1// Copyright 2022 The Go Authors. All rights reserved. 2// Use of this source code is governed by a BSD-style 3// license that can be found in the LICENSE file. 4 5// Code generated by generate.go. DO NOT EDIT. 6 7//go:build (!amd64 && !arm64 && !ppc64le && !s390x) || purego 8 9package nistec 10 11import ( 12 "crypto/internal/nistec/fiat" 13 "crypto/subtle" 14 "errors" 15 "sync" 16) 17 18// p256ElementLength is the length of an element of the base or scalar field, 19// which have the same bytes length for all NIST P curves. 20const p256ElementLength = 32 21 22// P256Point is a P256 point. The zero value is NOT valid. 23type P256Point struct { 24 // The point is represented in projective coordinates (X:Y:Z), 25 // where x = X/Z and y = Y/Z. 26 x, y, z *fiat.P256Element 27} 28 29// NewP256Point returns a new P256Point representing the point at infinity point. 30func NewP256Point() *P256Point { 31 return &P256Point{ 32 x: new(fiat.P256Element), 33 y: new(fiat.P256Element).One(), 34 z: new(fiat.P256Element), 35 } 36} 37 38// SetGenerator sets p to the canonical generator and returns p. 39func (p *P256Point) SetGenerator() *P256Point { 40 p.x.SetBytes([]byte{0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, 0x77, 0x3, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0, 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96}) 41 p.y.SetBytes([]byte{0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0xf, 0x9e, 0x16, 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce, 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}) 42 p.z.One() 43 return p 44} 45 46// Set sets p = q and returns p. 47func (p *P256Point) Set(q *P256Point) *P256Point { 48 p.x.Set(q.x) 49 p.y.Set(q.y) 50 p.z.Set(q.z) 51 return p 52} 53 54// SetBytes sets p to the compressed, uncompressed, or infinity value encoded in 55// b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on 56// the curve, it returns nil and an error, and the receiver is unchanged. 57// Otherwise, it returns p. 58func (p *P256Point) SetBytes(b []byte) (*P256Point, error) { 59 switch { 60 // Point at infinity. 61 case len(b) == 1 && b[0] == 0: 62 return p.Set(NewP256Point()), nil 63 64 // Uncompressed form. 65 case len(b) == 1+2*p256ElementLength && b[0] == 4: 66 x, err := new(fiat.P256Element).SetBytes(b[1 : 1+p256ElementLength]) 67 if err != nil { 68 return nil, err 69 } 70 y, err := new(fiat.P256Element).SetBytes(b[1+p256ElementLength:]) 71 if err != nil { 72 return nil, err 73 } 74 if err := p256CheckOnCurve(x, y); err != nil { 75 return nil, err 76 } 77 p.x.Set(x) 78 p.y.Set(y) 79 p.z.One() 80 return p, nil 81 82 // Compressed form. 83 case len(b) == 1+p256ElementLength && (b[0] == 2 || b[0] == 3): 84 x, err := new(fiat.P256Element).SetBytes(b[1:]) 85 if err != nil { 86 return nil, err 87 } 88 89 // y² = x³ - 3x + b 90 y := p256Polynomial(new(fiat.P256Element), x) 91 if !p256Sqrt(y, y) { 92 return nil, errors.New("invalid P256 compressed point encoding") 93 } 94 95 // Select the positive or negative root, as indicated by the least 96 // significant bit, based on the encoding type byte. 97 otherRoot := new(fiat.P256Element) 98 otherRoot.Sub(otherRoot, y) 99 cond := y.Bytes()[p256ElementLength-1]&1 ^ b[0]&1 100 y.Select(otherRoot, y, int(cond)) 101 102 p.x.Set(x) 103 p.y.Set(y) 104 p.z.One() 105 return p, nil 106 107 default: 108 return nil, errors.New("invalid P256 point encoding") 109 } 110} 111 112var _p256B *fiat.P256Element 113var _p256BOnce sync.Once 114 115func p256B() *fiat.P256Element { 116 _p256BOnce.Do(func() { 117 _p256B, _ = new(fiat.P256Element).SetBytes([]byte{0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc, 0x65, 0x1d, 0x6, 0xb0, 0xcc, 0x53, 0xb0, 0xf6, 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b}) 118 }) 119 return _p256B 120} 121 122// p256Polynomial sets y2 to x³ - 3x + b, and returns y2. 123func p256Polynomial(y2, x *fiat.P256Element) *fiat.P256Element { 124 y2.Square(x) 125 y2.Mul(y2, x) 126 127 threeX := new(fiat.P256Element).Add(x, x) 128 threeX.Add(threeX, x) 129 y2.Sub(y2, threeX) 130 131 return y2.Add(y2, p256B()) 132} 133 134func p256CheckOnCurve(x, y *fiat.P256Element) error { 135 // y² = x³ - 3x + b 136 rhs := p256Polynomial(new(fiat.P256Element), x) 137 lhs := new(fiat.P256Element).Square(y) 138 if rhs.Equal(lhs) != 1 { 139 return errors.New("P256 point not on curve") 140 } 141 return nil 142} 143 144// Bytes returns the uncompressed or infinity encoding of p, as specified in 145// SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at 146// infinity is shorter than all other encodings. 147func (p *P256Point) Bytes() []byte { 148 // This function is outlined to make the allocations inline in the caller 149 // rather than happen on the heap. 150 var out [1 + 2*p256ElementLength]byte 151 return p.bytes(&out) 152} 153 154func (p *P256Point) bytes(out *[1 + 2*p256ElementLength]byte) []byte { 155 if p.z.IsZero() == 1 { 156 return append(out[:0], 0) 157 } 158 159 zinv := new(fiat.P256Element).Invert(p.z) 160 x := new(fiat.P256Element).Mul(p.x, zinv) 161 y := new(fiat.P256Element).Mul(p.y, zinv) 162 163 buf := append(out[:0], 4) 164 buf = append(buf, x.Bytes()...) 165 buf = append(buf, y.Bytes()...) 166 return buf 167} 168 169// BytesX returns the encoding of the x-coordinate of p, as specified in SEC 1, 170// Version 2.0, Section 2.3.5, or an error if p is the point at infinity. 171func (p *P256Point) BytesX() ([]byte, error) { 172 // This function is outlined to make the allocations inline in the caller 173 // rather than happen on the heap. 174 var out [p256ElementLength]byte 175 return p.bytesX(&out) 176} 177 178func (p *P256Point) bytesX(out *[p256ElementLength]byte) ([]byte, error) { 179 if p.z.IsZero() == 1 { 180 return nil, errors.New("P256 point is the point at infinity") 181 } 182 183 zinv := new(fiat.P256Element).Invert(p.z) 184 x := new(fiat.P256Element).Mul(p.x, zinv) 185 186 return append(out[:0], x.Bytes()...), nil 187} 188 189// BytesCompressed returns the compressed or infinity encoding of p, as 190// specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the 191// point at infinity is shorter than all other encodings. 192func (p *P256Point) BytesCompressed() []byte { 193 // This function is outlined to make the allocations inline in the caller 194 // rather than happen on the heap. 195 var out [1 + p256ElementLength]byte 196 return p.bytesCompressed(&out) 197} 198 199func (p *P256Point) bytesCompressed(out *[1 + p256ElementLength]byte) []byte { 200 if p.z.IsZero() == 1 { 201 return append(out[:0], 0) 202 } 203 204 zinv := new(fiat.P256Element).Invert(p.z) 205 x := new(fiat.P256Element).Mul(p.x, zinv) 206 y := new(fiat.P256Element).Mul(p.y, zinv) 207 208 // Encode the sign of the y coordinate (indicated by the least significant 209 // bit) as the encoding type (2 or 3). 210 buf := append(out[:0], 2) 211 buf[0] |= y.Bytes()[p256ElementLength-1] & 1 212 buf = append(buf, x.Bytes()...) 213 return buf 214} 215 216// Add sets q = p1 + p2, and returns q. The points may overlap. 217func (q *P256Point) Add(p1, p2 *P256Point) *P256Point { 218 // Complete addition formula for a = -3 from "Complete addition formulas for 219 // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. 220 221 t0 := new(fiat.P256Element).Mul(p1.x, p2.x) // t0 := X1 * X2 222 t1 := new(fiat.P256Element).Mul(p1.y, p2.y) // t1 := Y1 * Y2 223 t2 := new(fiat.P256Element).Mul(p1.z, p2.z) // t2 := Z1 * Z2 224 t3 := new(fiat.P256Element).Add(p1.x, p1.y) // t3 := X1 + Y1 225 t4 := new(fiat.P256Element).Add(p2.x, p2.y) // t4 := X2 + Y2 226 t3.Mul(t3, t4) // t3 := t3 * t4 227 t4.Add(t0, t1) // t4 := t0 + t1 228 t3.Sub(t3, t4) // t3 := t3 - t4 229 t4.Add(p1.y, p1.z) // t4 := Y1 + Z1 230 x3 := new(fiat.P256Element).Add(p2.y, p2.z) // X3 := Y2 + Z2 231 t4.Mul(t4, x3) // t4 := t4 * X3 232 x3.Add(t1, t2) // X3 := t1 + t2 233 t4.Sub(t4, x3) // t4 := t4 - X3 234 x3.Add(p1.x, p1.z) // X3 := X1 + Z1 235 y3 := new(fiat.P256Element).Add(p2.x, p2.z) // Y3 := X2 + Z2 236 x3.Mul(x3, y3) // X3 := X3 * Y3 237 y3.Add(t0, t2) // Y3 := t0 + t2 238 y3.Sub(x3, y3) // Y3 := X3 - Y3 239 z3 := new(fiat.P256Element).Mul(p256B(), t2) // Z3 := b * t2 240 x3.Sub(y3, z3) // X3 := Y3 - Z3 241 z3.Add(x3, x3) // Z3 := X3 + X3 242 x3.Add(x3, z3) // X3 := X3 + Z3 243 z3.Sub(t1, x3) // Z3 := t1 - X3 244 x3.Add(t1, x3) // X3 := t1 + X3 245 y3.Mul(p256B(), y3) // Y3 := b * Y3 246 t1.Add(t2, t2) // t1 := t2 + t2 247 t2.Add(t1, t2) // t2 := t1 + t2 248 y3.Sub(y3, t2) // Y3 := Y3 - t2 249 y3.Sub(y3, t0) // Y3 := Y3 - t0 250 t1.Add(y3, y3) // t1 := Y3 + Y3 251 y3.Add(t1, y3) // Y3 := t1 + Y3 252 t1.Add(t0, t0) // t1 := t0 + t0 253 t0.Add(t1, t0) // t0 := t1 + t0 254 t0.Sub(t0, t2) // t0 := t0 - t2 255 t1.Mul(t4, y3) // t1 := t4 * Y3 256 t2.Mul(t0, y3) // t2 := t0 * Y3 257 y3.Mul(x3, z3) // Y3 := X3 * Z3 258 y3.Add(y3, t2) // Y3 := Y3 + t2 259 x3.Mul(t3, x3) // X3 := t3 * X3 260 x3.Sub(x3, t1) // X3 := X3 - t1 261 z3.Mul(t4, z3) // Z3 := t4 * Z3 262 t1.Mul(t3, t0) // t1 := t3 * t0 263 z3.Add(z3, t1) // Z3 := Z3 + t1 264 265 q.x.Set(x3) 266 q.y.Set(y3) 267 q.z.Set(z3) 268 return q 269} 270 271// Double sets q = p + p, and returns q. The points may overlap. 272func (q *P256Point) Double(p *P256Point) *P256Point { 273 // Complete addition formula for a = -3 from "Complete addition formulas for 274 // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. 275 276 t0 := new(fiat.P256Element).Square(p.x) // t0 := X ^ 2 277 t1 := new(fiat.P256Element).Square(p.y) // t1 := Y ^ 2 278 t2 := new(fiat.P256Element).Square(p.z) // t2 := Z ^ 2 279 t3 := new(fiat.P256Element).Mul(p.x, p.y) // t3 := X * Y 280 t3.Add(t3, t3) // t3 := t3 + t3 281 z3 := new(fiat.P256Element).Mul(p.x, p.z) // Z3 := X * Z 282 z3.Add(z3, z3) // Z3 := Z3 + Z3 283 y3 := new(fiat.P256Element).Mul(p256B(), t2) // Y3 := b * t2 284 y3.Sub(y3, z3) // Y3 := Y3 - Z3 285 x3 := new(fiat.P256Element).Add(y3, y3) // X3 := Y3 + Y3 286 y3.Add(x3, y3) // Y3 := X3 + Y3 287 x3.Sub(t1, y3) // X3 := t1 - Y3 288 y3.Add(t1, y3) // Y3 := t1 + Y3 289 y3.Mul(x3, y3) // Y3 := X3 * Y3 290 x3.Mul(x3, t3) // X3 := X3 * t3 291 t3.Add(t2, t2) // t3 := t2 + t2 292 t2.Add(t2, t3) // t2 := t2 + t3 293 z3.Mul(p256B(), z3) // Z3 := b * Z3 294 z3.Sub(z3, t2) // Z3 := Z3 - t2 295 z3.Sub(z3, t0) // Z3 := Z3 - t0 296 t3.Add(z3, z3) // t3 := Z3 + Z3 297 z3.Add(z3, t3) // Z3 := Z3 + t3 298 t3.Add(t0, t0) // t3 := t0 + t0 299 t0.Add(t3, t0) // t0 := t3 + t0 300 t0.Sub(t0, t2) // t0 := t0 - t2 301 t0.Mul(t0, z3) // t0 := t0 * Z3 302 y3.Add(y3, t0) // Y3 := Y3 + t0 303 t0.Mul(p.y, p.z) // t0 := Y * Z 304 t0.Add(t0, t0) // t0 := t0 + t0 305 z3.Mul(t0, z3) // Z3 := t0 * Z3 306 x3.Sub(x3, z3) // X3 := X3 - Z3 307 z3.Mul(t0, t1) // Z3 := t0 * t1 308 z3.Add(z3, z3) // Z3 := Z3 + Z3 309 z3.Add(z3, z3) // Z3 := Z3 + Z3 310 311 q.x.Set(x3) 312 q.y.Set(y3) 313 q.z.Set(z3) 314 return q 315} 316 317// Select sets q to p1 if cond == 1, and to p2 if cond == 0. 318func (q *P256Point) Select(p1, p2 *P256Point, cond int) *P256Point { 319 q.x.Select(p1.x, p2.x, cond) 320 q.y.Select(p1.y, p2.y, cond) 321 q.z.Select(p1.z, p2.z, cond) 322 return q 323} 324 325// A p256Table holds the first 15 multiples of a point at offset -1, so [1]P 326// is at table[0], [15]P is at table[14], and [0]P is implicitly the identity 327// point. 328type p256Table [15]*P256Point 329 330// Select selects the n-th multiple of the table base point into p. It works in 331// constant time by iterating over every entry of the table. n must be in [0, 15]. 332func (table *p256Table) Select(p *P256Point, n uint8) { 333 if n >= 16 { 334 panic("nistec: internal error: p256Table called with out-of-bounds value") 335 } 336 p.Set(NewP256Point()) 337 for i := uint8(1); i < 16; i++ { 338 cond := subtle.ConstantTimeByteEq(i, n) 339 p.Select(table[i-1], p, cond) 340 } 341} 342 343// ScalarMult sets p = scalar * q, and returns p. 344func (p *P256Point) ScalarMult(q *P256Point, scalar []byte) (*P256Point, error) { 345 // Compute a p256Table for the base point q. The explicit NewP256Point 346 // calls get inlined, letting the allocations live on the stack. 347 var table = p256Table{NewP256Point(), NewP256Point(), NewP256Point(), 348 NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point(), 349 NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point(), 350 NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point()} 351 table[0].Set(q) 352 for i := 1; i < 15; i += 2 { 353 table[i].Double(table[i/2]) 354 table[i+1].Add(table[i], q) 355 } 356 357 // Instead of doing the classic double-and-add chain, we do it with a 358 // four-bit window: we double four times, and then add [0-15]P. 359 t := NewP256Point() 360 p.Set(NewP256Point()) 361 for i, byte := range scalar { 362 // No need to double on the first iteration, as p is the identity at 363 // this point, and [N]∞ = ∞. 364 if i != 0 { 365 p.Double(p) 366 p.Double(p) 367 p.Double(p) 368 p.Double(p) 369 } 370 371 windowValue := byte >> 4 372 table.Select(t, windowValue) 373 p.Add(p, t) 374 375 p.Double(p) 376 p.Double(p) 377 p.Double(p) 378 p.Double(p) 379 380 windowValue = byte & 0b1111 381 table.Select(t, windowValue) 382 p.Add(p, t) 383 } 384 385 return p, nil 386} 387 388var p256GeneratorTable *[p256ElementLength * 2]p256Table 389var p256GeneratorTableOnce sync.Once 390 391// generatorTable returns a sequence of p256Tables. The first table contains 392// multiples of G. Each successive table is the previous table doubled four 393// times. 394func (p *P256Point) generatorTable() *[p256ElementLength * 2]p256Table { 395 p256GeneratorTableOnce.Do(func() { 396 p256GeneratorTable = new([p256ElementLength * 2]p256Table) 397 base := NewP256Point().SetGenerator() 398 for i := 0; i < p256ElementLength*2; i++ { 399 p256GeneratorTable[i][0] = NewP256Point().Set(base) 400 for j := 1; j < 15; j++ { 401 p256GeneratorTable[i][j] = NewP256Point().Add(p256GeneratorTable[i][j-1], base) 402 } 403 base.Double(base) 404 base.Double(base) 405 base.Double(base) 406 base.Double(base) 407 } 408 }) 409 return p256GeneratorTable 410} 411 412// ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and 413// returns p. 414func (p *P256Point) ScalarBaseMult(scalar []byte) (*P256Point, error) { 415 if len(scalar) != p256ElementLength { 416 return nil, errors.New("invalid scalar length") 417 } 418 tables := p.generatorTable() 419 420 // This is also a scalar multiplication with a four-bit window like in 421 // ScalarMult, but in this case the doublings are precomputed. The value 422 // [windowValue]G added at iteration k would normally get doubled 423 // (totIterations-k)×4 times, but with a larger precomputation we can 424 // instead add [2^((totIterations-k)×4)][windowValue]G and avoid the 425 // doublings between iterations. 426 t := NewP256Point() 427 p.Set(NewP256Point()) 428 tableIndex := len(tables) - 1 429 for _, byte := range scalar { 430 windowValue := byte >> 4 431 tables[tableIndex].Select(t, windowValue) 432 p.Add(p, t) 433 tableIndex-- 434 435 windowValue = byte & 0b1111 436 tables[tableIndex].Select(t, windowValue) 437 p.Add(p, t) 438 tableIndex-- 439 } 440 441 return p, nil 442} 443 444// p256Sqrt sets e to a square root of x. If x is not a square, p256Sqrt returns 445// false and e is unchanged. e and x can overlap. 446func p256Sqrt(e, x *fiat.P256Element) (isSquare bool) { 447 candidate := new(fiat.P256Element) 448 p256SqrtCandidate(candidate, x) 449 square := new(fiat.P256Element).Square(candidate) 450 if square.Equal(x) != 1 { 451 return false 452 } 453 e.Set(candidate) 454 return true 455} 456 457// p256SqrtCandidate sets z to a square root candidate for x. z and x must not overlap. 458func p256SqrtCandidate(z, x *fiat.P256Element) { 459 // Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate. 460 // 461 // The sequence of 7 multiplications and 253 squarings is derived from the 462 // following addition chain generated with github.com/mmcloughlin/addchain v0.4.0. 463 // 464 // _10 = 2*1 465 // _11 = 1 + _10 466 // _1100 = _11 << 2 467 // _1111 = _11 + _1100 468 // _11110000 = _1111 << 4 469 // _11111111 = _1111 + _11110000 470 // x16 = _11111111 << 8 + _11111111 471 // x32 = x16 << 16 + x16 472 // return ((x32 << 32 + 1) << 96 + 1) << 94 473 // 474 var t0 = new(fiat.P256Element) 475 476 z.Square(x) 477 z.Mul(x, z) 478 t0.Square(z) 479 for s := 1; s < 2; s++ { 480 t0.Square(t0) 481 } 482 z.Mul(z, t0) 483 t0.Square(z) 484 for s := 1; s < 4; s++ { 485 t0.Square(t0) 486 } 487 z.Mul(z, t0) 488 t0.Square(z) 489 for s := 1; s < 8; s++ { 490 t0.Square(t0) 491 } 492 z.Mul(z, t0) 493 t0.Square(z) 494 for s := 1; s < 16; s++ { 495 t0.Square(t0) 496 } 497 z.Mul(z, t0) 498 for s := 0; s < 32; s++ { 499 z.Square(z) 500 } 501 z.Mul(x, z) 502 for s := 0; s < 96; s++ { 503 z.Square(z) 504 } 505 z.Mul(x, z) 506 for s := 0; s < 94; s++ { 507 z.Square(z) 508 } 509} 510