + // tmp contains 64-bit words with the same 29,28,29-bit positions as an + // field element. So the top of an element of tmp might overlap with + // another element two positions down. The following loop eliminates + // this overlap. + tmp2[0] = uint32(tmp[0]) & bottom29Bits + + tmp2[1] = uint32(tmp[0]) >> 29 + tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits + tmp2[1] += uint32(tmp[1]) & bottom28Bits + carry = tmp2[1] >> 28 + tmp2[1] &= bottom28Bits + + for i := 2; i < 17; i++ { + tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25 + tmp2[i] += (uint32(tmp[i-1])) >> 28 + tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits + tmp2[i] += uint32(tmp[i]) & bottom29Bits + tmp2[i] += carry + carry = tmp2[i] >> 29 + tmp2[i] &= bottom29Bits + + i++ + if i == 17 { + break + } + tmp2[i] = uint32(tmp[i-2]>>32) >> 25 + tmp2[i] += uint32(tmp[i-1]) >> 29 + tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits + tmp2[i] += uint32(tmp[i]) & bottom28Bits + tmp2[i] += carry + carry = tmp2[i] >> 28 + tmp2[i] &= bottom28Bits + } + + tmp2[17] = uint32(tmp[15]>>32) >> 25 + tmp2[17] += uint32(tmp[16]) >> 29 + tmp2[17] += uint32(tmp[16]>>32) << 3 + tmp2[17] += carry + + // Montgomery elimination of terms: + // + // Since R is 2**257, we can divide by R with a bitwise shift if we can + // ensure that the right-most 257 bits are all zero. We can make that true + // by adding multiplies of p without affecting the value. + // + // So we eliminate limbs from right to left. Since the bottom 29 bits of p + // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0. + // We can do that for 8 further limbs and then right shift to eliminate the + // extra factor of R. + for i := 0; ; i += 2 { + tmp2[i+1] += tmp2[i] >> 29 + x = tmp2[i] & bottom29Bits + xMask = nonZeroToAllOnes(x) + tmp2[i] = 0 + + // The bounds calculations for this loop are tricky. Each iteration of + // the loop eliminates two words by adding values to words to their + // right. + // + // The following table contains the amounts added to each word (as an + // offset from the value of i at the top of the loop). The amounts are + // accounted for from the first and second half of the loop separately + // and are written as, for example, 28 to mean a value <2**28. + // + // Word: 3 4 5 6 7 8 9 10 + // Added in top half: 28 11 29 21 29 28 + // 28 29 + // 29 + // Added in bottom half: 29 10 28 21 28 28 + // 29 + // + // The value that is currently offset 7 will be offset 5 for the next + // iteration and then offset 3 for the iteration after that. Therefore + // the total value added will be the values added at 7, 5 and 3. + // + // The following table accumulates these values. The sums at the bottom + // are written as, for example, 29+28, to mean a value < 2**29+2**28. + // + // Word: 3 4 5 6 7 8 9 10 11 12 13 + // 28 11 10 29 21 29 28 28 28 28 28 + // 29 28 11 28 29 28 29 28 29 28 + // 29 28 21 21 29 21 29 21 + // 10 29 28 21 28 21 28 + // 28 29 28 29 28 29 28 + // 11 10 29 10 29 10 + // 29 28 11 28 11 + // 29 29 + // -------------------------------------------- + // 30+ 31+ 30+ 31+ 30+ + // 28+ 29+ 28+ 29+ 21+ + // 21+ 28+ 21+ 28+ 10 + // 10 21+ 10 21+ + // 11 11 + // + // So the greatest amount is added to tmp2[10] and tmp2[12]. If + // tmp2[10/12] has an initial value of <2**29, then the maximum value + // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32, + // as required. + tmp2[i+3] += (x << 10) & bottom28Bits + tmp2[i+4] += (x >> 18) + + tmp2[i+6] += (x << 21) & bottom29Bits + tmp2[i+7] += x >> 8 + + // At position 200, which is the starting bit position for word 7, we + // have a factor of 0xf000000 = 2**28 - 2**24. + tmp2[i+7] += 0x10000000 & xMask + tmp2[i+8] += (x - 1) & xMask + tmp2[i+7] -= (x << 24) & bottom28Bits + tmp2[i+8] -= x >> 4 + + tmp2[i+8] += 0x20000000 & xMask + tmp2[i+8] -= x + tmp2[i+8] += (x << 28) & bottom29Bits + tmp2[i+9] += ((x >> 1) - 1) & xMask + + if i+1 == p256Limbs { + break + } + tmp2[i+2] += tmp2[i+1] >> 28 + x = tmp2[i+1] & bottom28Bits + xMask = nonZeroToAllOnes(x) + tmp2[i+1] = 0 + + tmp2[i+4] += (x << 11) & bottom29Bits + tmp2[i+5] += (x >> 18) + + tmp2[i+7] += (x << 21) & bottom28Bits + tmp2[i+8] += x >> 7 + + // At position 199, which is the starting bit of the 8th word when + // dealing with a context starting on an odd word, we have a factor of + // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th + // word from i+1 is i+8. + tmp2[i+8] += 0x20000000 & xMask + tmp2[i+9] += (x - 1) & xMask + tmp2[i+8] -= (x << 25) & bottom29Bits + tmp2[i+9] -= x >> 4 + + tmp2[i+9] += 0x10000000 & xMask + tmp2[i+9] -= x + tmp2[i+10] += (x - 1) & xMask + } + + // We merge the right shift with a carry chain. The words above 2**257 have + // widths of 28,29,... which we need to correct when copying them down. + carry = 0 + for i := 0; i < 8; i++ { + // The maximum value of tmp2[i + 9] occurs on the first iteration and + // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is + // therefore safe. + out[i] = tmp2[i+9] + out[i] += carry + out[i] += (tmp2[i+10] << 28) & bottom29Bits + carry = out[i] >> 29 + out[i] &= bottom29Bits + + i++ + out[i] = tmp2[i+9] >> 1 + out[i] += carry + carry = out[i] >> 28 + out[i] &= bottom28Bits + } + + out[8] = tmp2[17] + out[8] += carry + carry = out[8] >> 29 + out[8] &= bottom29Bits + + p256ReduceCarry(out, carry) +} + +// p256Square sets out=in*in. +// +// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Square(out, in *[p256Limbs]uint32) { + var tmp [17]uint64 + + tmp[0] = uint64(in[0]) * uint64(in[0]) + tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1) + tmp[2] = uint64(in[0])*(uint64(in[2])<<1) + + uint64(in[1])*(uint64(in[1])<<1) + tmp[3] = uint64(in[0])*(uint64(in[3])<<1) + + uint64(in[1])*(uint64(in[2])<<1) + tmp[4] = uint64(in[0])*(uint64(in[4])<<1) + + uint64(in[1])*(uint64(in[3])<<2) + + uint64(in[2])*uint64(in[2]) + tmp[5] = uint64(in[0])*(uint64(in[5])<<1) + + uint64(in[1])*(uint64(in[4])<<1) + + uint64(in[2])*(uint64(in[3])<<1) + tmp[6] = uint64(in[0])*(uint64(in[6])<<1) + + uint64(in[1])*(uint64(in[5])<<2) + + uint64(in[2])*(uint64(in[4])<<1) + + uint64(in[3])*(uint64(in[3])<<1) + tmp[7] = uint64(in[0])*(uint64(in[7])<<1) + + uint64(in[1])*(uint64(in[6])<<1) + + uint64(in[2])*(uint64(in[5])<<1) + + uint64(in[3])*(uint64(in[4])<<1) + // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60, + // which is < 2**64 as required. + tmp[8] = uint64(in[0])*(uint64(in[8])<<1) + + uint64(in[1])*(uint64(in[7])<<2) + + uint64(in[2])*(uint64(in[6])<<1) + + uint64(in[3])*(uint64(in[5])<<2) + + uint64(in[4])*uint64(in[4]) + tmp[9] = uint64(in[1])*(uint64(in[8])<<1) + + uint64(in[2])*(uint64(in[7])<<1) + + uint64(in[3])*(uint64(in[6])<<1) + + uint64(in[4])*(uint64(in[5])<<1) + tmp[10] = uint64(in[2])*(uint64(in[8])<<1) + + uint64(in[3])*(uint64(in[7])<<2) + + uint64(in[4])*(uint64(in[6])<<1) + + uint64(in[5])*(uint64(in[5])<<1) + tmp[11] = uint64(in[3])*(uint64(in[8])<<1) + + uint64(in[4])*(uint64(in[7])<<1) + + uint64(in[5])*(uint64(in[6])<<1) + tmp[12] = uint64(in[4])*(uint64(in[8])<<1) + + uint64(in[5])*(uint64(in[7])<<2) + + uint64(in[6])*uint64(in[6]) + tmp[13] = uint64(in[5])*(uint64(in[8])<<1) + + uint64(in[6])*(uint64(in[7])<<1) + tmp[14] = uint64(in[6])*(uint64(in[8])<<1) + + uint64(in[7])*(uint64(in[7])<<1) + tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1) + tmp[16] = uint64(in[8]) * uint64(in[8]) + + p256ReduceDegree(out, tmp) +} + +// p256Mul sets out=in*in2. +// +// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and +// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Mul(out, in, in2 *[p256Limbs]uint32) { + var tmp [17]uint64 + + tmp[0] = uint64(in[0]) * uint64(in2[0]) + tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) + + uint64(in[1])*(uint64(in2[0])<<0) + tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) + + uint64(in[1])*(uint64(in2[1])<<1) + + uint64(in[2])*(uint64(in2[0])<<0) + tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) + + uint64(in[1])*(uint64(in2[2])<<0) + + uint64(in[2])*(uint64(in2[1])<<0) + + uint64(in[3])*(uint64(in2[0])<<0) + tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) + + uint64(in[1])*(uint64(in2[3])<<1) + + uint64(in[2])*(uint64(in2[2])<<0) + + uint64(in[3])*(uint64(in2[1])<<1) + + uint64(in[4])*(uint64(in2[0])<<0) + tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) + + uint64(in[1])*(uint64(in2[4])<<0) + + uint64(in[2])*(uint64(in2[3])<<0) + + uint64(in[3])*(uint64(in2[2])<<0) + + uint64(in[4])*(uint64(in2[1])<<0) + + uint64(in[5])*(uint64(in2[0])<<0) + tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) + + uint64(in[1])*(uint64(in2[5])<<1) + + uint64(in[2])*(uint64(in2[4])<<0) + + uint64(in[3])*(uint64(in2[3])<<1) + + uint64(in[4])*(uint64(in2[2])<<0) + + uint64(in[5])*(uint64(in2[1])<<1) + + uint64(in[6])*(uint64(in2[0])<<0) + tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) + + uint64(in[1])*(uint64(in2[6])<<0) + + uint64(in[2])*(uint64(in2[5])<<0) + + uint64(in[3])*(uint64(in2[4])<<0) + + uint64(in[4])*(uint64(in2[3])<<0) + + uint64(in[5])*(uint64(in2[2])<<0) + + uint64(in[6])*(uint64(in2[1])<<0) + + uint64(in[7])*(uint64(in2[0])<<0) + // tmp[8] has the greatest value but doesn't overflow. See logic in + // p256Square. + tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) + + uint64(in[1])*(uint64(in2[7])<<1) + + uint64(in[2])*(uint64(in2[6])<<0) + + uint64(in[3])*(uint64(in2[5])<<1) + + uint64(in[4])*(uint64(in2[4])<<0) + + uint64(in[5])*(uint64(in2[3])<<1) + + uint64(in[6])*(uint64(in2[2])<<0) + + uint64(in[7])*(uint64(in2[1])<<1) + + uint64(in[8])*(uint64(in2[0])<<0) + tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) + + uint64(in[2])*(uint64(in2[7])<<0) + + uint64(in[3])*(uint64(in2[6])<<0) + + uint64(in[4])*(uint64(in2[5])<<0) + + uint64(in[5])*(uint64(in2[4])<<0) + + uint64(in[6])*(uint64(in2[3])<<0) + + uint64(in[7])*(uint64(in2[2])<<0) + + uint64(in[8])*(uint64(in2[1])<<0) + tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) + + uint64(in[3])*(uint64(in2[7])<<1) + + uint64(in[4])*(uint64(in2[6])<<0) + + uint64(in[5])*(uint64(in2[5])<<1) + + uint64(in[6])*(uint64(in2[4])<<0) + + uint64(in[7])*(uint64(in2[3])<<1) + + uint64(in[8])*(uint64(in2[2])<<0) + tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) + + uint64(in[4])*(uint64(in2[7])<<0) + + uint64(in[5])*(uint64(in2[6])<<0) + + uint64(in[6])*(uint64(in2[5])<<0) + + uint64(in[7])*(uint64(in2[4])<<0) + + uint64(in[8])*(uint64(in2[3])<<0) + tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) + + uint64(in[5])*(uint64(in2[7])<<1) + + uint64(in[6])*(uint64(in2[6])<<0) + + uint64(in[7])*(uint64(in2[5])<<1) + + uint64(in[8])*(uint64(in2[4])<<0) + tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) + + uint64(in[6])*(uint64(in2[7])<<0) + + uint64(in[7])*(uint64(in2[6])<<0) + + uint64(in[8])*(uint64(in2[5])<<0) + tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) + + uint64(in[7])*(uint64(in2[7])<<1) + + uint64(in[8])*(uint64(in2[6])<<0) + tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) + + uint64(in[8])*(uint64(in2[7])<<0) + tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0) + + p256ReduceDegree(out, tmp) +} + +func p256Assign(out, in *[p256Limbs]uint32) { + *out = *in +} + +// p256Invert calculates |out| = |in|^{-1} +// +// Based on Fermat's Little Theorem: +// a^p = a (mod p) +// a^{p-1} = 1 (mod p) +// a^{p-2} = a^{-1} (mod p) +func p256Invert(out, in *[p256Limbs]uint32) { + var ftmp, ftmp2 [p256Limbs]uint32 + + // each e_I will hold |in|^{2^I - 1} + var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32 + + p256Square(&ftmp, in) // 2^1 + p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0 + p256Assign(&e2, &ftmp) + p256Square(&ftmp, &ftmp) // 2^3 - 2^1 + p256Square(&ftmp, &ftmp) // 2^4 - 2^2 + p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0 + p256Assign(&e4, &ftmp) + p256Square(&ftmp, &ftmp) // 2^5 - 2^1 + p256Square(&ftmp, &ftmp) // 2^6 - 2^2 + p256Square(&ftmp, &ftmp) // 2^7 - 2^3 + p256Square(&ftmp, &ftmp) // 2^8 - 2^4 + p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0 + p256Assign(&e8, &ftmp) + for i := 0; i < 8; i++ { + p256Square(&ftmp, &ftmp) + } // 2^16 - 2^8 + p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0 + p256Assign(&e16, &ftmp) + for i := 0; i < 16; i++ { + p256Square(&ftmp, &ftmp) + } // 2^32 - 2^16 + p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0 + p256Assign(&e32, &ftmp) + for i := 0; i < 32; i++ { + p256Square(&ftmp, &ftmp) + } // 2^64 - 2^32 + p256Assign(&e64, &ftmp) + p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0 + for i := 0; i < 192; i++ { + p256Square(&ftmp, &ftmp) + } // 2^256 - 2^224 + 2^192 + + p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0 + for i := 0; i < 16; i++ { + p256Square(&ftmp2, &ftmp2) + } // 2^80 - 2^16 + p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0 + for i := 0; i < 8; i++ { + p256Square(&ftmp2, &ftmp2) + } // 2^88 - 2^8 + p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0 + for i := 0; i < 4; i++ { + p256Square(&ftmp2, &ftmp2) + } // 2^92 - 2^4 + p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0 + p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1 + p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2 + p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0 + p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1 + p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2 + p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3 + + p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3 +} + +// p256Scalar3 sets out=3*out. +// +// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Scalar3(out *[p256Limbs]uint32) { + var carry uint32 + + for i := 0; ; i++ { + out[i] *= 3 + out[i] += carry + carry = out[i] >> 29 + out[i] &= bottom29Bits + + i++ + if i == p256Limbs { + break + } + + out[i] *= 3 + out[i] += carry + carry = out[i] >> 28 + out[i] &= bottom28Bits + } + + p256ReduceCarry(out, carry) +} + +// p256Scalar4 sets out=4*out. +// +// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Scalar4(out *[p256Limbs]uint32) { + var carry, nextCarry uint32 + + for i := 0; ; i++ { + nextCarry = out[i] >> 27 + out[i] <<= 2 + out[i] &= bottom29Bits + out[i] += carry + carry = nextCarry + (out[i] >> 29) + out[i] &= bottom29Bits + + i++ + if i == p256Limbs { + break + } + nextCarry = out[i] >> 26 + out[i] <<= 2 + out[i] &= bottom28Bits + out[i] += carry + carry = nextCarry + (out[i] >> 28) + out[i] &= bottom28Bits + } + + p256ReduceCarry(out, carry) +} + +// p256Scalar8 sets out=8*out. +// +// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29. +func p256Scalar8(out *[p256Limbs]uint32) { + var carry, nextCarry uint32 + + for i := 0; ; i++ { + nextCarry = out[i] >> 26 + out[i] <<= 3 + out[i] &= bottom29Bits + out[i] += carry + carry = nextCarry + (out[i] >> 29) + out[i] &= bottom29Bits + + i++ + if i == p256Limbs { + break + } + nextCarry = out[i] >> 25 + out[i] <<= 3 + out[i] &= bottom28Bits + out[i] += carry + carry = nextCarry + (out[i] >> 28) + out[i] &= bottom28Bits + } + + p256ReduceCarry(out, carry) +} + +// Group operations: +// +// Elements of the elliptic curve group are represented in Jacobian +// coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in +// Jacobian form. + +// p256PointDouble sets {xOut,yOut,zOut} = 2*{x,y,z}. +// +// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l +func p256PointDouble(xOut, yOut, zOut, x, y, z *[p256Limbs]uint32) { + var delta, gamma, alpha, beta, tmp, tmp2 [p256Limbs]uint32 + + p256Square(&delta, z) + p256Square(&gamma, y) + p256Mul(&beta, x, &gamma) + + p256Sum(&tmp, x, &delta) + p256Diff(&tmp2, x, &delta) + p256Mul(&alpha, &tmp, &tmp2) + p256Scalar3(&alpha) + + p256Sum(&tmp, y, z) + p256Square(&tmp, &tmp) + p256Diff(&tmp, &tmp, &gamma) + p256Diff(zOut, &tmp, &delta) + + p256Scalar4(&beta) + p256Square(xOut, &alpha) + p256Diff(xOut, xOut, &beta) + p256Diff(xOut, xOut, &beta) + + p256Diff(&tmp, &beta, xOut) + p256Mul(&tmp, &alpha, &tmp) + p256Square(&tmp2, &gamma) + p256Scalar8(&tmp2) + p256Diff(yOut, &tmp, &tmp2) +} + +// p256PointAddMixed sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,1}. +// (i.e. the second point is affine.) +// +// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl +// +// Note that this function does not handle P+P, infinity+P nor P+infinity +// correctly. +func p256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[p256Limbs]uint32) { + var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32 + + p256Square(&z1z1, z1) + p256Sum(&tmp, z1, z1) + + p256Mul(&u2, x2, &z1z1) + p256Mul(&z1z1z1, z1, &z1z1) + p256Mul(&s2, y2, &z1z1z1) + p256Diff(&h, &u2, x1) + p256Sum(&i, &h, &h) + p256Square(&i, &i) + p256Mul(&j, &h, &i) + p256Diff(&r, &s2, y1) + p256Sum(&r, &r, &r) + p256Mul(&v, x1, &i) + + p256Mul(zOut, &tmp, &h) + p256Square(&rr, &r) + p256Diff(xOut, &rr, &j) + p256Diff(xOut, xOut, &v) + p256Diff(xOut, xOut, &v) + + p256Diff(&tmp, &v, xOut) + p256Mul(yOut, &tmp, &r) + p256Mul(&tmp, y1, &j) + p256Diff(yOut, yOut, &tmp) + p256Diff(yOut, yOut, &tmp) +} + +// p256PointAdd sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,z2}. +// +// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl +// +// Note that this function does not handle P+P, infinity+P nor P+infinity +// correctly. +func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) { + var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32 + + p256Square(&z1z1, z1) + p256Square(&z2z2, z2) + p256Mul(&u1, x1, &z2z2) + + p256Sum(&tmp, z1, z2) + p256Square(&tmp, &tmp) + p256Diff(&tmp, &tmp, &z1z1) + p256Diff(&tmp, &tmp, &z2z2) + + p256Mul(&z2z2z2, z2, &z2z2) + p256Mul(&s1, y1, &z2z2z2) + + p256Mul(&u2, x2, &z1z1) + p256Mul(&z1z1z1, z1, &z1z1) + p256Mul(&s2, y2, &z1z1z1) + p256Diff(&h, &u2, &u1) + p256Sum(&i, &h, &h) + p256Square(&i, &i) + p256Mul(&j, &h, &i) + p256Diff(&r, &s2, &s1) + p256Sum(&r, &r, &r) + p256Mul(&v, &u1, &i) + + p256Mul(zOut, &tmp, &h) + p256Square(&rr, &r) + p256Diff(xOut, &rr, &j) + p256Diff(xOut, xOut, &v) + p256Diff(xOut, xOut, &v) + + p256Diff(&tmp, &v, xOut) + p256Mul(yOut, &tmp, &r) + p256Mul(&tmp, &s1, &j) + p256Diff(yOut, yOut, &tmp) + p256Diff(yOut, yOut, &tmp) +} + +// p256CopyConditional sets out=in if mask = 0xffffffff in constant time. +// +// On entry: mask is either 0 or 0xffffffff. +func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) { + for i := 0; i < p256Limbs; i++ { + tmp := mask & (in[i] ^ out[i]) + out[i] ^= tmp + } +} + +// p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table. +// On entry: index < 16, table[0] must be zero. +func p256SelectAffinePoint(xOut, yOut *[p256Limbs]uint32, table []uint32, index uint32) { + for i := range xOut { + xOut[i] = 0 + } + for i := range yOut { + yOut[i] = 0 + } + + for i := uint32(1); i < 16; i++ { + mask := i ^ index + mask |= mask >> 2 + mask |= mask >> 1 + mask &= 1 + mask-- + for j := range xOut { + xOut[j] |= table[0] & mask + table = table[1:] + } + for j := range yOut { + yOut[j] |= table[0] & mask + table = table[1:] + } + } +} + +// p256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of +// table. +// On entry: index < 16, table[0] must be zero. +func p256SelectJacobianPoint(xOut, yOut, zOut *[p256Limbs]uint32, table *[16][3][p256Limbs]uint32, index uint32) { + for i := range xOut { + xOut[i] = 0 + } + for i := range yOut { + yOut[i] = 0 + } + for i := range zOut { + zOut[i] = 0 + } + + // The implicit value at index 0 is all zero. We don't need to perform that + // iteration of the loop because we already set out_* to zero. + for i := uint32(1); i < 16; i++ { + mask := i ^ index + mask |= mask >> 2 + mask |= mask >> 1 + mask &= 1 + mask-- + for j := range xOut { + xOut[j] |= table[i][0][j] & mask + } + for j := range yOut { + yOut[j] |= table[i][1][j] & mask + } + for j := range zOut { + zOut[j] |= table[i][2][j] & mask + } + } +} + +// p256GetBit returns the bit'th bit of scalar. +func p256GetBit(scalar *[32]uint8, bit uint) uint32 { + return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1) +} + +// p256ScalarBaseMult sets {xOut,yOut,zOut} = scalar*G where scalar is a +// little-endian number. Note that the value of scalar must be less than the +// order of the group. +func p256ScalarBaseMult(xOut, yOut, zOut *[p256Limbs]uint32, scalar *[32]uint8) { + nIsInfinityMask := ^uint32(0) + var pIsNoninfiniteMask, mask, tableOffset uint32 + var px, py, tx, ty, tz [p256Limbs]uint32 + + for i := range xOut { + xOut[i] = 0 + } + for i := range yOut { + yOut[i] = 0 + } + for i := range zOut { + zOut[i] = 0 + } + + // The loop adds bits at positions 0, 64, 128 and 192, followed by + // positions 32,96,160 and 224 and does this 32 times. + for i := uint(0); i < 32; i++ { + if i != 0 { + p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) + } + tableOffset = 0 + for j := uint(0); j <= 32; j += 32 { + bit0 := p256GetBit(scalar, 31-i+j) + bit1 := p256GetBit(scalar, 95-i+j) + bit2 := p256GetBit(scalar, 159-i+j) + bit3 := p256GetBit(scalar, 223-i+j) + index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3) + + p256SelectAffinePoint(&px, &py, p256Precomputed[tableOffset:], index) + tableOffset += 30 * p256Limbs + + // Since scalar is less than the order of the group, we know that + // {xOut,yOut,zOut} != {px,py,1}, unless both are zero, which we handle + // below. + p256PointAddMixed(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py) + // The result of pointAddMixed is incorrect if {xOut,yOut,zOut} is zero + // (a.k.a. the point at infinity). We handle that situation by + // copying the point from the table. + p256CopyConditional(xOut, &px, nIsInfinityMask) + p256CopyConditional(yOut, &py, nIsInfinityMask) + p256CopyConditional(zOut, &p256One, nIsInfinityMask) + + // Equally, the result is also wrong if the point from the table is + // zero, which happens when the index is zero. We handle that by + // only copying from {tx,ty,tz} to {xOut,yOut,zOut} if index != 0. + pIsNoninfiniteMask = nonZeroToAllOnes(index) + mask = pIsNoninfiniteMask & ^nIsInfinityMask + p256CopyConditional(xOut, &tx, mask) + p256CopyConditional(yOut, &ty, mask) + p256CopyConditional(zOut, &tz, mask) + // If p was not zero, then n is now non-zero. + nIsInfinityMask &= ^pIsNoninfiniteMask + } + } +} + +// p256PointToAffine converts a Jacobian point to an affine point. If the input +// is the point at infinity then it returns (0, 0) in constant time. +func p256PointToAffine(xOut, yOut, x, y, z *[p256Limbs]uint32) { + var zInv, zInvSq [p256Limbs]uint32 + + p256Invert(&zInv, z) + p256Square(&zInvSq, &zInv) + p256Mul(xOut, x, &zInvSq) + p256Mul(&zInv, &zInv, &zInvSq) + p256Mul(yOut, y, &zInv) +} + +// p256ToAffine returns a pair of *big.Int containing the affine representation +// of {x,y,z}. +func p256ToAffine(x, y, z *[p256Limbs]uint32) (xOut, yOut *big.Int) { + var xx, yy [p256Limbs]uint32 + p256PointToAffine(&xx, &yy, x, y, z) + return p256ToBig(&xx), p256ToBig(&yy) +} + +// p256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}. +func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8) { + var px, py, pz, tx, ty, tz [p256Limbs]uint32 + var precomp [16][3][p256Limbs]uint32 + var nIsInfinityMask, index, pIsNoninfiniteMask, mask uint32 + + // We precompute 0,1,2,... times {x,y}. + precomp[1][0] = *x + precomp[1][1] = *y + precomp[1][2] = p256One + + for i := 2; i < 16; i += 2 { + p256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2]) + p256PointAddMixed(&precomp[i+1][0], &precomp[i+1][1], &precomp[i+1][2], &precomp[i][0], &precomp[i][1], &precomp[i][2], x, y) + } + + for i := range xOut { + xOut[i] = 0 + } + for i := range yOut { + yOut[i] = 0 + } + for i := range zOut { + zOut[i] = 0 + } + nIsInfinityMask = ^uint32(0) + + // We add in a window of four bits each iteration and do this 64 times. + for i := 0; i < 64; i++ { + if i != 0 { + p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) + p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) + p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) + p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut) + } + + index = uint32(scalar[31-i/2]) + if (i & 1) == 1 { + index &= 15 + } else { + index >>= 4 + } + + // See the comments in scalarBaseMult about handling infinities. + p256SelectJacobianPoint(&px, &py, &pz, &precomp, index) + p256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz) + p256CopyConditional(xOut, &px, nIsInfinityMask) + p256CopyConditional(yOut, &py, nIsInfinityMask) + p256CopyConditional(zOut, &pz, nIsInfinityMask) + + pIsNoninfiniteMask = nonZeroToAllOnes(index) + mask = pIsNoninfiniteMask & ^nIsInfinityMask + p256CopyConditional(xOut, &tx, mask) + p256CopyConditional(yOut, &ty, mask) + p256CopyConditional(zOut, &tz, mask) + nIsInfinityMask &= ^pIsNoninfiniteMask + } +} + +// p256FromBig sets out = R*in. +func p256FromBig(out *[p256Limbs]uint32, in *big.Int) { + tmp := new(big.Int).Lsh(in, 257) + tmp.Mod(tmp, p256.P) + + for i := 0; i < p256Limbs; i++ { + if bits := tmp.Bits(); len(bits) > 0 { + out[i] = uint32(bits[0]) & bottom29Bits + } else { + out[i] = 0 + } + tmp.Rsh(tmp, 29) + + i++ + if i == p256Limbs { + break + } + + if bits := tmp.Bits(); len(bits) > 0 { + out[i] = uint32(bits[0]) & bottom28Bits + } else { + out[i] = 0 + } + tmp.Rsh(tmp, 28) + } +} + +// p256ToBig returns a *big.Int containing the value of in. +func p256ToBig(in *[p256Limbs]uint32) *big.Int { + result, tmp := new(big.Int), new(big.Int) + + result.SetInt64(int64(in[p256Limbs-1])) + for i := p256Limbs - 2; i >= 0; i-- { + if (i & 1) == 0 { + result.Lsh(result, 29) + } else { + result.Lsh(result, 28) + } + tmp.SetInt64(int64(in[i])) + result.Add(result, tmp) + } + + result.Mul(result, p256RInverse) + result.Mod(result, p256.P) + return result +} |