gitee.com/zhaochuninhefei/gmgo@v0.0.31-0.20240209061119-069254a02979/sm2soft/p256.go

// Copyright (c) 2022 zhaochun
// gmgo is licensed under Mulan PSL v2.
// You can use this software according to the terms and conditions of the Mulan PSL v2.
// You may obtain a copy of Mulan PSL v2 at:
//          http://license.coscl.org.cn/MulanPSL2
// THIS SOFTWARE IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTIES OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO NON-INFRINGEMENT, MERCHANTABILITY OR FIT FOR A PARTICULAR PURPOSE.
// See the Mulan PSL v2 for more details.

/*
sm2soft 是sm2的纯软实现，基于tjfoc国密算法库`tjfoc/gmsm`做了少量修改。
对应版权声明: thrid_licenses/github.com/tjfoc/gmsm/版权声明
*/

package sm2soft

/*
sm2/p256.go 定义sm2P256椭圆曲线
*/

import (
	"crypto/elliptic"
	"math/big"
	"sync"
)

// SM2P256椭圆曲线
type sm2P256Curve struct {
	RInverse *big.Int
	*elliptic.CurveParams
	a, b, gx, gy sm2P256FieldElement
}

var initonce sync.Once
var sm2P256 sm2P256Curve

type sm2P256FieldElement [9]uint32
type sm2P256LargeFieldElement [17]uint64

const (
	bottom28Bits = 0xFFFFFFF
	bottom29Bits = 0x1FFFFFFF
)

// 初始化sm2椭圆曲线 sm2P256
func initP256Sm2() {
	sm2P256.CurveParams = &elliptic.CurveParams{Name: "SM2-P-256"} // sm2
	A, _ := new(big.Int).SetString("FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFC", 16)
	//SM2椭圆曲线公钥密码算法推荐曲线参数
	sm2P256.P, _ = new(big.Int).SetString("FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFF", 16)
	sm2P256.N, _ = new(big.Int).SetString("FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFF7203DF6B21C6052B53BBF40939D54123", 16)
	sm2P256.B, _ = new(big.Int).SetString("28E9FA9E9D9F5E344D5A9E4BCF6509A7F39789F515AB8F92DDBCBD414D940E93", 16)
	sm2P256.Gx, _ = new(big.Int).SetString("32C4AE2C1F1981195F9904466A39C9948FE30BBFF2660BE1715A4589334C74C7", 16)
	sm2P256.Gy, _ = new(big.Int).SetString("BC3736A2F4F6779C59BDCEE36B692153D0A9877CC62A474002DF32E52139F0A0", 16)
	sm2P256.RInverse, _ = new(big.Int).SetString("7ffffffd80000002fffffffe000000017ffffffe800000037ffffffc80000002", 16)
	sm2P256.BitSize = 256
	sm2P256FromBig(&sm2P256.a, A)
	sm2P256FromBig(&sm2P256.gx, sm2P256.Gx)
	sm2P256FromBig(&sm2P256.gy, sm2P256.Gy)
	sm2P256FromBig(&sm2P256.b, sm2P256.B)
}

// P256Sm2 获取SM2P256椭圆曲线
// 该曲线是一个单例
func P256Sm2() elliptic.Curve {
	// 为确保sm2P256是一个单例，这里只初始化一次sm2P256
	initonce.Do(initP256Sm2)
	return sm2P256
}

// Params 获取SM2P256椭圆曲线的曲线参数
func (curve sm2P256Curve) Params() *elliptic.CurveParams {
	return sm2P256.CurveParams
}

// IsOnCurve y^2 = x^3 + ax + b
// 判断给定的X,Y座标是否在sm2P256Curve曲线上
func (curve sm2P256Curve) IsOnCurve(X, Y *big.Int) bool {
	var a, x, y, y2, x3 sm2P256FieldElement

	sm2P256FromBig(&x, X)
	sm2P256FromBig(&y, Y)

	sm2P256Square(&x3, &x)       // x3 = x ^ 2
	sm2P256Mul(&x3, &x3, &x)     // x3 = x ^ 2 * x
	sm2P256Mul(&a, &curve.a, &x) // a = a * x
	sm2P256Add(&x3, &x3, &a)
	sm2P256Add(&x3, &x3, &curve.b)

	sm2P256Square(&y2, &y) // y2 = y ^ 2
	return sm2P256ToBig(&x3).Cmp(sm2P256ToBig(&y2)) == 0
}

func zForAffine(x, y *big.Int) *big.Int {
	z := new(big.Int)
	if x.Sign() != 0 || y.Sign() != 0 {
		z.SetInt64(1)
	}
	return z
}

func (curve sm2P256Curve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
	var X1, Y1, Z1, X2, Y2, Z2, X3, Y3, Z3 sm2P256FieldElement

	z1 := zForAffine(x1, y1)
	z2 := zForAffine(x2, y2)
	sm2P256FromBig(&X1, x1)
	sm2P256FromBig(&Y1, y1)
	sm2P256FromBig(&Z1, z1)
	sm2P256FromBig(&X2, x2)
	sm2P256FromBig(&Y2, y2)
	sm2P256FromBig(&Z2, z2)
	sm2P256PointAdd(&X1, &Y1, &Z1, &X2, &Y2, &Z2, &X3, &Y3, &Z3)
	return sm2P256ToAffine(&X3, &Y3, &Z3)
}

func (curve sm2P256Curve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
	var X1, Y1, Z1 sm2P256FieldElement

	z1 := zForAffine(x1, y1)
	sm2P256FromBig(&X1, x1)
	sm2P256FromBig(&Y1, y1)
	sm2P256FromBig(&Z1, z1)
	sm2P256PointDouble(&X1, &Y1, &Z1, &X1, &Y1, &Z1)
	return sm2P256ToAffine(&X1, &Y1, &Z1)
}

func (curve sm2P256Curve) ScalarMult(x1, y1 *big.Int, k []byte) (*big.Int, *big.Int) {
	var X, Y, Z, X1, Y1 sm2P256FieldElement
	sm2P256FromBig(&X1, x1)
	sm2P256FromBig(&Y1, y1)
	scalar := sm2GenrateWNaf(k)
	scalarReversed := WNafReversed(scalar)
	sm2P256ScalarMult(&X, &Y, &Z, &X1, &Y1, scalarReversed)
	return sm2P256ToAffine(&X, &Y, &Z)
}

func (curve sm2P256Curve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
	var scalarReversed [32]byte
	var X, Y, Z sm2P256FieldElement

	sm2P256GetScalar(&scalarReversed, k)
	sm2P256ScalarBaseMult(&X, &Y, &Z, &scalarReversed)
	return sm2P256ToAffine(&X, &Y, &Z)
}

var sm2P256Precomputed = [9 * 2 * 15 * 2]uint32{
	0x830053d, 0x328990f, 0x6c04fe1, 0xc0f72e5, 0x1e19f3c, 0x666b093, 0x175a87b, 0xec38276, 0x222cf4b,
	0x185a1bba, 0x354e593, 0x1295fac1, 0xf2bc469, 0x47c60fa, 0xc19b8a9, 0xf63533e, 0x903ae6b, 0xc79acba,
	0x15b061a4, 0x33e020b, 0xdffb34b, 0xfcf2c8, 0x16582e08, 0x262f203, 0xfb34381, 0xa55452, 0x604f0ff,
	0x41f1f90, 0xd64ced2, 0xee377bf, 0x75f05f0, 0x189467ae, 0xe2244e, 0x1e7700e8, 0x3fbc464, 0x9612d2e,
	0x1341b3b8, 0xee84e23, 0x1edfa5b4, 0x14e6030, 0x19e87be9, 0x92f533c, 0x1665d96c, 0x226653e, 0xa238d3e,
	0xf5c62c, 0x95bb7a, 0x1f0e5a41, 0x28789c3, 0x1f251d23, 0x8726609, 0xe918910, 0x8096848, 0xf63d028,
	0x152296a1, 0x9f561a8, 0x14d376fb, 0x898788a, 0x61a95fb, 0xa59466d, 0x159a003d, 0x1ad1698, 0x93cca08,
	0x1b314662, 0x706e006, 0x11ce1e30, 0x97b710, 0x172fbc0d, 0x8f50158, 0x11c7ffe7, 0xd182cce, 0xc6ad9e8,
	0x12ea31b2, 0xc4e4f38, 0x175b0d96, 0xec06337, 0x75a9c12, 0xb001fdf, 0x93e82f5, 0x34607de, 0xb8035ed,
	0x17f97924, 0x75cf9e6, 0xdceaedd, 0x2529924, 0x1a10c5ff, 0xb1a54dc, 0x19464d8, 0x2d1997, 0xde6a110,
	0x1e276ee5, 0x95c510c, 0x1aca7c7a, 0xfe48aca, 0x121ad4d9, 0xe4132c6, 0x8239b9d, 0x40ea9cd, 0x816c7b,
	0x632d7a4, 0xa679813, 0x5911fcf, 0x82b0f7c, 0x57b0ad5, 0xbef65, 0xd541365, 0x7f9921f, 0xc62e7a,
	0x3f4b32d, 0x58e50e1, 0x6427aed, 0xdcdda67, 0xe8c2d3e, 0x6aa54a4, 0x18df4c35, 0x49a6a8e, 0x3cd3d0c,
	0xd7adf2, 0xcbca97, 0x1bda5f2d, 0x3258579, 0x606b1e6, 0x6fc1b5b, 0x1ac27317, 0x503ca16, 0xa677435,
	0x57bc73, 0x3992a42, 0xbab987b, 0xfab25eb, 0x128912a4, 0x90a1dc4, 0x1402d591, 0x9ffbcfc, 0xaa48856,
	0x7a7c2dc, 0xcefd08a, 0x1b29bda6, 0xa785641, 0x16462d8c, 0x76241b7, 0x79b6c3b, 0x204ae18, 0xf41212b,
	0x1f567a4d, 0xd6ce6db, 0xedf1784, 0x111df34, 0x85d7955, 0x55fc189, 0x1b7ae265, 0xf9281ac, 0xded7740,
	0xf19468b, 0x83763bb, 0x8ff7234, 0x3da7df8, 0x9590ac3, 0xdc96f2a, 0x16e44896, 0x7931009, 0x99d5acc,
	0x10f7b842, 0xaef5e84, 0xc0310d7, 0xdebac2c, 0x2a7b137, 0x4342344, 0x19633649, 0x3a10624, 0x4b4cb56,
	0x1d809c59, 0xac007f, 0x1f0f4bcd, 0xa1ab06e, 0xc5042cf, 0x82c0c77, 0x76c7563, 0x22c30f3, 0x3bf1568,
	0x7a895be, 0xfcca554, 0x12e90e4c, 0x7b4ab5f, 0x13aeb76b, 0x5887e2c, 0x1d7fe1e3, 0x908c8e3, 0x95800ee,
	0xb36bd54, 0xf08905d, 0x4e73ae8, 0xf5a7e48, 0xa67cb0, 0x50e1067, 0x1b944a0a, 0xf29c83a, 0xb23cfb9,
	0xbe1db1, 0x54de6e8, 0xd4707f2, 0x8ebcc2d, 0x2c77056, 0x1568ce4, 0x15fcc849, 0x4069712, 0xe2ed85f,
	0x2c5ff09, 0x42a6929, 0x628e7ea, 0xbd5b355, 0xaf0bd79, 0xaa03699, 0xdb99816, 0x4379cef, 0x81d57b,
	0x11237f01, 0xe2a820b, 0xfd53b95, 0x6beb5ee, 0x1aeb790c, 0xe470d53, 0x2c2cfee, 0x1c1d8d8, 0xa520fc4,
	0x1518e034, 0xa584dd4, 0x29e572b, 0xd4594fc, 0x141a8f6f, 0x8dfccf3, 0x5d20ba3, 0x2eb60c3, 0x9f16eb0,
	0x11cec356, 0xf039f84, 0x1b0990c1, 0xc91e526, 0x10b65bae, 0xf0616e8, 0x173fa3ff, 0xec8ccf9, 0xbe32790,
	0x11da3e79, 0xe2f35c7, 0x908875c, 0xdacf7bd, 0x538c165, 0x8d1487f, 0x7c31aed, 0x21af228, 0x7e1689d,
	0xdfc23ca, 0x24f15dc, 0x25ef3c4, 0x35248cd, 0x99a0f43, 0xa4b6ecc, 0xd066b3, 0x2481152, 0x37a7688,
	0x15a444b6, 0xb62300c, 0x4b841b, 0xa655e79, 0xd53226d, 0xbeb348a, 0x127f3c2, 0xb989247, 0x71a277d,
	0x19e9dfcb, 0xb8f92d0, 0xe2d226c, 0x390a8b0, 0x183cc462, 0x7bd8167, 0x1f32a552, 0x5e02db4, 0xa146ee9,
	0x1a003957, 0x1c95f61, 0x1eeec155, 0x26f811f, 0xf9596ba, 0x3082bfb, 0x96df083, 0x3e3a289, 0x7e2d8be,
	0x157a63e0, 0x99b8941, 0x1da7d345, 0xcc6cd0, 0x10beed9a, 0x48e83c0, 0x13aa2e25, 0x7cad710, 0x4029988,
	0x13dfa9dd, 0xb94f884, 0x1f4adfef, 0xb88543, 0x16f5f8dc, 0xa6a67f4, 0x14e274e2, 0x5e56cf4, 0x2f24ef,
	0x1e9ef967, 0xfe09bad, 0xfe079b3, 0xcc0ae9e, 0xb3edf6d, 0x3e961bc, 0x130d7831, 0x31043d6, 0xba986f9,
	0x1d28055, 0x65240ca, 0x4971fa3, 0x81b17f8, 0x11ec34a5, 0x8366ddc, 0x1471809, 0xfa5f1c6, 0xc911e15,
	0x8849491, 0xcf4c2e2, 0x14471b91, 0x39f75be, 0x445c21e, 0xf1585e9, 0x72cc11f, 0x4c79f0c, 0xe5522e1,
	0x1874c1ee, 0x4444211, 0x7914884, 0x3d1b133, 0x25ba3c, 0x4194f65, 0x1c0457ef, 0xac4899d, 0xe1fa66c,
	0x130a7918, 0x9b8d312, 0x4b1c5c8, 0x61ccac3, 0x18c8aa6f, 0xe93cb0a, 0xdccb12c, 0xde10825, 0x969737d,
	0xf58c0c3, 0x7cee6a9, 0xc2c329a, 0xc7f9ed9, 0x107b3981, 0x696a40e, 0x152847ff, 0x4d88754, 0xb141f47,
	0x5a16ffe, 0x3a7870a, 0x18667659, 0x3b72b03, 0xb1c9435, 0x9285394, 0xa00005a, 0x37506c, 0x2edc0bb,
	0x19afe392, 0xeb39cac, 0x177ef286, 0xdf87197, 0x19f844ed, 0x31fe8, 0x15f9bfd, 0x80dbec, 0x342e96e,
	0x497aced, 0xe88e909, 0x1f5fa9ba, 0x530a6ee, 0x1ef4e3f1, 0x69ffd12, 0x583006d, 0x2ecc9b1, 0x362db70,
	0x18c7bdc5, 0xf4bb3c5, 0x1c90b957, 0xf067c09, 0x9768f2b, 0xf73566a, 0x1939a900, 0x198c38a, 0x202a2a1,
	0x4bbf5a6, 0x4e265bc, 0x1f44b6e7, 0x185ca49, 0xa39e81b, 0x24aff5b, 0x4acc9c2, 0x638bdd3, 0xb65b2a8,
	0x6def8be, 0xb94537a, 0x10b81dee, 0xe00ec55, 0x2f2cdf7, 0xc20622d, 0x2d20f36, 0xe03c8c9, 0x898ea76,
	0x8e3921b, 0x8905bff, 0x1e94b6c8, 0xee7ad86, 0x154797f2, 0xa620863, 0x3fbd0d9, 0x1f3caab, 0x30c24bd,
	0x19d3892f, 0x59c17a2, 0x1ab4b0ae, 0xf8714ee, 0x90c4098, 0xa9c800d, 0x1910236b, 0xea808d3, 0x9ae2f31,
	0x1a15ad64, 0xa48c8d1, 0x184635a4, 0xb725ef1, 0x11921dcc, 0x3f866df, 0x16c27568, 0xbdf580a, 0xb08f55c,
	0x186ee1c, 0xb1627fa, 0x34e82f6, 0x933837e, 0xf311be5, 0xfedb03b, 0x167f72cd, 0xa5469c0, 0x9c82531,
	0xb92a24b, 0x14fdc8b, 0x141980d1, 0xbdc3a49, 0x7e02bb1, 0xaf4e6dd, 0x106d99e1, 0xd4616fc, 0x93c2717,
	0x1c0a0507, 0xc6d5fed, 0x9a03d8b, 0xa1d22b0, 0x127853e3, 0xc4ac6b8, 0x1a048cf7, 0x9afb72c, 0x65d485d,
	0x72d5998, 0xe9fa744, 0xe49e82c, 0x253cf80, 0x5f777ce, 0xa3799a5, 0x17270cbb, 0xc1d1ef0, 0xdf74977,
	0x114cb859, 0xfa8e037, 0xb8f3fe5, 0xc734cc6, 0x70d3d61, 0xeadac62, 0x12093dd0, 0x9add67d, 0x87200d6,
	0x175bcbb, 0xb29b49f, 0x1806b79c, 0x12fb61f, 0x170b3a10, 0x3aaf1cf, 0xa224085, 0x79d26af, 0x97759e2,
	0x92e19f1, 0xb32714d, 0x1f00d9f1, 0xc728619, 0x9e6f627, 0xe745e24, 0x18ea4ace, 0xfc60a41, 0x125f5b2,
	0xc3cf512, 0x39ed486, 0xf4d15fa, 0xf9167fd, 0x1c1f5dd5, 0xc21a53e, 0x1897930, 0x957a112, 0x21059a0,
	0x1f9e3ddc, 0xa4dfced, 0x8427f6f, 0x726fbe7, 0x1ea658f8, 0x2fdcd4c, 0x17e9b66f, 0xb2e7c2e, 0x39923bf,
	0x1bae104, 0x3973ce5, 0xc6f264c, 0x3511b84, 0x124195d7, 0x11996bd, 0x20be23d, 0xdc437c4, 0x4b4f16b,
	0x11902a0, 0x6c29cc9, 0x1d5ffbe6, 0xdb0b4c7, 0x10144c14, 0x2f2b719, 0x301189, 0x2343336, 0xa0bf2ac,
}

func sm2P256GetScalar(b *[32]byte, a []byte) {
	var scalarBytes []byte

	n := new(big.Int).SetBytes(a)
	if n.Cmp(sm2P256.N) >= 0 {
		n.Mod(n, sm2P256.N)
		scalarBytes = n.Bytes()
	} else {
		scalarBytes = a
	}
	for i, v := range scalarBytes {
		b[len(scalarBytes)-(1+i)] = v
	}
}

func sm2P256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *sm2P256FieldElement) {
	var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp sm2P256FieldElement

	sm2P256Square(&z1z1, z1)
	sm2P256Add(&tmp, z1, z1)

	sm2P256Mul(&u2, x2, &z1z1)
	sm2P256Mul(&z1z1z1, z1, &z1z1)
	sm2P256Mul(&s2, y2, &z1z1z1)
	sm2P256Sub(&h, &u2, x1)
	sm2P256Add(&i, &h, &h)
	sm2P256Square(&i, &i)
	sm2P256Mul(&j, &h, &i)
	sm2P256Sub(&r, &s2, y1)
	sm2P256Add(&r, &r, &r)
	sm2P256Mul(&v, x1, &i)

	sm2P256Mul(zOut, &tmp, &h)
	sm2P256Square(&rr, &r)
	sm2P256Sub(xOut, &rr, &j)
	sm2P256Sub(xOut, xOut, &v)
	sm2P256Sub(xOut, xOut, &v)

	sm2P256Sub(&tmp, &v, xOut)
	sm2P256Mul(yOut, &tmp, &r)
	sm2P256Mul(&tmp, y1, &j)
	sm2P256Sub(yOut, yOut, &tmp)
	sm2P256Sub(yOut, yOut, &tmp)
}

// sm2P256CopyConditional sets out=in if mask = 0xffffffff in constant time.
//
// On entry: mask is either 0 or 0xffffffff.
func sm2P256CopyConditional(out, in *sm2P256FieldElement, mask uint32) {
	for i := 0; i < 9; i++ {
		tmp := mask & (in[i] ^ out[i])
		out[i] ^= tmp
	}
}

// sm2P256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
// On entry: index < 16, table[0] must be zero.
func sm2P256SelectAffinePoint(xOut, yOut *sm2P256FieldElement, 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:]
		}
	}
}

// sm2P256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of
// table.
// On entry: index < 16, table[0] must be zero.
func sm2P256SelectJacobianPoint(xOut, yOut, zOut *sm2P256FieldElement, table *[16][3]sm2P256FieldElement, 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
		}
	}
}

// sm2P256GetBit returns the bit'th bit of scalar.
func sm2P256GetBit(scalar *[32]uint8, bit uint) uint32 {
	return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1)
}

// sm2P256ScalarBaseMult 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 sm2P256ScalarBaseMult(xOut, yOut, zOut *sm2P256FieldElement, scalar *[32]uint8) {
	nIsInfinityMask := ^uint32(0)
	var px, py, tx, ty, tz sm2P256FieldElement
	var pIsNoninfiniteMask, mask, tableOffset 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 {
			sm2P256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
		}
		tableOffset = 0
		for j := uint(0); j <= 32; j += 32 {
			bit0 := sm2P256GetBit(scalar, 31-i+j)
			bit1 := sm2P256GetBit(scalar, 95-i+j)
			bit2 := sm2P256GetBit(scalar, 159-i+j)
			bit3 := sm2P256GetBit(scalar, 223-i+j)
			index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3)

			sm2P256SelectAffinePoint(&px, &py, sm2P256Precomputed[tableOffset:], index)
			tableOffset += 30 * 9

			// 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.
			sm2P256PointAddMixed(&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.
			sm2P256CopyConditional(xOut, &px, nIsInfinityMask)
			sm2P256CopyConditional(yOut, &py, nIsInfinityMask)
			sm2P256CopyConditional(zOut, &sm2P256Factor[1], 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
			sm2P256CopyConditional(xOut, &tx, mask)
			sm2P256CopyConditional(yOut, &ty, mask)
			sm2P256CopyConditional(zOut, &tz, mask)
			// If p was not zero, then n is now non-zero.
			nIsInfinityMask &^= pIsNoninfiniteMask
		}
	}
}

func sm2P256PointToAffine(xOut, yOut, x, y, z *sm2P256FieldElement) {
	var zInv, zInvSq sm2P256FieldElement

	zz := sm2P256ToBig(z)
	zz.ModInverse(zz, sm2P256.P)
	sm2P256FromBig(&zInv, zz)

	sm2P256Square(&zInvSq, &zInv)
	sm2P256Mul(xOut, x, &zInvSq)
	sm2P256Mul(&zInv, &zInv, &zInvSq)
	sm2P256Mul(yOut, y, &zInv)
}

func sm2P256ToAffine(x, y, z *sm2P256FieldElement) (xOut, yOut *big.Int) {
	var xx, yy sm2P256FieldElement

	sm2P256PointToAffine(&xx, &yy, x, y, z)
	return sm2P256ToBig(&xx), sm2P256ToBig(&yy)
}

var sm2P256Factor = []sm2P256FieldElement{
	{0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0},
	{0x2, 0x0, 0x1FFFFF00, 0x7FF, 0x0, 0x0, 0x0, 0x2000000, 0x0},
	{0x4, 0x0, 0x1FFFFE00, 0xFFF, 0x0, 0x0, 0x0, 0x4000000, 0x0},
	{0x6, 0x0, 0x1FFFFD00, 0x17FF, 0x0, 0x0, 0x0, 0x6000000, 0x0},
	{0x8, 0x0, 0x1FFFFC00, 0x1FFF, 0x0, 0x0, 0x0, 0x8000000, 0x0},
	{0xA, 0x0, 0x1FFFFB00, 0x27FF, 0x0, 0x0, 0x0, 0xA000000, 0x0},
	{0xC, 0x0, 0x1FFFFA00, 0x2FFF, 0x0, 0x0, 0x0, 0xC000000, 0x0},
	{0xE, 0x0, 0x1FFFF900, 0x37FF, 0x0, 0x0, 0x0, 0xE000000, 0x0},
	{0x10, 0x0, 0x1FFFF800, 0x3FFF, 0x0, 0x0, 0x0, 0x0, 0x01},
}

func sm2P256Scalar(b *sm2P256FieldElement, a int) {
	sm2P256Mul(b, b, &sm2P256Factor[a])
}

// (x3, y3, z3) = (x1, y1, z1) + (x2, y2, z2)
func sm2P256PointAdd(x1, y1, z1, x2, y2, z2, x3, y3, z3 *sm2P256FieldElement) {
	var u1, u2, z22, z12, z23, z13, s1, s2, h, h2, r, r2, tm sm2P256FieldElement

	if sm2P256ToBig(z1).Sign() == 0 {
		sm2P256Dup(x3, x2)
		sm2P256Dup(y3, y2)
		sm2P256Dup(z3, z2)
		return
	}

	if sm2P256ToBig(z2).Sign() == 0 {
		sm2P256Dup(x3, x1)
		sm2P256Dup(y3, y1)
		sm2P256Dup(z3, z1)
		return
	}

	sm2P256Square(&z12, z1) // z12 = z1 ^ 2
	sm2P256Square(&z22, z2) // z22 = z2 ^ 2

	sm2P256Mul(&z13, &z12, z1) // z13 = z1 ^ 3
	sm2P256Mul(&z23, &z22, z2) // z23 = z2 ^ 3

	sm2P256Mul(&u1, x1, &z22) // u1 = x1 * z2 ^ 2
	sm2P256Mul(&u2, x2, &z12) // u2 = x2 * z1 ^ 2

	sm2P256Mul(&s1, y1, &z23) // s1 = y1 * z2 ^ 3
	sm2P256Mul(&s2, y2, &z13) // s2 = y2 * z1 ^ 3

	if sm2P256ToBig(&u1).Cmp(sm2P256ToBig(&u2)) == 0 &&
		sm2P256ToBig(&s1).Cmp(sm2P256ToBig(&s2)) == 0 {
		sm2P256PointDouble(x1, y1, z1, x1, y1, z1)
	}

	sm2P256Sub(&h, &u2, &u1) // h = u2 - u1
	sm2P256Sub(&r, &s2, &s1) // r = s2 - s1

	sm2P256Square(&r2, &r) // r2 = r ^ 2
	sm2P256Square(&h2, &h) // h2 = h ^ 2

	sm2P256Mul(&tm, &h2, &h) // tm = h ^ 3
	sm2P256Sub(x3, &r2, &tm)
	sm2P256Mul(&tm, &u1, &h2)
	sm2P256Scalar(&tm, 2)   // tm = 2 * (u1 * h ^ 2)
	sm2P256Sub(x3, x3, &tm) // x3 = r ^ 2 - h ^ 3 - 2 * u1 * h ^ 2

	sm2P256Mul(&tm, &u1, &h2) // tm = u1 * h ^ 2
	sm2P256Sub(&tm, &tm, x3)  // tm = u1 * h ^ 2 - x3
	sm2P256Mul(y3, &r, &tm)
	sm2P256Mul(&tm, &h2, &h)  // tm = h ^ 3
	sm2P256Mul(&tm, &tm, &s1) // tm = s1 * h ^ 3
	sm2P256Sub(y3, y3, &tm)   // y3 = r * (u1 * h ^ 2 - x3) - s1 * h ^ 3

	sm2P256Mul(z3, z1, z2)
	sm2P256Mul(z3, z3, &h) // z3 = z1 * z3 * h
}

// (x3, y3, z3) = (x1, y1, z1)- (x2, y2, z2)
func sm2P256PointSub(x1, y1, z1, x2, y2, z2, x3, y3, z3 *sm2P256FieldElement) {
	var u1, u2, z22, z12, z23, z13, s1, s2, h, h2, r, r2, tm sm2P256FieldElement
	y := sm2P256ToBig(y2)
	zero := new(big.Int).SetInt64(0)
	y.Sub(zero, y)
	sm2P256FromBig(y2, y)

	if sm2P256ToBig(z1).Sign() == 0 {
		sm2P256Dup(x3, x2)
		sm2P256Dup(y3, y2)
		sm2P256Dup(z3, z2)
		return
	}

	if sm2P256ToBig(z2).Sign() == 0 {
		sm2P256Dup(x3, x1)
		sm2P256Dup(y3, y1)
		sm2P256Dup(z3, z1)
		return
	}

	sm2P256Square(&z12, z1) // z12 = z1 ^ 2
	sm2P256Square(&z22, z2) // z22 = z2 ^ 2

	sm2P256Mul(&z13, &z12, z1) // z13 = z1 ^ 3
	sm2P256Mul(&z23, &z22, z2) // z23 = z2 ^ 3

	sm2P256Mul(&u1, x1, &z22) // u1 = x1 * z2 ^ 2
	sm2P256Mul(&u2, x2, &z12) // u2 = x2 * z1 ^ 2

	sm2P256Mul(&s1, y1, &z23) // s1 = y1 * z2 ^ 3
	sm2P256Mul(&s2, y2, &z13) // s2 = y2 * z1 ^ 3

	if sm2P256ToBig(&u1).Cmp(sm2P256ToBig(&u2)) == 0 &&
		sm2P256ToBig(&s1).Cmp(sm2P256ToBig(&s2)) == 0 {
		sm2P256PointDouble(x1, y1, z1, x1, y1, z1)
	}

	sm2P256Sub(&h, &u2, &u1) // h = u2 - u1
	sm2P256Sub(&r, &s2, &s1) // r = s2 - s1

	sm2P256Square(&r2, &r) // r2 = r ^ 2
	sm2P256Square(&h2, &h) // h2 = h ^ 2

	sm2P256Mul(&tm, &h2, &h) // tm = h ^ 3
	sm2P256Sub(x3, &r2, &tm)
	sm2P256Mul(&tm, &u1, &h2)
	sm2P256Scalar(&tm, 2)   // tm = 2 * (u1 * h ^ 2)
	sm2P256Sub(x3, x3, &tm) // x3 = r ^ 2 - h ^ 3 - 2 * u1 * h ^ 2

	sm2P256Mul(&tm, &u1, &h2) // tm = u1 * h ^ 2
	sm2P256Sub(&tm, &tm, x3)  // tm = u1 * h ^ 2 - x3
	sm2P256Mul(y3, &r, &tm)
	sm2P256Mul(&tm, &h2, &h)  // tm = h ^ 3
	sm2P256Mul(&tm, &tm, &s1) // tm = s1 * h ^ 3
	sm2P256Sub(y3, y3, &tm)   // y3 = r * (u1 * h ^ 2 - x3) - s1 * h ^ 3

	sm2P256Mul(z3, z1, z2)
	sm2P256Mul(z3, z3, &h) // z3 = z1 * z3 * h
}

func sm2P256PointDouble(x3, y3, z3, x, y, z *sm2P256FieldElement) {
	var s, m, m2, x2, y2, z2, z4, y4, az4 sm2P256FieldElement

	sm2P256Square(&x2, x) // x2 = x ^ 2
	sm2P256Square(&y2, y) // y2 = y ^ 2
	sm2P256Square(&z2, z) // z2 = z ^ 2

	sm2P256Square(&z4, z)   // z4 = z ^ 2
	sm2P256Mul(&z4, &z4, z) // z4 = z ^ 3
	sm2P256Mul(&z4, &z4, z) // z4 = z ^ 4

	sm2P256Square(&y4, y)   // y4 = y ^ 2
	sm2P256Mul(&y4, &y4, y) // y4 = y ^ 3
	sm2P256Mul(&y4, &y4, y) // y4 = y ^ 4
	sm2P256Scalar(&y4, 8)   // y4 = 8 * y ^ 4

	sm2P256Mul(&s, x, &y2)
	sm2P256Scalar(&s, 4) // s = 4 * x * y ^ 2

	sm2P256Dup(&m, &x2)
	sm2P256Scalar(&m, 3)
	sm2P256Mul(&az4, &sm2P256.a, &z4)
	sm2P256Add(&m, &m, &az4) // m = 3 * x ^ 2 + a * z ^ 4

	sm2P256Square(&m2, &m) // m2 = m ^ 2

	sm2P256Add(z3, y, z)
	sm2P256Square(z3, z3)
	sm2P256Sub(z3, z3, &z2)
	sm2P256Sub(z3, z3, &y2) // z' = (y + z) ^2 - z ^ 2 - y ^ 2

	sm2P256Sub(x3, &m2, &s)
	sm2P256Sub(x3, x3, &s) // x' = m2 - 2 * s

	sm2P256Sub(y3, &s, x3)
	sm2P256Mul(y3, y3, &m)
	sm2P256Sub(y3, y3, &y4) // y' = m * (s - x') - 8 * y ^ 4
}

// p256Zero31 is 0 mod p.
var sm2P256Zero31 = sm2P256FieldElement{0x7FFFFFF8, 0x3FFFFFFC, 0x800003FC, 0x3FFFDFFC, 0x7FFFFFFC, 0x3FFFFFFC, 0x7FFFFFFC, 0x37FFFFFC, 0x7FFFFFFC}

// c = a + b
func sm2P256Add(c, a, b *sm2P256FieldElement) {
	carry := uint32(0)
	for i := 0; ; i++ {
		c[i] = a[i] + b[i]
		c[i] += carry
		carry = c[i] >> 29
		c[i] &= bottom29Bits
		i++
		if i == 9 {
			break
		}
		c[i] = a[i] + b[i]
		c[i] += carry
		carry = c[i] >> 28
		c[i] &= bottom28Bits
	}
	sm2P256ReduceCarry(c, carry)
}

// c = a - b
func sm2P256Sub(c, a, b *sm2P256FieldElement) {
	var carry uint32

	for i := 0; ; i++ {
		c[i] = a[i] - b[i]
		c[i] += sm2P256Zero31[i]
		c[i] += carry
		carry = c[i] >> 29
		c[i] &= bottom29Bits
		i++
		if i == 9 {
			break
		}
		c[i] = a[i] - b[i]
		c[i] += sm2P256Zero31[i]
		c[i] += carry
		carry = c[i] >> 28
		c[i] &= bottom28Bits
	}
	sm2P256ReduceCarry(c, carry)
}

// c = a * b
func sm2P256Mul(c, a, b *sm2P256FieldElement) {
	var tmp sm2P256LargeFieldElement

	tmp[0] = uint64(a[0]) * uint64(b[0])
	tmp[1] = uint64(a[0])*(uint64(b[1])<<0) +
		uint64(a[1])*(uint64(b[0])<<0)
	tmp[2] = uint64(a[0])*(uint64(b[2])<<0) +
		uint64(a[1])*(uint64(b[1])<<1) +
		uint64(a[2])*(uint64(b[0])<<0)
	tmp[3] = uint64(a[0])*(uint64(b[3])<<0) +
		uint64(a[1])*(uint64(b[2])<<0) +
		uint64(a[2])*(uint64(b[1])<<0) +
		uint64(a[3])*(uint64(b[0])<<0)
	tmp[4] = uint64(a[0])*(uint64(b[4])<<0) +
		uint64(a[1])*(uint64(b[3])<<1) +
		uint64(a[2])*(uint64(b[2])<<0) +
		uint64(a[3])*(uint64(b[1])<<1) +
		uint64(a[4])*(uint64(b[0])<<0)
	tmp[5] = uint64(a[0])*(uint64(b[5])<<0) +
		uint64(a[1])*(uint64(b[4])<<0) +
		uint64(a[2])*(uint64(b[3])<<0) +
		uint64(a[3])*(uint64(b[2])<<0) +
		uint64(a[4])*(uint64(b[1])<<0) +
		uint64(a[5])*(uint64(b[0])<<0)
	tmp[6] = uint64(a[0])*(uint64(b[6])<<0) +
		uint64(a[1])*(uint64(b[5])<<1) +
		uint64(a[2])*(uint64(b[4])<<0) +
		uint64(a[3])*(uint64(b[3])<<1) +
		uint64(a[4])*(uint64(b[2])<<0) +
		uint64(a[5])*(uint64(b[1])<<1) +
		uint64(a[6])*(uint64(b[0])<<0)
	tmp[7] = uint64(a[0])*(uint64(b[7])<<0) +
		uint64(a[1])*(uint64(b[6])<<0) +
		uint64(a[2])*(uint64(b[5])<<0) +
		uint64(a[3])*(uint64(b[4])<<0) +
		uint64(a[4])*(uint64(b[3])<<0) +
		uint64(a[5])*(uint64(b[2])<<0) +
		uint64(a[6])*(uint64(b[1])<<0) +
		uint64(a[7])*(uint64(b[0])<<0)
	// tmp[8] has the greatest value but doesn't overflow. See logic in
	// p256Square.
	tmp[8] = uint64(a[0])*(uint64(b[8])<<0) +
		uint64(a[1])*(uint64(b[7])<<1) +
		uint64(a[2])*(uint64(b[6])<<0) +
		uint64(a[3])*(uint64(b[5])<<1) +
		uint64(a[4])*(uint64(b[4])<<0) +
		uint64(a[5])*(uint64(b[3])<<1) +
		uint64(a[6])*(uint64(b[2])<<0) +
		uint64(a[7])*(uint64(b[1])<<1) +
		uint64(a[8])*(uint64(b[0])<<0)
	tmp[9] = uint64(a[1])*(uint64(b[8])<<0) +
		uint64(a[2])*(uint64(b[7])<<0) +
		uint64(a[3])*(uint64(b[6])<<0) +
		uint64(a[4])*(uint64(b[5])<<0) +
		uint64(a[5])*(uint64(b[4])<<0) +
		uint64(a[6])*(uint64(b[3])<<0) +
		uint64(a[7])*(uint64(b[2])<<0) +
		uint64(a[8])*(uint64(b[1])<<0)
	tmp[10] = uint64(a[2])*(uint64(b[8])<<0) +
		uint64(a[3])*(uint64(b[7])<<1) +
		uint64(a[4])*(uint64(b[6])<<0) +
		uint64(a[5])*(uint64(b[5])<<1) +
		uint64(a[6])*(uint64(b[4])<<0) +
		uint64(a[7])*(uint64(b[3])<<1) +
		uint64(a[8])*(uint64(b[2])<<0)
	tmp[11] = uint64(a[3])*(uint64(b[8])<<0) +
		uint64(a[4])*(uint64(b[7])<<0) +
		uint64(a[5])*(uint64(b[6])<<0) +
		uint64(a[6])*(uint64(b[5])<<0) +
		uint64(a[7])*(uint64(b[4])<<0) +
		uint64(a[8])*(uint64(b[3])<<0)
	tmp[12] = uint64(a[4])*(uint64(b[8])<<0) +
		uint64(a[5])*(uint64(b[7])<<1) +
		uint64(a[6])*(uint64(b[6])<<0) +
		uint64(a[7])*(uint64(b[5])<<1) +
		uint64(a[8])*(uint64(b[4])<<0)
	tmp[13] = uint64(a[5])*(uint64(b[8])<<0) +
		uint64(a[6])*(uint64(b[7])<<0) +
		uint64(a[7])*(uint64(b[6])<<0) +
		uint64(a[8])*(uint64(b[5])<<0)
	tmp[14] = uint64(a[6])*(uint64(b[8])<<0) +
		uint64(a[7])*(uint64(b[7])<<1) +
		uint64(a[8])*(uint64(b[6])<<0)
	tmp[15] = uint64(a[7])*(uint64(b[8])<<0) +
		uint64(a[8])*(uint64(b[7])<<0)
	tmp[16] = uint64(a[8]) * (uint64(b[8]) << 0)
	sm2P256ReduceDegree(c, &tmp)
}

// b = a * a
func sm2P256Square(b, a *sm2P256FieldElement) {
	var tmp sm2P256LargeFieldElement

	tmp[0] = uint64(a[0]) * uint64(a[0])
	tmp[1] = uint64(a[0]) * (uint64(a[1]) << 1)
	tmp[2] = uint64(a[0])*(uint64(a[2])<<1) +
		uint64(a[1])*(uint64(a[1])<<1)
	tmp[3] = uint64(a[0])*(uint64(a[3])<<1) +
		uint64(a[1])*(uint64(a[2])<<1)
	tmp[4] = uint64(a[0])*(uint64(a[4])<<1) +
		uint64(a[1])*(uint64(a[3])<<2) +
		uint64(a[2])*uint64(a[2])
	tmp[5] = uint64(a[0])*(uint64(a[5])<<1) +
		uint64(a[1])*(uint64(a[4])<<1) +
		uint64(a[2])*(uint64(a[3])<<1)
	tmp[6] = uint64(a[0])*(uint64(a[6])<<1) +
		uint64(a[1])*(uint64(a[5])<<2) +
		uint64(a[2])*(uint64(a[4])<<1) +
		uint64(a[3])*(uint64(a[3])<<1)
	tmp[7] = uint64(a[0])*(uint64(a[7])<<1) +
		uint64(a[1])*(uint64(a[6])<<1) +
		uint64(a[2])*(uint64(a[5])<<1) +
		uint64(a[3])*(uint64(a[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(a[0])*(uint64(a[8])<<1) +
		uint64(a[1])*(uint64(a[7])<<2) +
		uint64(a[2])*(uint64(a[6])<<1) +
		uint64(a[3])*(uint64(a[5])<<2) +
		uint64(a[4])*uint64(a[4])
	tmp[9] = uint64(a[1])*(uint64(a[8])<<1) +
		uint64(a[2])*(uint64(a[7])<<1) +
		uint64(a[3])*(uint64(a[6])<<1) +
		uint64(a[4])*(uint64(a[5])<<1)
	tmp[10] = uint64(a[2])*(uint64(a[8])<<1) +
		uint64(a[3])*(uint64(a[7])<<2) +
		uint64(a[4])*(uint64(a[6])<<1) +
		uint64(a[5])*(uint64(a[5])<<1)
	tmp[11] = uint64(a[3])*(uint64(a[8])<<1) +
		uint64(a[4])*(uint64(a[7])<<1) +
		uint64(a[5])*(uint64(a[6])<<1)
	tmp[12] = uint64(a[4])*(uint64(a[8])<<1) +
		uint64(a[5])*(uint64(a[7])<<2) +
		uint64(a[6])*uint64(a[6])
	tmp[13] = uint64(a[5])*(uint64(a[8])<<1) +
		uint64(a[6])*(uint64(a[7])<<1)
	tmp[14] = uint64(a[6])*(uint64(a[8])<<1) +
		uint64(a[7])*(uint64(a[7])<<1)
	tmp[15] = uint64(a[7]) * (uint64(a[8]) << 1)
	tmp[16] = uint64(a[8]) * uint64(a[8])
	sm2P256ReduceDegree(b, &tmp)
}

// 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
}

var sm2P256Carry = [8 * 9]uint32{
	0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0,
	0x2, 0x0, 0x1FFFFF00, 0x7FF, 0x0, 0x0, 0x0, 0x2000000, 0x0,
	0x4, 0x0, 0x1FFFFE00, 0xFFF, 0x0, 0x0, 0x0, 0x4000000, 0x0,
	0x6, 0x0, 0x1FFFFD00, 0x17FF, 0x0, 0x0, 0x0, 0x6000000, 0x0,
	0x8, 0x0, 0x1FFFFC00, 0x1FFF, 0x0, 0x0, 0x0, 0x8000000, 0x0,
	0xA, 0x0, 0x1FFFFB00, 0x27FF, 0x0, 0x0, 0x0, 0xA000000, 0x0,
	0xC, 0x0, 0x1FFFFA00, 0x2FFF, 0x0, 0x0, 0x0, 0xC000000, 0x0,
	0xE, 0x0, 0x1FFFF900, 0x37FF, 0x0, 0x0, 0x0, 0xE000000, 0x0,
}

// carry < 2 ^ 3
func sm2P256ReduceCarry(a *sm2P256FieldElement, carry uint32) {
	a[0] += sm2P256Carry[carry*9+0]
	a[2] += sm2P256Carry[carry*9+2]
	a[3] += sm2P256Carry[carry*9+3]
	a[7] += sm2P256Carry[carry*9+7]
}

func sm2P256ReduceDegree(a *sm2P256FieldElement, b *sm2P256LargeFieldElement) {
	var tmp [18]uint32
	var carry, x, xMask uint32

	// tmp
	// 0  | 1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  |  9 | 10 ...
	// 29 | 28 | 29 | 28 | 29 | 28 | 29 | 28 | 29 | 28 | 29 ...
	tmp[0] = uint32(b[0]) & bottom29Bits
	tmp[1] = uint32(b[0]) >> 29
	tmp[1] |= (uint32(b[0]>>32) << 3) & bottom28Bits
	tmp[1] += uint32(b[1]) & bottom28Bits
	carry = tmp[1] >> 28
	tmp[1] &= bottom28Bits
	for i := 2; i < 17; i++ {
		tmp[i] = (uint32(b[i-2] >> 32)) >> 25
		tmp[i] += (uint32(b[i-1])) >> 28
		tmp[i] += (uint32(b[i-1]>>32) << 4) & bottom29Bits
		tmp[i] += uint32(b[i]) & bottom29Bits
		tmp[i] += carry
		carry = tmp[i] >> 29
		tmp[i] &= bottom29Bits

		i++
		if i == 17 {
			break
		}
		tmp[i] = uint32(b[i-2]>>32) >> 25
		tmp[i] += uint32(b[i-1]) >> 29
		tmp[i] += ((uint32(b[i-1] >> 32)) << 3) & bottom28Bits
		tmp[i] += uint32(b[i]) & bottom28Bits
		tmp[i] += carry
		carry = tmp[i] >> 28
		tmp[i] &= bottom28Bits
	}
	tmp[17] = uint32(b[15]>>32) >> 25
	tmp[17] += uint32(b[16]) >> 29
	tmp[17] += uint32(b[16]>>32) << 3
	tmp[17] += carry

	for i := 0; ; i += 2 {

		tmp[i+1] += tmp[i] >> 29
		x = tmp[i] & bottom29Bits
		tmp[i] = 0
		if x > 0 {
			set4 := uint32(0)
			set7 := uint32(0)
			xMask = nonZeroToAllOnes(x)
			tmp[i+2] += (x << 7) & bottom29Bits
			tmp[i+3] += x >> 22
			if tmp[i+3] < 0x10000000 {
				set4 = 1
				tmp[i+3] += 0x10000000 & xMask
				tmp[i+3] -= (x << 10) & bottom28Bits
			} else {
				tmp[i+3] -= (x << 10) & bottom28Bits
			}
			if tmp[i+4] < 0x20000000 {
				tmp[i+4] += 0x20000000 & xMask
				tmp[i+4] -= set4 // 借位
				tmp[i+4] -= x >> 18
				if tmp[i+5] < 0x10000000 {
					tmp[i+5] += 0x10000000 & xMask
					tmp[i+5] -= 1 // 借位
					if tmp[i+6] < 0x20000000 {
						set7 = 1
						tmp[i+6] += 0x20000000 & xMask
						tmp[i+6] -= 1 // 借位
					} else {
						tmp[i+6] -= 1 // 借位
					}
				} else {
					tmp[i+5] -= 1
				}
			} else {
				tmp[i+4] -= set4 // 借位
				tmp[i+4] -= x >> 18
			}
			if tmp[i+7] < 0x10000000 {
				tmp[i+7] += 0x10000000 & xMask
				tmp[i+7] -= set7
				tmp[i+7] -= (x << 24) & bottom28Bits
				tmp[i+8] += (x << 28) & bottom29Bits
				if tmp[i+8] < 0x20000000 {
					tmp[i+8] += 0x20000000 & xMask
					tmp[i+8] -= 1
					tmp[i+8] -= x >> 4
					tmp[i+9] += ((x >> 1) - 1) & xMask
				} else {
					tmp[i+8] -= 1
					tmp[i+8] -= x >> 4
					tmp[i+9] += (x >> 1) & xMask
				}
			} else {
				tmp[i+7] -= set7 // 借位
				tmp[i+7] -= (x << 24) & bottom28Bits
				tmp[i+8] += (x << 28) & bottom29Bits
				if tmp[i+8] < 0x20000000 {
					tmp[i+8] += 0x20000000 & xMask
					tmp[i+8] -= x >> 4
					tmp[i+9] += ((x >> 1) - 1) & xMask
				} else {
					tmp[i+8] -= x >> 4
					tmp[i+9] += (x >> 1) & xMask
				}
			}

		}

		if i+1 == 9 {
			break
		}

		tmp[i+2] += tmp[i+1] >> 28
		x = tmp[i+1] & bottom28Bits
		tmp[i+1] = 0
		if x > 0 {
			set5 := uint32(0)
			set8 := uint32(0)
			set9 := uint32(0)
			xMask = nonZeroToAllOnes(x)
			tmp[i+3] += (x << 7) & bottom28Bits
			tmp[i+4] += x >> 21
			if tmp[i+4] < 0x20000000 {
				set5 = 1
				tmp[i+4] += 0x20000000 & xMask
				tmp[i+4] -= (x << 11) & bottom29Bits
			} else {
				tmp[i+4] -= (x << 11) & bottom29Bits
			}
			if tmp[i+5] < 0x10000000 {
				tmp[i+5] += 0x10000000 & xMask
				tmp[i+5] -= set5 // 借位
				tmp[i+5] -= x >> 18
				if tmp[i+6] < 0x20000000 {
					tmp[i+6] += 0x20000000 & xMask
					tmp[i+6] -= 1 // 借位
					if tmp[i+7] < 0x10000000 {
						set8 = 1
						tmp[i+7] += 0x10000000 & xMask
						tmp[i+7] -= 1 // 借位
					} else {
						tmp[i+7] -= 1 // 借位
					}
				} else {
					tmp[i+6] -= 1 // 借位
				}
			} else {
				tmp[i+5] -= set5 // 借位
				tmp[i+5] -= x >> 18
			}
			if tmp[i+8] < 0x20000000 {
				set9 = 1
				tmp[i+8] += 0x20000000 & xMask
				tmp[i+8] -= set8
				tmp[i+8] -= (x << 25) & bottom29Bits
			} else {
				tmp[i+8] -= set8
				tmp[i+8] -= (x << 25) & bottom29Bits
			}
			if tmp[i+9] < 0x10000000 {
				tmp[i+9] += 0x10000000 & xMask
				tmp[i+9] -= set9 // 借位
				tmp[i+9] -= x >> 4
				tmp[i+10] += (x - 1) & xMask
			} else {
				tmp[i+9] -= set9 // 借位
				tmp[i+9] -= x >> 4
				tmp[i+10] += x & xMask
			}
		}
	}

	carry = uint32(0)
	for i := 0; i < 8; i++ {
		a[i] = tmp[i+9]
		a[i] += carry
		a[i] += (tmp[i+10] << 28) & bottom29Bits
		carry = a[i] >> 29
		a[i] &= bottom29Bits

		i++
		a[i] = tmp[i+9] >> 1
		a[i] += carry
		carry = a[i] >> 28
		a[i] &= bottom28Bits
	}
	a[8] = tmp[17]
	a[8] += carry
	carry = a[8] >> 29
	a[8] &= bottom29Bits
	sm2P256ReduceCarry(a, carry)
}

// b = a
func sm2P256Dup(b, a *sm2P256FieldElement) {
	*b = *a
}

// X = a * R mod P
func sm2P256FromBig(X *sm2P256FieldElement, a *big.Int) {
	x := new(big.Int).Lsh(a, 257)
	x.Mod(x, sm2P256.P)
	for i := 0; i < 9; i++ {
		if bits := x.Bits(); len(bits) > 0 {
			X[i] = uint32(bits[0]) & bottom29Bits
		} else {
			X[i] = 0
		}
		x.Rsh(x, 29)
		i++
		if i == 9 {
			break
		}
		if bits := x.Bits(); len(bits) > 0 {
			X[i] = uint32(bits[0]) & bottom28Bits
		} else {
			X[i] = 0
		}
		x.Rsh(x, 28)
	}
}

// X = r * R mod P
// r = X * R' mod P
func sm2P256ToBig(X *sm2P256FieldElement) *big.Int {
	r, tm := new(big.Int), new(big.Int)
	r.SetInt64(int64(X[8]))
	for i := 7; i >= 0; i-- {
		if (i & 1) == 0 {
			r.Lsh(r, 29)
		} else {
			r.Lsh(r, 28)
		}
		tm.SetInt64(int64(X[i]))
		r.Add(r, tm)
	}
	r.Mul(r, sm2P256.RInverse)
	r.Mod(r, sm2P256.P)
	return r
}
func WNafReversed(wnaf []int8) []int8 {
	// wnafRev := make([]int8, len(wnaf), len(wnaf))
	wnafRev := make([]int8, len(wnaf))
	for i, v := range wnaf {
		wnafRev[len(wnaf)-(1+i)] = v
	}
	return wnafRev
}
func sm2GenrateWNaf(b []byte) []int8 {
	n := new(big.Int).SetBytes(b)
	var k *big.Int
	if n.Cmp(sm2P256.N) >= 0 {
		n.Mod(n, sm2P256.N)
		k = n
	} else {
		k = n
	}
	// wnaf := make([]int8, k.BitLen()+1, k.BitLen()+1)
	wnaf := make([]int8, k.BitLen()+1)
	if k.Sign() == 0 {
		return wnaf
	}
	var width, pow2, sign int
	width, pow2, sign = 4, 16, 8
	var mask int64 = 15
	var carry bool
	var length, pos int
	for pos <= k.BitLen() {
		if k.Bit(pos) == boolToUint(carry) {
			pos++
			continue
		}
		k.Rsh(k, uint(pos))
		var digit int
		digit = int(k.Int64() & mask)
		if carry {
			digit++
		}
		carry = (digit & sign) != 0
		if carry {
			digit -= pow2
		}
		length += pos
		wnaf[length] = int8(digit)
		pos = width
	}
	if len(wnaf) > length+1 {
		// t := make([]int8, length+1, length+1)
		t := make([]int8, length+1)
		copy(t, wnaf[0:length+1])
		wnaf = t
	}
	return wnaf
}
func boolToUint(b bool) uint {
	if b {
		return 1
	}
	return 0
}
func abs(a int8) uint32 {
	if a < 0 {
		return uint32(-a)
	}
	return uint32(a)
}

func sm2P256ScalarMult(xOut, yOut, zOut, x, y *sm2P256FieldElement, scalar []int8) {
	var precomp [16][3]sm2P256FieldElement
	var px, py, pz, tx, ty, tz sm2P256FieldElement
	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] = sm2P256Factor[1]

	for i := 2; i < 8; i += 2 {
		sm2P256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2])
		sm2P256PointAddMixed(&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)
	var zeroes int16
	for i := 0; i < len(scalar); i++ {
		if scalar[i] == 0 {
			zeroes++
			continue
		}
		if zeroes > 0 {
			for ; zeroes > 0; zeroes-- {
				sm2P256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
			}
		}
		index = abs(scalar[i])
		sm2P256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
		sm2P256SelectJacobianPoint(&px, &py, &pz, &precomp, index)
		if scalar[i] > 0 {
			sm2P256PointAdd(xOut, yOut, zOut, &px, &py, &pz, &tx, &ty, &tz)
		} else {
			sm2P256PointSub(xOut, yOut, zOut, &px, &py, &pz, &tx, &ty, &tz)
		}
		sm2P256CopyConditional(xOut, &px, nIsInfinityMask)
		sm2P256CopyConditional(yOut, &py, nIsInfinityMask)
		sm2P256CopyConditional(zOut, &pz, nIsInfinityMask)
		pIsNoninfiniteMask = nonZeroToAllOnes(index)
		mask = pIsNoninfiniteMask & ^nIsInfinityMask
		sm2P256CopyConditional(xOut, &tx, mask)
		sm2P256CopyConditional(yOut, &ty, mask)
		sm2P256CopyConditional(zOut, &tz, mask)
		nIsInfinityMask &^= pIsNoninfiniteMask
	}
	if zeroes > 0 {
		for ; zeroes > 0; zeroes-- {
			sm2P256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
		}
	}
}