diff options
| author | Tianon Gravi <admwiggin@gmail.com> | 2015-01-15 11:54:00 -0700 |
|---|---|---|
| committer | Tianon Gravi <admwiggin@gmail.com> | 2015-01-15 11:54:00 -0700 |
| commit | f154da9e12608589e8d5f0508f908a0c3e88a1bb (patch) | |
| tree | f8255d51e10c6f1e0ed69702200b966c9556a431 /src/pkg/crypto/elliptic | |
| parent | 8d8329ed5dfb9622c82a9fbec6fd99a580f9c9f6 (diff) | |
| download | golang-upstream/1.4.tar.gz | |
Imported Upstream version 1.4upstream/1.4
Diffstat (limited to 'src/pkg/crypto/elliptic')
| -rw-r--r-- | src/pkg/crypto/elliptic/elliptic.go | 373 | ||||
| -rw-r--r-- | src/pkg/crypto/elliptic/elliptic_test.go | 458 | ||||
| -rw-r--r-- | src/pkg/crypto/elliptic/p224.go | 765 | ||||
| -rw-r--r-- | src/pkg/crypto/elliptic/p224_test.go | 47 | ||||
| -rw-r--r-- | src/pkg/crypto/elliptic/p256.go | 1186 |
5 files changed, 0 insertions, 2829 deletions
diff --git a/src/pkg/crypto/elliptic/elliptic.go b/src/pkg/crypto/elliptic/elliptic.go deleted file mode 100644 index ba673f80c..000000000 --- a/src/pkg/crypto/elliptic/elliptic.go +++ /dev/null @@ -1,373 +0,0 @@ -// 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/pkg/crypto/elliptic/elliptic_test.go b/src/pkg/crypto/elliptic/elliptic_test.go deleted file mode 100644 index 4dc27c92b..000000000 --- a/src/pkg/crypto/elliptic/elliptic_test.go +++ /dev/null @@ -1,458 +0,0 @@ -// 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", - }, - { - "13479767645505654746623887797783387853576174193480695826442858012671", - "6670c20afcceaea672c97f75e2e9dd5c8460e54bb38538ebb4bd30eb", - "f280d8008d07a4caf54271f993527d46ff3ff46fd1190a3f1faa4f74", - }, - { - "205688069665150753842126177372015544874550518966168735589597183", - "eca934247425cfd949b795cb5ce1eff401550386e28d1a4c5a8eb", - "d4c01040dba19628931bc8855370317c722cbd9ca6156985f1c2e9ce", - }, - { - "13479966930919337728895168462090683249159702977113823384618282123295", - "ef353bf5c73cd551b96d596fbc9a67f16d61dd9fe56af19de1fba9cd", - "21771b9cdce3e8430c09b3838be70b48c21e15bc09ee1f2d7945b91f", - }, - { - "50210731791415612487756441341851895584393717453129007497216", - "4036052a3091eb481046ad3289c95d3ac905ca0023de2c03ecd451cf", - "d768165a38a2b96f812586a9d59d4136035d9c853a5bf2e1c86a4993", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368041", - "fcc7f2b45df1cd5a3c0c0731ca47a8af75cfb0347e8354eefe782455", - "f2a28eefd8b345832116f1e574f2c6b2c895aa8c24941f40d8b80ad1", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368042", - "a1e81c04f30ce201c7c9ace785ed44cc33b455a022f2acdbc6cae83c", - "230e093c24f638f533dac6e2b6d01da3b5e7f45429315ca93fb8e634", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368043", - "c9ff61b040874c0568479216824a15eab1a838a797d189746226e4cc", - "156729f1a003647030666054e208180f8f7b0df2249e44fba5931fff", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368044", - "b8357c3a6ceef288310e17b8bfeff9200846ca8c1942497c484403bc", - "eb610599f95942df1082e4f9426d086fb9c6231ae8b24933aab5db", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368045", - "b6ec4fe1777382404ef679997ba8d1cc5cd8e85349259f590c4c66d", - "cc662b9bcba6f94ee4ff1c9c10bd6ddd0d138df2d099a282152a4b7f", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368046", - "baa4d8635511a7d288aebeedd12ce529ff102c91f97f867e21916bf9", - "6865a0b8a607f0b04b13d1cb0aa992a5a97f5ee8ca1849efb9ed8678", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368047", - "a53640c83dc208603ded83e4ecf758f24c357d7cf48088b2ce01e9fa", - "2a7eb328dbe663b5a468b5bc97a040a3745396ba636b964370dc3352", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368048", - "34e8e17a430e43289793c383fac9774247b40e9ebd3366981fcfaeca", - "dad7e608e380480434ea641cc82c82cbc92801469c8db0204f13489a", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368049", - "6e31ee1dc137f81b056752e4deab1443a481033e9b4c93a3044f4f7a", - "df82220fc7a4021549165325725f94c3410ddb56c54e161fc9ef62ee", - }, - { - "26959946667150639794667015087019625940457807714424391721682722368050", - "ef53b6294aca431f0f3c22dc82eb9050324f1d88d377e716448e507c", - "df4aefffbf6d1699c930481cd102127c9a3d992048ab05929b6e5927", - }, - { - "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/pkg/crypto/elliptic/p224.go b/src/pkg/crypto/elliptic/p224.go deleted file mode 100644 index 1f7ff3f9d..000000000 --- a/src/pkg/crypto/elliptic/p224.go +++ /dev/null @@ -1,765 +0,0 @@ -// 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/pkg/crypto/elliptic/p224_test.go b/src/pkg/crypto/elliptic/p224_test.go deleted file mode 100644 index 4b26d1610..000000000 --- a/src/pkg/crypto/elliptic/p224_test.go +++ /dev/null @@ -1,47 +0,0 @@ -// 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/pkg/crypto/elliptic/p256.go b/src/pkg/crypto/elliptic/p256.go deleted file mode 100644 index 82be51e62..000000000 --- a/src/pkg/crypto/elliptic/p256.go +++ /dev/null @@ -1,1186 +0,0 @@ -// 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 -} |
