yaole
1278 lines
42 KiB
Go
1278 lines
42 KiB
Go
//go:build (!amd64 && !arm64 && !generic32) || generic64 || generic
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// +build !amd64,!arm64,!generic32 generic64 generic
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// SM2CurveParam implement Curve, with accelerated by Montgomery
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// Pure Go code, with [5]uint64 as a GFp elements
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// TODO still 32bit,
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package ec256
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import (
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"math/big"
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)
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var (
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// RInverse contains 1/R mod p - the inverse of the Montgomery constant
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// (2**257).
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c256RInverse = bigFromBase16("7ffffffd80000002fffffffe000000017ffffffe800000037ffffffc80000002")
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)
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func init() {
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// fmt.Println("generic64")
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}
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// c256GetScalar endian-swaps the big-endian scalar value from in and writes it
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// to out. If the scalar is equal or greater than the order of the group, it's
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// reduced modulo that order.
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func c256GetScalar(out *[32]byte, in []byte) {
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n := new(big.Int).SetBytes(in)
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var scalarBytes []byte
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if n.Cmp(c256.N) >= 0 || len(in) > len(out) {
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n.Mod(n, c256.N)
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scalarBytes = n.Bytes()
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} else {
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scalarBytes = in
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}
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for i, v := range scalarBytes {
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out[len(scalarBytes)-(1+i)] = v
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}
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}
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// func (SM2CurveParam) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
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// z1 := zForAffine(x1, y1)
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// z2 := zForAffine(x2, y2)
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// return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
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// }
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// ScalarBaseMult overload the Curveparams's ScalarBaseMult method
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func (SM2CurveParam) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
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var scalarReversed [32]byte
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c256GetScalar(&scalarReversed, scalar)
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var x1, y1, z1 [c256Limbs]uint32
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c256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed)
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return c256ToAffine(&x1, &y1, &z1)
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}
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// ScalarMult overload the Curveparams's ScalarMult method
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func (SM2CurveParam) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) {
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var scalarReversed [32]byte
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c256GetScalar(&scalarReversed, scalar)
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var px, py, x1, y1, z1 [c256Limbs]uint32
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c256FromBig(&px, bigX)
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c256FromBig(&py, bigY)
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c256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed)
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x, y = c256ToAffine(&x1, &y1, &z1)
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return
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}
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// func (curve SM2CurveParam) CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int) {
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func CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int) {
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var scalarReversed [32]byte
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var px, py, x1, y1, z1, x2, y2, z2 [c256Limbs]uint32
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c256FromBig(&px, bigX)
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c256FromBig(&py, bigY)
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c256GetScalar(&scalarReversed, scalar)
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c256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed)
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// must set 0, for baseScalar may less than 32 bytes
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for i := range scalarReversed {
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scalarReversed[i] = 0
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}
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c256GetScalar(&scalarReversed, baseScalar)
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c256ScalarBaseMult(&x2, &y2, &z2, &scalarReversed)
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c256PointAdd(&x1, &y1, &z1, &x1, &y1, &z1, &x2, &y2, &z2)
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x, y = c256ToAffine(&x1, &y1, &z1)
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return
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}
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// Field elements are represented as nine, unsigned 32-bit words.
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//
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// The value of an field element is:
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// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
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//
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// That is, each limb is alternately 29 or 28-bits wide in little-endian
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// order.
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//
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// This means that a field element hits 2**257, rather than 2**256 as we would
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// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
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// problems when multiplying as terms end up one bit short of a limb which
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// would require much bit-shifting to correct.
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//
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// Finally, the values stored in a field element are in Montgomery form. So the
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// value |y| is stored as (y*R) mod p, where p is the C-256 prime and R is
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// 2**257.
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const (
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c256Limbs = 9
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bottom29Bits = 0x1fffffff
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)
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const bottom12Bits = 0xfff
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const bottom28Bits = 0xfffffff
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var (
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// c256One is the number 1 as a field element. -- note is Montgomery form of 1,
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// i.e. = 1*R mod p
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c256One = [c256Limbs]uint32{2, 0, 0x1fffff00, 0x7ff, 0, 0, 0, 0x2000000, 0}
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c256Zero = [c256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
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// c256P is the prime modulus as a field element.
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c256P = [c256Limbs]uint32{0x1fffffff, 0xfffffff, 0x7f, 0xffffc00, 0x1fffffff, 0xfffffff, 0x1fffffff, 0xeffffff, 0xfffffff}
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// c2562P is the twice prime modulus as a field element.
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c2562P = [c256Limbs]uint32{0x1ffffffe, 0xfffffff, 0xff, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fffffff, 0xdffffff, 0x1fffffff}
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)
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// c256Precomputed contains precomputed values to aid the calculation of scalar
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// multiples of the base point, G. It's actually two, equal length, tables
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// concatenated.
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//
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// The first table contains (x,y) field element pairs for 16 multiples of the
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// base point, G.
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//
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// Index | Index (binary) | Value
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// 0 | 0000 | 0G (all zeros, omitted)
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// 1 | 0001 | G
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// 2 | 0010 | 2**64G
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// 3 | 0011 | 2**64G + G
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// 4 | 0100 | 2**128G
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// 5 | 0101 | 2**128G + G
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// 6 | 0110 | 2**128G + 2**64G
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// 7 | 0111 | 2**128G + 2**64G + G
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// 8 | 1000 | 2**192G
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// 9 | 1001 | 2**192G + G
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// 10 | 1010 | 2**192G + 2**64G
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// 11 | 1011 | 2**192G + 2**64G + G
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// 12 | 1100 | 2**192G + 2**128G
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// 13 | 1101 | 2**192G + 2**128G + G
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// 14 | 1110 | 2**192G + 2**128G + 2**64G
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// 15 | 1111 | 2**192G + 2**128G + 2**64G + G
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//
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// The second table follows the same style, but the terms are 2**32G,
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// 2**96G, 2**160G, 2**224G.
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//
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// This is ~2KB of data.
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var c256Precomputed = [c256Limbs * 2 * 15 * 2]uint32{
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0x830053d, 0x328990f, 0x6c04fe1, 0xc0f72e5, 0x1e19f3c, 0x666b093, 0x175a87b, 0xec38276, 0x222cf4b,
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0x185a1bba, 0x354e593, 0x1295fac1, 0xf2bc469, 0x47c60fa, 0xc19b8a9, 0xf63533e, 0x903ae6b, 0xc79acba,
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0x15b061a4, 0x33e020b, 0xdffb34b, 0xfcf2c8, 0x16582e08, 0x262f203, 0xfb34381, 0xa55452, 0x604f0ff,
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0x41f1f90, 0xd64ced2, 0xee377bf, 0x75f05f0, 0x189467ae, 0xe2244e, 0x1e7700e8, 0x3fbc464, 0x9612d2e,
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0x1341b3b8, 0xee84e23, 0x1edfa5b4, 0x14e6030, 0x19e87be9, 0x92f533c, 0x1665d96c, 0x226653e, 0xa238d3e,
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0xf5c62c, 0x95bb7a, 0x1f0e5a41, 0x28789c3, 0x1f251d23, 0x8726609, 0xe918910, 0x8096848, 0xf63d028,
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0x152296a1, 0x9f561a8, 0x14d376fb, 0x898788a, 0x61a95fb, 0xa59466d, 0x159a003d, 0x1ad1698, 0x93cca08,
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0x1b314662, 0x706e006, 0x11ce1e30, 0x97b710, 0x172fbc0d, 0x8f50158, 0x11c7ffe7, 0xd182cce, 0xc6ad9e8,
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0x12ea31b2, 0xc4e4f38, 0x175b0d96, 0xec06337, 0x75a9c12, 0xb001fdf, 0x93e82f5, 0x34607de, 0xb8035ed,
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0x17f97924, 0x75cf9e6, 0xdceaedd, 0x2529924, 0x1a10c5ff, 0xb1a54dc, 0x19464d8, 0x2d1997, 0xde6a110,
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0x1e276ee5, 0x95c510c, 0x1aca7c7a, 0xfe48aca, 0x121ad4d9, 0xe4132c6, 0x8239b9d, 0x40ea9cd, 0x816c7b,
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0x632d7a4, 0xa679813, 0x5911fcf, 0x82b0f7c, 0x57b0ad5, 0xbef65, 0xd541365, 0x7f9921f, 0xc62e7a,
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0x3f4b32d, 0x58e50e1, 0x6427aed, 0xdcdda67, 0xe8c2d3e, 0x6aa54a4, 0x18df4c35, 0x49a6a8e, 0x3cd3d0c,
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0xd7adf2, 0xcbca97, 0x1bda5f2d, 0x3258579, 0x606b1e6, 0x6fc1b5b, 0x1ac27317, 0x503ca16, 0xa677435,
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0x57bc73, 0x3992a42, 0xbab987b, 0xfab25eb, 0x128912a4, 0x90a1dc4, 0x1402d591, 0x9ffbcfc, 0xaa48856,
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0x7a7c2dc, 0xcefd08a, 0x1b29bda6, 0xa785641, 0x16462d8c, 0x76241b7, 0x79b6c3b, 0x204ae18, 0xf41212b,
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0x1f567a4d, 0xd6ce6db, 0xedf1784, 0x111df34, 0x85d7955, 0x55fc189, 0x1b7ae265, 0xf9281ac, 0xded7740,
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0xf19468b, 0x83763bb, 0x8ff7234, 0x3da7df8, 0x9590ac3, 0xdc96f2a, 0x16e44896, 0x7931009, 0x99d5acc,
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0x10f7b842, 0xaef5e84, 0xc0310d7, 0xdebac2c, 0x2a7b137, 0x4342344, 0x19633649, 0x3a10624, 0x4b4cb56,
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0x1d809c59, 0xac007f, 0x1f0f4bcd, 0xa1ab06e, 0xc5042cf, 0x82c0c77, 0x76c7563, 0x22c30f3, 0x3bf1568,
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0x7a895be, 0xfcca554, 0x12e90e4c, 0x7b4ab5f, 0x13aeb76b, 0x5887e2c, 0x1d7fe1e3, 0x908c8e3, 0x95800ee,
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0xb36bd54, 0xf08905d, 0x4e73ae8, 0xf5a7e48, 0xa67cb0, 0x50e1067, 0x1b944a0a, 0xf29c83a, 0xb23cfb9,
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0xbe1db1, 0x54de6e8, 0xd4707f2, 0x8ebcc2d, 0x2c77056, 0x1568ce4, 0x15fcc849, 0x4069712, 0xe2ed85f,
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0x2c5ff09, 0x42a6929, 0x628e7ea, 0xbd5b355, 0xaf0bd79, 0xaa03699, 0xdb99816, 0x4379cef, 0x81d57b,
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0x11237f01, 0xe2a820b, 0xfd53b95, 0x6beb5ee, 0x1aeb790c, 0xe470d53, 0x2c2cfee, 0x1c1d8d8, 0xa520fc4,
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0x1518e034, 0xa584dd4, 0x29e572b, 0xd4594fc, 0x141a8f6f, 0x8dfccf3, 0x5d20ba3, 0x2eb60c3, 0x9f16eb0,
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0x11cec356, 0xf039f84, 0x1b0990c1, 0xc91e526, 0x10b65bae, 0xf0616e8, 0x173fa3ff, 0xec8ccf9, 0xbe32790,
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0x11da3e79, 0xe2f35c7, 0x908875c, 0xdacf7bd, 0x538c165, 0x8d1487f, 0x7c31aed, 0x21af228, 0x7e1689d,
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0xdfc23ca, 0x24f15dc, 0x25ef3c4, 0x35248cd, 0x99a0f43, 0xa4b6ecc, 0xd066b3, 0x2481152, 0x37a7688,
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0x15a444b6, 0xb62300c, 0x4b841b, 0xa655e79, 0xd53226d, 0xbeb348a, 0x127f3c2, 0xb989247, 0x71a277d,
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0x19e9dfcb, 0xb8f92d0, 0xe2d226c, 0x390a8b0, 0x183cc462, 0x7bd8167, 0x1f32a552, 0x5e02db4, 0xa146ee9,
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0x1a003957, 0x1c95f61, 0x1eeec155, 0x26f811f, 0xf9596ba, 0x3082bfb, 0x96df083, 0x3e3a289, 0x7e2d8be,
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0x157a63e0, 0x99b8941, 0x1da7d345, 0xcc6cd0, 0x10beed9a, 0x48e83c0, 0x13aa2e25, 0x7cad710, 0x4029988,
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0x13dfa9dd, 0xb94f884, 0x1f4adfef, 0xb88543, 0x16f5f8dc, 0xa6a67f4, 0x14e274e2, 0x5e56cf4, 0x2f24ef,
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0x1e9ef967, 0xfe09bad, 0xfe079b3, 0xcc0ae9e, 0xb3edf6d, 0x3e961bc, 0x130d7831, 0x31043d6, 0xba986f9,
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0x1d28055, 0x65240ca, 0x4971fa3, 0x81b17f8, 0x11ec34a5, 0x8366ddc, 0x1471809, 0xfa5f1c6, 0xc911e15,
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0x8849491, 0xcf4c2e2, 0x14471b91, 0x39f75be, 0x445c21e, 0xf1585e9, 0x72cc11f, 0x4c79f0c, 0xe5522e1,
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0x1874c1ee, 0x4444211, 0x7914884, 0x3d1b133, 0x25ba3c, 0x4194f65, 0x1c0457ef, 0xac4899d, 0xe1fa66c,
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0x130a7918, 0x9b8d312, 0x4b1c5c8, 0x61ccac3, 0x18c8aa6f, 0xe93cb0a, 0xdccb12c, 0xde10825, 0x969737d,
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0xf58c0c3, 0x7cee6a9, 0xc2c329a, 0xc7f9ed9, 0x107b3981, 0x696a40e, 0x152847ff, 0x4d88754, 0xb141f47,
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0x5a16ffe, 0x3a7870a, 0x18667659, 0x3b72b03, 0xb1c9435, 0x9285394, 0xa00005a, 0x37506c, 0x2edc0bb,
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0x19afe392, 0xeb39cac, 0x177ef286, 0xdf87197, 0x19f844ed, 0x31fe8, 0x15f9bfd, 0x80dbec, 0x342e96e,
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0x497aced, 0xe88e909, 0x1f5fa9ba, 0x530a6ee, 0x1ef4e3f1, 0x69ffd12, 0x583006d, 0x2ecc9b1, 0x362db70,
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0x18c7bdc5, 0xf4bb3c5, 0x1c90b957, 0xf067c09, 0x9768f2b, 0xf73566a, 0x1939a900, 0x198c38a, 0x202a2a1,
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0x4bbf5a6, 0x4e265bc, 0x1f44b6e7, 0x185ca49, 0xa39e81b, 0x24aff5b, 0x4acc9c2, 0x638bdd3, 0xb65b2a8,
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0x6def8be, 0xb94537a, 0x10b81dee, 0xe00ec55, 0x2f2cdf7, 0xc20622d, 0x2d20f36, 0xe03c8c9, 0x898ea76,
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0x8e3921b, 0x8905bff, 0x1e94b6c8, 0xee7ad86, 0x154797f2, 0xa620863, 0x3fbd0d9, 0x1f3caab, 0x30c24bd,
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0x19d3892f, 0x59c17a2, 0x1ab4b0ae, 0xf8714ee, 0x90c4098, 0xa9c800d, 0x1910236b, 0xea808d3, 0x9ae2f31,
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0x1a15ad64, 0xa48c8d1, 0x184635a4, 0xb725ef1, 0x11921dcc, 0x3f866df, 0x16c27568, 0xbdf580a, 0xb08f55c,
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0x186ee1c, 0xb1627fa, 0x34e82f6, 0x933837e, 0xf311be5, 0xfedb03b, 0x167f72cd, 0xa5469c0, 0x9c82531,
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0xb92a24b, 0x14fdc8b, 0x141980d1, 0xbdc3a49, 0x7e02bb1, 0xaf4e6dd, 0x106d99e1, 0xd4616fc, 0x93c2717,
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0x1c0a0507, 0xc6d5fed, 0x9a03d8b, 0xa1d22b0, 0x127853e3, 0xc4ac6b8, 0x1a048cf7, 0x9afb72c, 0x65d485d,
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0x72d5998, 0xe9fa744, 0xe49e82c, 0x253cf80, 0x5f777ce, 0xa3799a5, 0x17270cbb, 0xc1d1ef0, 0xdf74977,
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0x114cb859, 0xfa8e037, 0xb8f3fe5, 0xc734cc6, 0x70d3d61, 0xeadac62, 0x12093dd0, 0x9add67d, 0x87200d6,
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0x175bcbb, 0xb29b49f, 0x1806b79c, 0x12fb61f, 0x170b3a10, 0x3aaf1cf, 0xa224085, 0x79d26af, 0x97759e2,
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0x92e19f1, 0xb32714d, 0x1f00d9f1, 0xc728619, 0x9e6f627, 0xe745e24, 0x18ea4ace, 0xfc60a41, 0x125f5b2,
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0xc3cf512, 0x39ed486, 0xf4d15fa, 0xf9167fd, 0x1c1f5dd5, 0xc21a53e, 0x1897930, 0x957a112, 0x21059a0,
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0x1f9e3ddc, 0xa4dfced, 0x8427f6f, 0x726fbe7, 0x1ea658f8, 0x2fdcd4c, 0x17e9b66f, 0xb2e7c2e, 0x39923bf,
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0x1bae104, 0x3973ce5, 0xc6f264c, 0x3511b84, 0x124195d7, 0x11996bd, 0x20be23d, 0xdc437c4, 0x4b4f16b,
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0x11902a0, 0x6c29cc9, 0x1d5ffbe6, 0xdb0b4c7, 0x10144c14, 0x2f2b719, 0x301189, 0x2343336, 0xa0bf2ac,
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}
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// Field element operations:
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// nonZeroToAllOnes returns:
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//
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// 0xffffffff for 0 < x <= 2**31
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// 0 for x == 0 or x > 2**31.
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func nonZeroToAllOnes(x uint32) uint32 {
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return ((x - 1) >> 31) - 1
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}
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// c256ReduceCarry adds a multiple of p in order to cancel |carry|,
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// which is a term at 2**257.
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// p = 2^256 - 2^224 + 2^96 + 2^64 -1
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// 2**257 = [c256Limbs]uint32{2, 0, 0x1fffff00, 0x7ff, 0, 0, 0, 0x2000000, 0}
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//
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// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
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// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
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func c256ReduceCarry(inout *[c256Limbs]uint32, carry uint32) {
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carryMask := nonZeroToAllOnes(carry)
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inout[0] += carry << 1
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inout[2] += 0x20000000 & carryMask
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inout[2] -= carry << 8
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inout[3] += carry << 11
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inout[3] -= 1 & carryMask
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inout[7] += carry << 25
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}
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// c256Sum sets out = in+in2.
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//
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// On entry, in[i]+in2[i] must not overflow a 32-bit word.
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// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
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func c256Sum(out, in, in2 *[c256Limbs]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 == c256Limbs {
|
|
break
|
|
}
|
|
|
|
out[i] = in[i] + in2[i]
|
|
out[i] += carry
|
|
carry = out[i] >> 28
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
c256ReduceCarry(out, carry)
|
|
}
|
|
|
|
const (
|
|
two30m2 = 1<<30 - 1<<2
|
|
two31m2 = 1<<31 - 1<<2
|
|
two30m27m2 = 1<<30 - 1<<27 - 1<<2
|
|
two30m13m2 = 1<<30 - 1<<13 - 1<<2
|
|
two31p10m2 = 1<<31 + 1<<10 - 1<<2
|
|
two31m3 = 1<<31 - 1<<3
|
|
)
|
|
|
|
// c256Zero31 is 0 mod p.
|
|
var c256Zero31 = [c256Limbs]uint32{two31m3, two30m2, two31p10m2, two30m13m2, two31m2, two30m2, two31m2, two30m27m2, two31m2}
|
|
|
|
// c256Diff 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 c256Diff(out, in, in2 *[c256Limbs]uint32) {
|
|
var carry uint32
|
|
|
|
for i := 0; ; i++ {
|
|
out[i] = in[i] - in2[i]
|
|
out[i] += c256Zero31[i]
|
|
out[i] += carry
|
|
carry = out[i] >> 29
|
|
out[i] &= bottom29Bits
|
|
|
|
i++
|
|
if i == c256Limbs {
|
|
break
|
|
}
|
|
|
|
out[i] = in[i] - in2[i]
|
|
out[i] += c256Zero31[i]
|
|
out[i] += carry
|
|
carry = out[i] >> 28
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
c256ReduceCarry(out, carry)
|
|
}
|
|
|
|
// c256ReduceDegree 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 c256ReduceDegree(out *[c256Limbs]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
|
|
|
|
// for l := 0; l < 18; l++ {
|
|
// fmt.Printf("%08x ", l)
|
|
// }
|
|
// fmt.Println("\n----------------------------------------------------------------------------")
|
|
// 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.
|
|
// c256P = [c256Limbs]uint32{0x1fffffff, 0xfffffff, 0x7f, 0xffffc00, 0x1fffffff, 0xfffffff, 0x1fffffff, 0xeffffff, 0xfffffff}
|
|
// 0 1 2 3 4 5 6 7 8
|
|
// =={2**29-1, 2**28-1, 2**7-1, 2**28-2**10, 2**29-1, 2**28-1, 2**29-1, 2**28-2**24-1, 2**28-1}
|
|
for i := 0; i < 8; i += 2 {
|
|
// tmp2[i] maybe > 2**29, fix it first
|
|
tmp2[i+1] += tmp2[i] >> 29
|
|
|
|
x = tmp2[i] & bottom29Bits // x is the really digital of tmp2[0], now plus x*p
|
|
xMask = nonZeroToAllOnes(x)
|
|
tmp2[i] = 0
|
|
|
|
// carry = x
|
|
// tmp2[i+2] += x + 2**7 * x - x = 2**7 * x
|
|
// tmp2[10] may overflow
|
|
// var xxx uint64 = uint64(x)
|
|
// xxx = xxx << 7
|
|
// xxx += uint64(tmp2[i+2])
|
|
// tmp2[i+2] = uint32(xxx & bottom29Bits)
|
|
// tmp2[i+3] += uint32(xxx >> 29)
|
|
tmp2[i+2] += (x << 7) & bottom29Bits
|
|
|
|
// carry = x >> 22
|
|
// tmp2[i+3] += x>>22 + 2**28*x - 2**10*x = 2**28 + x>>22 - 2**10*x + (x-1)*2**28
|
|
tmp2[i+3] += (x >> 22)
|
|
tmp2[i+3] += 0x10000000 & xMask
|
|
tmp2[i+3] -= (x << 10) & bottom28Bits
|
|
|
|
// carry = x-1-(x>>18)
|
|
// tmp2[i+4] += (2**29 - 1) * x + x - 1 - (x>>18) = 2**29 - 1 - (x>>18) + (x-1)*2**29
|
|
tmp2[i+4] += 0x1fffffff & xMask
|
|
tmp2[i+4] -= (x >> 18)
|
|
// fmt.Printf("xxx=%016x\n", xxx)
|
|
// fmt.Printf("x=%08x\n", x)
|
|
|
|
// carry = x-1
|
|
// tmp2[i+5] += x-1 + 2**28*x - x = 2**28 - 1 + (x-1)*2**28
|
|
tmp2[i+5] += 0xfffffff & xMask
|
|
|
|
// carry = x-1
|
|
// tmp2[i+6] += x-1 + 2**29*x - x = 2**29 - 1 + (x-1)*2**29
|
|
tmp2[i+6] += 0x1fffffff & xMask
|
|
|
|
// carry = x-1
|
|
// tmp2[i+7] += x-1 + 2**28*x - 2**24*x - x = 2**28-1 - 2**24*x + (x-1)*2**28
|
|
tmp2[i+7] += 0xfffffff & xMask
|
|
tmp2[i+7] -= (x << 24) & bottom28Bits
|
|
|
|
// carry = x-1 - (x>>4)
|
|
// tmp2[i+8] += x-1-(x>>4) + 2**28*x - x = 2**28*x - 1 - (x>>4)
|
|
///////////////////////////////
|
|
// tmp2[i+8] += (x << 28) & bottom29Bits
|
|
// tmp2[i+8] -= (x >> 4)
|
|
// tmp2[i+8] -= 1 & xMask
|
|
// tmp2[i+9] += x >> 1
|
|
// var xxx uint64 = uint64(x) << 28
|
|
// xxx -= uint64(1 & xMask)
|
|
// xxx -= uint64(x) >> 4
|
|
// tmp2[i+8] += uint32(xxx & bottom29Bits)
|
|
// tmp2[i+9] += uint32(xxx >> 29)
|
|
|
|
tmp2[i+8] += 0xfffffff & xMask
|
|
tmp2[i+8] -= x >> 4
|
|
tmp2[i+8] += ((x - 1) << 28) & bottom29Bits & xMask
|
|
tmp2[i+9] += ((x - 1) >> 1) & xMask
|
|
|
|
// half-----------------------------------------------------------
|
|
|
|
// 28
|
|
// 0xfffffff, 0x1fffffff, 0x7f, 0x1ffff800, 0xfffffff, 0x1fffffff, 0xfffffff, 0x1dffffff, 0xfffffff
|
|
// 2**28-1 2**29-1 2**7-1 2**29-2**11 2**28-1 2**29-1 2**28-1 2**29-2**25-1 2**28-1
|
|
// 1 2 3 4 5 6 7 8 9
|
|
tmp2[i+2] += tmp2[i+1] >> 28
|
|
x = tmp2[i+1] & bottom28Bits
|
|
xMask = nonZeroToAllOnes(x)
|
|
tmp2[i+1] = 0
|
|
|
|
tmp2[i+3] += (x << 7) & bottom28Bits
|
|
|
|
// xxx = uint64(x >> 21)
|
|
// xxx += (uint64(x) << 29)
|
|
// xxx -= uint64(x >> 11)
|
|
// tmp2[i+4] += uint32(xxx & bottom29Bits)
|
|
// tmp2[i+5] += uint32(xxx >> 29)
|
|
|
|
tmp2[i+4] += (x >> 21)
|
|
tmp2[i+4] += 0x20000000 & xMask
|
|
tmp2[i+4] -= (x << 11) & bottom29Bits
|
|
// fmt.Printf("xxx=%016x\n", xxx)
|
|
// fmt.Printf("x=%08x\n", x)
|
|
// fmt.Printf("tmp2[%d]=%08x\n", i+4, tmp2[i+4])
|
|
|
|
// carry = x-1 - x>>18
|
|
tmp2[i+5] += 0xfffffff & xMask
|
|
tmp2[i+5] -= (x >> 18)
|
|
|
|
tmp2[i+6] += 0x1fffffff & xMask
|
|
|
|
tmp2[i+7] += 0xfffffff & xMask
|
|
|
|
//carry = x-1
|
|
// + 2**29*x -2**25*x -1 = 2**29 - 1 - 2**25*x + (x-1)*2**29
|
|
tmp2[i+8] += 0x1fffffff & xMask
|
|
tmp2[i+8] -= (x << 25) & bottom29Bits
|
|
|
|
// carry = x-1 - x>>4
|
|
// 2**28*x - 1 - x>>4
|
|
// xxx = uint64(x)
|
|
// xxx = (xxx << 28) - 1 - (xxx >> 4)
|
|
// tmp2[i+9] += uint32(xxx & bottom28Bits)
|
|
// tmp2[i+10] += uint32(xxx >> 28)
|
|
// fmt.Printf("tmp2[i+9] = %08x ", tmp2[i+9])
|
|
tmp2[i+9] += 0xfffffff & xMask
|
|
tmp2[i+9] -= x >> 4
|
|
// fmt.Printf("%08x\n", tmp2[i+9])
|
|
|
|
// fmt.Println(i, tmp2[i+10])
|
|
tmp2[i+10] += (x - 1) & xMask
|
|
|
|
}
|
|
// for last half
|
|
tmp2[9] += tmp2[8] >> 29
|
|
|
|
x = tmp2[8] & bottom29Bits // x is the really digital of tmp2[0], now plus x*p
|
|
xMask = nonZeroToAllOnes(x)
|
|
tmp2[8] = 0
|
|
|
|
// carry = x
|
|
// tmp2[i+2] += x + 2**7 * x - x = 2**7 * x
|
|
// tmp2[10] may overflow // tmp2[8+2] += (x << 7) & bottom29Bits
|
|
// xxx := uint64(x)<<7 + uint64(tmp2[10])
|
|
// tmp2[10] = uint32(xxx & bottom29Bits)
|
|
// tmp2[11] += uint32(xxx >> 29)
|
|
tmp2[11] += tmp2[10]>>29 + x>>22
|
|
tmp2[10] = tmp2[10]&bottom29Bits + (x<<7)&bottom29Bits
|
|
|
|
// carry already handled
|
|
// tmp2[8+3] += x>>22 + 2**28*x - 2**10*x = 2**28 + x>>22 - 2**10*x + (x-1)*2**28
|
|
tmp2[8+3] += 0x10000000 & xMask
|
|
tmp2[8+3] -= (x << 10) & bottom28Bits
|
|
|
|
// carry = x-1-(x>>18)
|
|
// tmp2[8+4] += (2**29 - 1) * x + x - 1 - (x>>18) = 2**29 - 1 - (x>>18) + (x-1)*2**29
|
|
tmp2[8+4] += 0x1fffffff & xMask
|
|
tmp2[8+4] -= (x >> 18)
|
|
// fmt.Printf("xxx=%016x\n", xxx)
|
|
// fmt.Printf("x=%08x\n", x)
|
|
|
|
// carry = x-1
|
|
// tmp2[8+5] += x-1 + 2**28*x - x = 2**28 - 1 + (x-1)*2**28
|
|
tmp2[8+5] += 0xfffffff & xMask
|
|
|
|
// carry = x-1
|
|
// tmp2[8+6] += x-1 + 2**29*x - x = 2**29 - 1 + (x-1)*2**29
|
|
tmp2[8+6] += 0x1fffffff & xMask
|
|
|
|
// carry = x-1
|
|
// tmp2[8+7] += x-1 + 2**28*x - 2**24*x - x = 2**28-1 - 2**24*x + (x-1)*2**28
|
|
tmp2[8+7] += 0xfffffff & xMask
|
|
tmp2[8+7] -= (x << 24) & bottom28Bits
|
|
|
|
// carry = x-1 - (x>>4)
|
|
// tmp2[8+8] += x-1-(x>>4) + 2**28*x - x = 2**28*x - 1 - (x>>4)
|
|
///////////////////////////////
|
|
// tmp2[8+8] += (x << 28) & bottom29Bits
|
|
// tmp2[8+8] -= (x >> 4)
|
|
// tmp2[8+8] -= 1 & xMask
|
|
// tmp2[8+9] += x >> 1
|
|
// var xxx uint64 = uint64(x) << 28
|
|
// xxx -= uint64(1 & xMask)
|
|
// xxx -= uint64(x) >> 4
|
|
// tmp2[8+8] += uint32(xxx & bottom29Bits)
|
|
// tmp2[8+9] += uint32(xxx >> 29)
|
|
|
|
tmp2[8+8] += 0xfffffff & xMask
|
|
tmp2[8+8] -= (x >> 4)
|
|
tmp2[8+8] += ((x - 1) << 28) & bottom29Bits & xMask
|
|
tmp2[8+9] += ((x - 1) >> 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]
|
|
// fmt.Printf("%d %08x\n", i, out[i])
|
|
out[i] += carry
|
|
// fmt.Printf("%d %08x\n", i, out[i])
|
|
out[i] += (tmp2[i+10] << 28) & bottom29Bits
|
|
// fmt.Printf("%d %08x\n", i, out[i])
|
|
carry = out[i] >> 29
|
|
out[i] &= bottom29Bits
|
|
|
|
i++
|
|
out[i] = tmp2[i+9] >> 1
|
|
// fmt.Printf("%d %08x\n", i, out[i])
|
|
out[i] += carry
|
|
// fmt.Printf("%d %08x\n", i, out[i])
|
|
carry = out[i] >> 28
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
out[8] = tmp2[17]
|
|
out[8] += carry
|
|
carry = out[8] >> 29
|
|
out[8] &= bottom29Bits
|
|
c256ReduceCarry(out, carry)
|
|
}
|
|
|
|
// c256Square 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 c256Square(out, in *[c256Limbs]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])
|
|
|
|
c256ReduceDegree(out, tmp)
|
|
}
|
|
|
|
// c256Mul 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 c256Mul(out, in, in2 *[c256Limbs]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
|
|
// c256Square.
|
|
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)
|
|
|
|
c256ReduceDegree(out, tmp)
|
|
}
|
|
|
|
func c256Assign(out, in *[c256Limbs]uint32) {
|
|
*out = *in
|
|
}
|
|
|
|
// c256Invert 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)
|
|
//
|
|
// The Loop is fixed, so the consumed time is constant
|
|
// TODO: can be faster?
|
|
func c256Invert(out, in *[c256Limbs]uint32) {
|
|
// set ftmp = 1 * R
|
|
ftmp := [c256Limbs]uint32{0x2, 0x0, 0x1fffff00, 0x7ff, 0x0, 0x0, 0x0, 0x2000000, 0}
|
|
// idx[0] is highest bit
|
|
var idx = [256]int{
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
|
|
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
|
|
}
|
|
for _, d := range idx {
|
|
c256Square(&ftmp, &ftmp)
|
|
if d == 1 {
|
|
c256Mul(&ftmp, &ftmp, in)
|
|
}
|
|
}
|
|
c256Assign(out, &ftmp)
|
|
}
|
|
|
|
// c256Scalar3 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 c256Scalar3(out *[c256Limbs]uint32) {
|
|
var carry uint32
|
|
|
|
for i := 0; ; i++ {
|
|
out[i] *= 3
|
|
out[i] += carry
|
|
carry = out[i] >> 29
|
|
out[i] &= bottom29Bits
|
|
|
|
i++
|
|
if i == c256Limbs {
|
|
break
|
|
}
|
|
|
|
out[i] *= 3
|
|
out[i] += carry
|
|
carry = out[i] >> 28
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
c256ReduceCarry(out, carry)
|
|
}
|
|
|
|
// c256Scalar4 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 c256Scalar4(out *[c256Limbs]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 == c256Limbs {
|
|
break
|
|
}
|
|
nextCarry = out[i] >> 26
|
|
out[i] <<= 2
|
|
out[i] &= bottom28Bits
|
|
out[i] += carry
|
|
carry = nextCarry + (out[i] >> 28)
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
c256ReduceCarry(out, carry)
|
|
}
|
|
|
|
// c256Scalar8 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 c256Scalar8(out *[c256Limbs]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 == c256Limbs {
|
|
break
|
|
}
|
|
nextCarry = out[i] >> 25
|
|
out[i] <<= 3
|
|
out[i] &= bottom28Bits
|
|
out[i] += carry
|
|
carry = nextCarry + (out[i] >> 28)
|
|
out[i] &= bottom28Bits
|
|
}
|
|
|
|
c256ReduceCarry(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.
|
|
|
|
// c256PointDouble 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 c256PointDouble(xOut, yOut, zOut, x, y, z *[c256Limbs]uint32) {
|
|
var delta, gamma, alpha, beta, tmp, tmp2 [c256Limbs]uint32
|
|
|
|
c256Square(&delta, z)
|
|
c256Square(&gamma, y)
|
|
c256Mul(&beta, x, &gamma)
|
|
|
|
c256Sum(&tmp, x, &delta)
|
|
c256Diff(&tmp2, x, &delta)
|
|
c256Mul(&alpha, &tmp, &tmp2)
|
|
c256Scalar3(&alpha)
|
|
|
|
c256Sum(&tmp, y, z)
|
|
c256Square(&tmp, &tmp)
|
|
c256Diff(&tmp, &tmp, &gamma)
|
|
c256Diff(zOut, &tmp, &delta)
|
|
|
|
c256Scalar4(&beta)
|
|
c256Square(xOut, &alpha)
|
|
c256Diff(xOut, xOut, &beta)
|
|
c256Diff(xOut, xOut, &beta)
|
|
|
|
c256Diff(&tmp, &beta, xOut)
|
|
c256Mul(&tmp, &alpha, &tmp)
|
|
c256Square(&tmp2, &gamma)
|
|
c256Scalar8(&tmp2)
|
|
c256Diff(yOut, &tmp, &tmp2)
|
|
}
|
|
|
|
// c256PointAddMixed 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 c256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[c256Limbs]uint32) {
|
|
var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [c256Limbs]uint32
|
|
|
|
c256Square(&z1z1, z1)
|
|
c256Sum(&tmp, z1, z1)
|
|
|
|
c256Mul(&u2, x2, &z1z1)
|
|
c256Mul(&z1z1z1, z1, &z1z1)
|
|
c256Mul(&s2, y2, &z1z1z1)
|
|
c256Diff(&h, &u2, x1)
|
|
c256Sum(&i, &h, &h)
|
|
c256Square(&i, &i)
|
|
c256Mul(&j, &h, &i)
|
|
c256Diff(&r, &s2, y1)
|
|
c256Sum(&r, &r, &r)
|
|
c256Mul(&v, x1, &i)
|
|
|
|
c256Mul(zOut, &tmp, &h)
|
|
c256Square(&rr, &r)
|
|
c256Diff(xOut, &rr, &j)
|
|
c256Diff(xOut, xOut, &v)
|
|
c256Diff(xOut, xOut, &v)
|
|
|
|
c256Diff(&tmp, &v, xOut)
|
|
c256Mul(yOut, &tmp, &r)
|
|
c256Mul(&tmp, y1, &j)
|
|
c256Diff(yOut, yOut, &tmp)
|
|
c256Diff(yOut, yOut, &tmp)
|
|
}
|
|
|
|
// c256PointAdd 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 c256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[c256Limbs]uint32) {
|
|
var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [c256Limbs]uint32
|
|
|
|
c256Square(&z1z1, z1)
|
|
c256Square(&z2z2, z2)
|
|
c256Mul(&u1, x1, &z2z2)
|
|
|
|
c256Sum(&tmp, z1, z2)
|
|
c256Square(&tmp, &tmp)
|
|
c256Diff(&tmp, &tmp, &z1z1)
|
|
c256Diff(&tmp, &tmp, &z2z2)
|
|
|
|
c256Mul(&z2z2z2, z2, &z2z2)
|
|
c256Mul(&s1, y1, &z2z2z2)
|
|
|
|
c256Mul(&u2, x2, &z1z1)
|
|
c256Mul(&z1z1z1, z1, &z1z1)
|
|
c256Mul(&s2, y2, &z1z1z1)
|
|
c256Diff(&h, &u2, &u1)
|
|
c256Sum(&i, &h, &h)
|
|
c256Square(&i, &i)
|
|
c256Mul(&j, &h, &i)
|
|
c256Diff(&r, &s2, &s1)
|
|
c256Sum(&r, &r, &r)
|
|
c256Mul(&v, &u1, &i)
|
|
|
|
c256Mul(zOut, &tmp, &h)
|
|
c256Square(&rr, &r)
|
|
c256Diff(xOut, &rr, &j)
|
|
c256Diff(xOut, xOut, &v)
|
|
c256Diff(xOut, xOut, &v)
|
|
|
|
c256Diff(&tmp, &v, xOut)
|
|
c256Mul(yOut, &tmp, &r)
|
|
c256Mul(&tmp, &s1, &j)
|
|
c256Diff(yOut, yOut, &tmp)
|
|
c256Diff(yOut, yOut, &tmp)
|
|
}
|
|
|
|
// c256CopyConditional sets out=in if mask = 0xffffffff in constant time.
|
|
//
|
|
// On entry: mask is either 0 or 0xffffffff.
|
|
func c256CopyConditional(out, in *[c256Limbs]uint32, mask uint32) {
|
|
for i := 0; i < c256Limbs; i++ {
|
|
tmp := mask & (in[i] ^ out[i])
|
|
out[i] ^= tmp
|
|
}
|
|
}
|
|
|
|
// c256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
|
|
// On entry: index < 16, table[0] must be zero.
|
|
func c256SelectAffinePoint(xOut, yOut *[c256Limbs]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:]
|
|
}
|
|
}
|
|
}
|
|
|
|
// c256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of
|
|
// table.
|
|
// On entry: index < 16, table[0] must be zero.
|
|
func c256SelectJacobianPoint(xOut, yOut, zOut *[c256Limbs]uint32, table *[16][3][c256Limbs]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
|
|
}
|
|
}
|
|
}
|
|
|
|
// c256GetBit returns the bit'th bit of scalar.
|
|
func c256GetBit(scalar *[32]uint8, bit uint) uint32 {
|
|
return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1)
|
|
}
|
|
|
|
// c256ScalarBaseMult 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 c256ScalarBaseMult(xOut, yOut, zOut *[c256Limbs]uint32, scalar *[32]uint8) {
|
|
nIsInfinityMask := ^uint32(0)
|
|
var pIsNoninfiniteMask, mask, tableOffset uint32
|
|
var px, py, tx, ty, tz [c256Limbs]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 {
|
|
c256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
|
|
}
|
|
tableOffset = 0
|
|
for j := uint(0); j <= 32; j += 32 {
|
|
bit0 := c256GetBit(scalar, 31-i+j)
|
|
bit1 := c256GetBit(scalar, 95-i+j)
|
|
bit2 := c256GetBit(scalar, 159-i+j)
|
|
bit3 := c256GetBit(scalar, 223-i+j)
|
|
index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3)
|
|
|
|
c256SelectAffinePoint(&px, &py, c256Precomputed[tableOffset:], index)
|
|
tableOffset += 30 * c256Limbs
|
|
|
|
// 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.
|
|
c256PointAddMixed(&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.
|
|
c256CopyConditional(xOut, &px, nIsInfinityMask)
|
|
c256CopyConditional(yOut, &py, nIsInfinityMask)
|
|
c256CopyConditional(zOut, &c256One, 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
|
|
c256CopyConditional(xOut, &tx, mask)
|
|
c256CopyConditional(yOut, &ty, mask)
|
|
c256CopyConditional(zOut, &tz, mask)
|
|
// If p was not zero, then n is now non-zero.
|
|
nIsInfinityMask &^= pIsNoninfiniteMask
|
|
}
|
|
}
|
|
}
|
|
|
|
// c256PointToAffine 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 c256PointToAffine(xOut, yOut, x, y, z *[c256Limbs]uint32) {
|
|
var zInv, zInvSq [c256Limbs]uint32
|
|
c256Invert(&zInv, z)
|
|
c256Square(&zInvSq, &zInv)
|
|
c256Mul(xOut, x, &zInvSq)
|
|
c256Mul(&zInv, &zInv, &zInvSq)
|
|
c256Mul(yOut, y, &zInv)
|
|
}
|
|
|
|
// c256ToAffine returns a pair of *big.Int containing the affine representation
|
|
// of {x,y,z}.
|
|
func c256ToAffine(x, y, z *[c256Limbs]uint32) (xOut, yOut *big.Int) {
|
|
var xx, yy [c256Limbs]uint32
|
|
c256PointToAffine(&xx, &yy, x, y, z)
|
|
return c256ToBig(&xx), c256ToBig(&yy)
|
|
}
|
|
|
|
// c256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}.
|
|
func c256ScalarMult(xOut, yOut, zOut, x, y *[c256Limbs]uint32, scalar *[32]uint8) {
|
|
var px, py, pz, tx, ty, tz [c256Limbs]uint32
|
|
var precomp [16][3][c256Limbs]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] = c256One
|
|
|
|
for i := 2; i < 16; i += 2 {
|
|
c256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2])
|
|
c256PointAddMixed(&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 {
|
|
c256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
|
|
c256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
|
|
c256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
|
|
c256PointDouble(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.
|
|
c256SelectJacobianPoint(&px, &py, &pz, &precomp, index)
|
|
c256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz)
|
|
c256CopyConditional(xOut, &px, nIsInfinityMask)
|
|
c256CopyConditional(yOut, &py, nIsInfinityMask)
|
|
c256CopyConditional(zOut, &pz, nIsInfinityMask)
|
|
|
|
pIsNoninfiniteMask = nonZeroToAllOnes(index)
|
|
mask = pIsNoninfiniteMask & ^nIsInfinityMask
|
|
c256CopyConditional(xOut, &tx, mask)
|
|
c256CopyConditional(yOut, &ty, mask)
|
|
c256CopyConditional(zOut, &tz, mask)
|
|
nIsInfinityMask &^= pIsNoninfiniteMask
|
|
}
|
|
}
|
|
|
|
// c256FromBig sets out = R*in.
|
|
func c256FromBig(out *[c256Limbs]uint32, in *big.Int) {
|
|
tmp := new(big.Int).Lsh(in, 257)
|
|
tmp.Mod(tmp, c256.P)
|
|
|
|
for i := 0; i < c256Limbs; 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 == c256Limbs {
|
|
break
|
|
}
|
|
|
|
if bits := tmp.Bits(); len(bits) > 0 {
|
|
out[i] = uint32(bits[0]) & bottom28Bits
|
|
} else {
|
|
out[i] = 0
|
|
}
|
|
tmp.Rsh(tmp, 28)
|
|
}
|
|
}
|
|
|
|
// c256ToBig returns a *big.Int containing the value of in.
|
|
func c256ToBig(in *[c256Limbs]uint32) *big.Int {
|
|
result, tmp := new(big.Int), new(big.Int)
|
|
|
|
result.SetInt64(int64(in[c256Limbs-1]))
|
|
for i := c256Limbs - 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, c256RInverse)
|
|
result.Mod(result, c256.P)
|
|
return result
|
|
}
|