1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956
|
// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
// Copyright 2013-2014 The btcsuite developers
// Use of this source code is governed by an ISC
// license that can be found in the LICENSE file.
package btcec
// References:
// [SECG]: Recommended Elliptic Curve Domain Parameters
// http://www.secg.org/sec2-v2.pdf
//
// [GECC]: Guide to Elliptic Curve Cryptography (Hankerson, Menezes, Vanstone)
// This package operates, internally, on Jacobian coordinates. For a given
// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
// calculation can be performed within the transform (as in ScalarMult and
// ScalarBaseMult). But even for Add and Double, it's faster to apply and
// reverse the transform than to operate in affine coordinates.
import (
"crypto/elliptic"
"math/big"
"sync"
)
var (
// fieldOne is simply the integer 1 in field representation. It is
// used to avoid needing to create it multiple times during the internal
// arithmetic.
fieldOne = new(fieldVal).SetInt(1)
)
// KoblitzCurve supports a koblitz curve implementation that fits the ECC Curve
// interface from crypto/elliptic.
type KoblitzCurve struct {
*elliptic.CurveParams
q *big.Int
H int // cofactor of the curve.
// byteSize is simply the bit size / 8 and is provided for convenience
// since it is calculated repeatedly.
byteSize int
// bytePoints
bytePoints *[32][256][3]fieldVal
// The next 6 values are used specifically for endomorphism
// optimizations in ScalarMult.
// lambda must fulfill lambda^3 = 1 mod N where N is the order of G.
lambda *big.Int
// beta must fulfill beta^3 = 1 mod P where P is the prime field of the
// curve.
beta *fieldVal
// See the EndomorphismVectors in gensecp256k1.go to see how these are
// derived.
a1 *big.Int
b1 *big.Int
a2 *big.Int
b2 *big.Int
}
// Params returns the parameters for the curve.
func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
return curve.CurveParams
}
// bigAffineToField takes an affine point (x, y) as big integers and converts
// it to an affine point as field values.
func (curve *KoblitzCurve) bigAffineToField(x, y *big.Int) (*fieldVal, *fieldVal) {
x3, y3 := new(fieldVal), new(fieldVal)
x3.SetByteSlice(x.Bytes())
y3.SetByteSlice(y.Bytes())
return x3, y3
}
// fieldJacobianToBigAffine takes a Jacobian point (x, y, z) as field values and
// converts it to an affine point as big integers.
func (curve *KoblitzCurve) fieldJacobianToBigAffine(x, y, z *fieldVal) (*big.Int, *big.Int) {
// Inversions are expensive and both point addition and point doubling
// are faster when working with points that have a z value of one. So,
// if the point needs to be converted to affine, go ahead and normalize
// the point itself at the same time as the calculation is the same.
var zInv, tempZ fieldVal
zInv.Set(z).Inverse() // zInv = Z^-1
tempZ.SquareVal(&zInv) // tempZ = Z^-2
x.Mul(&tempZ) // X = X/Z^2 (mag: 1)
y.Mul(tempZ.Mul(&zInv)) // Y = Y/Z^3 (mag: 1)
z.SetInt(1) // Z = 1 (mag: 1)
// Normalize the x and y values.
x.Normalize()
y.Normalize()
// Convert the field values for the now affine point to big.Ints.
x3, y3 := new(big.Int), new(big.Int)
x3.SetBytes(x.Bytes()[:])
y3.SetBytes(y.Bytes()[:])
return x3, y3
}
// IsOnCurve returns boolean if the point (x,y) is on the curve.
// Part of the elliptic.Curve interface. This function differs from the
// crypto/elliptic algorithm since a = 0 not -3.
func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
// Convert big ints to field values for faster arithmetic.
fx, fy := curve.bigAffineToField(x, y)
// Elliptic curve equation for secp256k1 is: y^2 = x^3 + 7
y2 := new(fieldVal).SquareVal(fy).Normalize()
result := new(fieldVal).SquareVal(fx).Mul(fx).AddInt(7).Normalize()
return y2.Equals(result)
}
// addZ1AndZ2EqualsOne adds two Jacobian points that are already known to have
// z values of 1 and stores the result in (x3, y3, z3). That is to say
// (x1, y1, 1) + (x2, y2, 1) = (x3, y3, z3). It performs faster addition than
// the generic add routine since less arithmetic is needed due to the ability to
// avoid the z value multiplications.
func (curve *KoblitzCurve) addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
//
// In particular it performs the calculations using the following:
// H = X2-X1, HH = H^2, I = 4*HH, J = H*I, r = 2*(Y2-Y1), V = X1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = 2*H
//
// This results in a cost of 4 field multiplications, 2 field squarings,
// 6 field additions, and 5 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography.
x1.Normalize()
y1.Normalize()
x2.Normalize()
y2.Normalize()
if x1.Equals(x2) {
if y1.Equals(y2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, i, j, r, v fieldVal
var negJ, neg2V, negX3 fieldVal
h.Set(x1).Negate(1).Add(x2) // H = X2-X1 (mag: 3)
i.SquareVal(&h).MulInt(4) // I = 4*H^2 (mag: 4)
j.Mul2(&h, &i) // J = H*I (mag: 1)
r.Set(y1).Negate(1).Add(y2).MulInt(2) // r = 2*(Y2-Y1) (mag: 6)
v.Mul2(x1, &i) // V = X1*I (mag: 1)
negJ.Set(&j).Negate(1) // negJ = -J (mag: 2)
neg2V.Set(&v).MulInt(2).Negate(2) // neg2V = -(2*V) (mag: 3)
x3.Set(&r).Square().Add(&negJ).Add(&neg2V) // X3 = r^2-J-2*V (mag: 6)
negX3.Set(x3).Negate(6) // negX3 = -X3 (mag: 7)
j.Mul(y1).MulInt(2).Negate(2) // J = -(2*Y1*J) (mag: 3)
y3.Set(&v).Add(&negX3).Mul(&r).Add(&j) // Y3 = r*(V-X3)-2*Y1*J (mag: 4)
z3.Set(&h).MulInt(2) // Z3 = 2*H (mag: 6)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// addZ1EqualsZ2 adds two Jacobian points that are already known to have the
// same z value and stores the result in (x3, y3, z3). That is to say
// (x1, y1, z1) + (x2, y2, z1) = (x3, y3, z3). It performs faster addition than
// the generic add routine since less arithmetic is needed due to the known
// equivalence.
func (curve *KoblitzCurve) addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using a slightly modified version
// of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
//
// In particular it performs the calculations using the following:
// A = X2-X1, B = A^2, C=Y2-Y1, D = C^2, E = X1*B, F = X2*B
// X3 = D-E-F, Y3 = C*(E-X3)-Y1*(F-E), Z3 = Z1*A
//
// This results in a cost of 5 field multiplications, 2 field squarings,
// 9 field additions, and 0 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography.
x1.Normalize()
y1.Normalize()
x2.Normalize()
y2.Normalize()
if x1.Equals(x2) {
if y1.Equals(y2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var a, b, c, d, e, f fieldVal
var negX1, negY1, negE, negX3 fieldVal
negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
a.Set(&negX1).Add(x2) // A = X2-X1 (mag: 3)
b.SquareVal(&a) // B = A^2 (mag: 1)
c.Set(&negY1).Add(y2) // C = Y2-Y1 (mag: 3)
d.SquareVal(&c) // D = C^2 (mag: 1)
e.Mul2(x1, &b) // E = X1*B (mag: 1)
negE.Set(&e).Negate(1) // negE = -E (mag: 2)
f.Mul2(x2, &b) // F = X2*B (mag: 1)
x3.Add2(&e, &f).Negate(3).Add(&d) // X3 = D-E-F (mag: 5)
negX3.Set(x3).Negate(5).Normalize() // negX3 = -X3 (mag: 1)
y3.Set(y1).Mul(f.Add(&negE)).Negate(3) // Y3 = -(Y1*(F-E)) (mag: 4)
y3.Add(e.Add(&negX3).Mul(&c)) // Y3 = C*(E-X3)+Y3 (mag: 5)
z3.Mul2(z1, &a) // Z3 = Z1*A (mag: 1)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
}
// addZ2EqualsOne adds two Jacobian points when the second point is already
// known to have a z value of 1 (and the z value for the first point is not 1)
// and stores the result in (x3, y3, z3). That is to say (x1, y1, z1) +
// (x2, y2, 1) = (x3, y3, z3). It performs faster addition than the generic
// add routine since less arithmetic is needed due to the ability to avoid
// multiplications by the second point's z value.
func (curve *KoblitzCurve) addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
//
// In particular it performs the calculations using the following:
// Z1Z1 = Z1^2, U2 = X2*Z1Z1, S2 = Y2*Z1*Z1Z1, H = U2-X1, HH = H^2,
// I = 4*HH, J = H*I, r = 2*(S2-Y1), V = X1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = (Z1+H)^2-Z1Z1-HH
//
// This results in a cost of 7 field multiplications, 4 field squarings,
// 9 field additions, and 4 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity per the group law for elliptic curve cryptography. Since
// any number of Jacobian coordinates can represent the same affine
// point, the x and y values need to be converted to like terms. Due to
// the assumption made for this function that the second point has a z
// value of 1 (z2=1), the first point is already "converted".
var z1z1, u2, s2 fieldVal
x1.Normalize()
y1.Normalize()
z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
if x1.Equals(&u2) {
if y1.Equals(&s2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, hh, i, j, r, rr, v fieldVal
var negX1, negY1, negX3 fieldVal
negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
h.Add2(&u2, &negX1) // H = U2-X1 (mag: 3)
hh.SquareVal(&h) // HH = H^2 (mag: 1)
i.Set(&hh).MulInt(4) // I = 4 * HH (mag: 4)
j.Mul2(&h, &i) // J = H*I (mag: 1)
negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
r.Set(&s2).Add(&negY1).MulInt(2) // r = 2*(S2-Y1) (mag: 6)
rr.SquareVal(&r) // rr = r^2 (mag: 1)
v.Mul2(x1, &i) // V = X1*I (mag: 1)
x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
y3.Set(y1).Mul(&j).MulInt(2).Negate(2) // Y3 = -(2*Y1*J) (mag: 3)
y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
z3.Add2(z1, &h).Square() // Z3 = (Z1+H)^2 (mag: 1)
z3.Add(z1z1.Add(&hh).Negate(2)) // Z3 = Z3-(Z1Z1+HH) (mag: 4)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// addGeneric adds two Jacobian points (x1, y1, z1) and (x2, y2, z2) without any
// assumptions about the z values of the two points and stores the result in
// (x3, y3, z3). That is to say (x1, y1, z1) + (x2, y2, z2) = (x3, y3, z3). It
// is the slowest of the add routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
// To compute the point addition efficiently, this implementation splits
// the equation into intermediate elements which are used to minimize
// the number of field multiplications using the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
//
// In particular it performs the calculations using the following:
// Z1Z1 = Z1^2, Z2Z2 = Z2^2, U1 = X1*Z2Z2, U2 = X2*Z1Z1, S1 = Y1*Z2*Z2Z2
// S2 = Y2*Z1*Z1Z1, H = U2-U1, I = (2*H)^2, J = H*I, r = 2*(S2-S1)
// V = U1*I
// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*S1*J, Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H
//
// This results in a cost of 11 field multiplications, 5 field squarings,
// 9 field additions, and 4 integer multiplications.
// When the x coordinates are the same for two points on the curve, the
// y coordinates either must be the same, in which case it is point
// doubling, or they are opposite and the result is the point at
// infinity. Since any number of Jacobian coordinates can represent the
// same affine point, the x and y values need to be converted to like
// terms.
var z1z1, z2z2, u1, u2, s1, s2 fieldVal
z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
z2z2.SquareVal(z2) // Z2Z2 = Z2^2 (mag: 1)
u1.Set(x1).Mul(&z2z2).Normalize() // U1 = X1*Z2Z2 (mag: 1)
u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
s1.Set(y1).Mul(&z2z2).Mul(z2).Normalize() // S1 = Y1*Z2*Z2Z2 (mag: 1)
s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
if u1.Equals(&u2) {
if s1.Equals(&s2) {
// Since x1 == x2 and y1 == y2, point doubling must be
// done, otherwise the addition would end up dividing
// by zero.
curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
return
}
// Since x1 == x2 and y1 == -y2, the sum is the point at
// infinity per the group law.
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Calculate X3, Y3, and Z3 according to the intermediate elements
// breakdown above.
var h, i, j, r, rr, v fieldVal
var negU1, negS1, negX3 fieldVal
negU1.Set(&u1).Negate(1) // negU1 = -U1 (mag: 2)
h.Add2(&u2, &negU1) // H = U2-U1 (mag: 3)
i.Set(&h).MulInt(2).Square() // I = (2*H)^2 (mag: 2)
j.Mul2(&h, &i) // J = H*I (mag: 1)
negS1.Set(&s1).Negate(1) // negS1 = -S1 (mag: 2)
r.Set(&s2).Add(&negS1).MulInt(2) // r = 2*(S2-S1) (mag: 6)
rr.SquareVal(&r) // rr = r^2 (mag: 1)
v.Mul2(&u1, &i) // V = U1*I (mag: 1)
x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
y3.Mul2(&s1, &j).MulInt(2).Negate(2) // Y3 = -(2*S1*J) (mag: 3)
y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
z3.Add2(z1, z2).Square() // Z3 = (Z1+Z2)^2 (mag: 1)
z3.Add(z1z1.Add(&z2z2).Negate(2)) // Z3 = Z3-(Z1Z1+Z2Z2) (mag: 4)
z3.Mul(&h) // Z3 = Z3*H (mag: 1)
// Normalize the resulting field values to a magnitude of 1 as needed.
x3.Normalize()
y3.Normalize()
}
// addJacobian adds the passed Jacobian points (x1, y1, z1) and (x2, y2, z2)
// together and stores the result in (x3, y3, z3).
func (curve *KoblitzCurve) addJacobian(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
// A point at infinity is the identity according to the group law for
// elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
if (x1.IsZero() && y1.IsZero()) || z1.IsZero() {
x3.Set(x2)
y3.Set(y2)
z3.Set(z2)
return
}
if (x2.IsZero() && y2.IsZero()) || z2.IsZero() {
x3.Set(x1)
y3.Set(y1)
z3.Set(z1)
return
}
// Faster point addition can be achieved when certain assumptions are
// met. For example, when both points have the same z value, arithmetic
// on the z values can be avoided. This section thus checks for these
// conditions and calls an appropriate add function which is accelerated
// by using those assumptions.
z1.Normalize()
z2.Normalize()
isZ1One := z1.Equals(fieldOne)
isZ2One := z2.Equals(fieldOne)
switch {
case isZ1One && isZ2One:
curve.addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
return
case z1.Equals(z2):
curve.addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3)
return
case isZ2One:
curve.addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
return
}
// None of the above assumptions are true, so fall back to generic
// point addition.
curve.addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3)
}
// Add returns the sum of (x1,y1) and (x2,y2). Part of the elliptic.Curve
// interface.
func (curve *KoblitzCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
// A point at infinity is the identity according to the group law for
// elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
if x1.Sign() == 0 && y1.Sign() == 0 {
return x2, y2
}
if x2.Sign() == 0 && y2.Sign() == 0 {
return x1, y1
}
// Convert the affine coordinates from big integers to field values
// and do the point addition in Jacobian projective space.
fx1, fy1 := curve.bigAffineToField(x1, y1)
fx2, fy2 := curve.bigAffineToField(x2, y2)
fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
fOne := new(fieldVal).SetInt(1)
curve.addJacobian(fx1, fy1, fOne, fx2, fy2, fOne, fx3, fy3, fz3)
// Convert the Jacobian coordinate field values back to affine big
// integers.
return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}
// doubleZ1EqualsOne performs point doubling on the passed Jacobian point
// when the point is already known to have a z value of 1 and stores
// the result in (x3, y3, z3). That is to say (x3, y3, z3) = 2*(x1, y1, 1). It
// performs faster point doubling than the generic routine since less arithmetic
// is needed due to the ability to avoid multiplication by the z value.
func (curve *KoblitzCurve) doubleZ1EqualsOne(x1, y1, x3, y3, z3 *fieldVal) {
// This function uses the assumptions that z1 is 1, thus the point
// doubling formulas reduce to:
//
// X3 = (3*X1^2)^2 - 8*X1*Y1^2
// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
// Z3 = 2*Y1
//
// To compute the above efficiently, this implementation splits the
// equation into intermediate elements which are used to minimize the
// number of field multiplications in favor of field squarings which
// are roughly 35% faster than field multiplications with the current
// implementation at the time this was written.
//
// This uses a slightly modified version of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-mdbl-2007-bl
//
// In particular it performs the calculations using the following:
// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
// Z3 = 2*Y1
//
// This results in a cost of 1 field multiplication, 5 field squarings,
// 6 field additions, and 5 integer multiplications.
var a, b, c, d, e, f fieldVal
z3.Set(y1).MulInt(2) // Z3 = 2*Y1 (mag: 2)
a.SquareVal(x1) // A = X1^2 (mag: 1)
b.SquareVal(y1) // B = Y1^2 (mag: 1)
c.SquareVal(&b) // C = B^2 (mag: 1)
b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
f.SquareVal(&e) // F = E^2 (mag: 1)
x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
x3.Add(&f) // X3 = F+X3 (mag: 18)
f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
// Normalize the field values back to a magnitude of 1.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// doubleGeneric performs point doubling on the passed Jacobian point without
// any assumptions about the z value and stores the result in (x3, y3, z3).
// That is to say (x3, y3, z3) = 2*(x1, y1, z1). It is the slowest of the point
// doubling routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) doubleGeneric(x1, y1, z1, x3, y3, z3 *fieldVal) {
// Point doubling formula for Jacobian coordinates for the secp256k1
// curve:
// X3 = (3*X1^2)^2 - 8*X1*Y1^2
// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
// Z3 = 2*Y1*Z1
//
// To compute the above efficiently, this implementation splits the
// equation into intermediate elements which are used to minimize the
// number of field multiplications in favor of field squarings which
// are roughly 35% faster than field multiplications with the current
// implementation at the time this was written.
//
// This uses a slightly modified version of the method shown at:
// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
//
// In particular it performs the calculations using the following:
// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
// Z3 = 2*Y1*Z1
//
// This results in a cost of 1 field multiplication, 5 field squarings,
// 6 field additions, and 5 integer multiplications.
var a, b, c, d, e, f fieldVal
z3.Mul2(y1, z1).MulInt(2) // Z3 = 2*Y1*Z1 (mag: 2)
a.SquareVal(x1) // A = X1^2 (mag: 1)
b.SquareVal(y1) // B = Y1^2 (mag: 1)
c.SquareVal(&b) // C = B^2 (mag: 1)
b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
f.SquareVal(&e) // F = E^2 (mag: 1)
x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
x3.Add(&f) // X3 = F+X3 (mag: 18)
f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
// Normalize the field values back to a magnitude of 1.
x3.Normalize()
y3.Normalize()
z3.Normalize()
}
// doubleJacobian doubles the passed Jacobian point (x1, y1, z1) and stores the
// result in (x3, y3, z3).
func (curve *KoblitzCurve) doubleJacobian(x1, y1, z1, x3, y3, z3 *fieldVal) {
// Doubling a point at infinity is still infinity.
if y1.IsZero() || z1.IsZero() {
x3.SetInt(0)
y3.SetInt(0)
z3.SetInt(0)
return
}
// Slightly faster point doubling can be achieved when the z value is 1
// by avoiding the multiplication on the z value. This section calls
// a point doubling function which is accelerated by using that
// assumption when possible.
if z1.Normalize().Equals(fieldOne) {
curve.doubleZ1EqualsOne(x1, y1, x3, y3, z3)
return
}
// Fall back to generic point doubling which works with arbitrary z
// values.
curve.doubleGeneric(x1, y1, z1, x3, y3, z3)
}
// Double returns 2*(x1,y1). Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
if y1.Sign() == 0 {
return new(big.Int), new(big.Int)
}
// Convert the affine coordinates from big integers to field values
// and do the point doubling in Jacobian projective space.
fx1, fy1 := curve.bigAffineToField(x1, y1)
fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
fOne := new(fieldVal).SetInt(1)
curve.doubleJacobian(fx1, fy1, fOne, fx3, fy3, fz3)
// Convert the Jacobian coordinate field values back to affine big
// integers.
return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}
// splitK returns a balanced length-two representation of k and their signs.
// This is algorithm 3.74 from [GECC].
//
// One thing of note about this algorithm is that no matter what c1 and c2 are,
// the final equation of k = k1 + k2 * lambda (mod n) will hold. This is
// provable mathematically due to how a1/b1/a2/b2 are computed.
//
// c1 and c2 are chosen to minimize the max(k1,k2).
func (curve *KoblitzCurve) splitK(k []byte) ([]byte, []byte, int, int) {
// All math here is done with big.Int, which is slow.
// At some point, it might be useful to write something similar to
// fieldVal but for N instead of P as the prime field if this ends up
// being a bottleneck.
bigIntK := new(big.Int)
c1, c2 := new(big.Int), new(big.Int)
tmp1, tmp2 := new(big.Int), new(big.Int)
k1, k2 := new(big.Int), new(big.Int)
bigIntK.SetBytes(k)
// c1 = round(b2 * k / n) from step 4.
// Rounding isn't really necessary and costs too much, hence skipped
c1.Mul(curve.b2, bigIntK)
c1.Div(c1, curve.N)
// c2 = round(b1 * k / n) from step 4 (sign reversed to optimize one step)
// Rounding isn't really necessary and costs too much, hence skipped
c2.Mul(curve.b1, bigIntK)
c2.Div(c2, curve.N)
// k1 = k - c1 * a1 - c2 * a2 from step 5 (note c2's sign is reversed)
tmp1.Mul(c1, curve.a1)
tmp2.Mul(c2, curve.a2)
k1.Sub(bigIntK, tmp1)
k1.Add(k1, tmp2)
// k2 = - c1 * b1 - c2 * b2 from step 5 (note c2's sign is reversed)
tmp1.Mul(c1, curve.b1)
tmp2.Mul(c2, curve.b2)
k2.Sub(tmp2, tmp1)
// Note Bytes() throws out the sign of k1 and k2. This matters
// since k1 and/or k2 can be negative. Hence, we pass that
// back separately.
return k1.Bytes(), k2.Bytes(), k1.Sign(), k2.Sign()
}
// moduloReduce reduces k from more than 32 bytes to 32 bytes and under. This
// is done by doing a simple modulo curve.N. We can do this since G^N = 1 and
// thus any other valid point on the elliptic curve has the same order.
func (curve *KoblitzCurve) moduloReduce(k []byte) []byte {
// Since the order of G is curve.N, we can use a much smaller number
// by doing modulo curve.N
if len(k) > curve.byteSize {
// Reduce k by performing modulo curve.N.
tmpK := new(big.Int).SetBytes(k)
tmpK.Mod(tmpK, curve.N)
return tmpK.Bytes()
}
return k
}
// NAF takes a positive integer k and returns the Non-Adjacent Form (NAF) as two
// byte slices. The first is where 1s will be. The second is where -1s will
// be. NAF is convenient in that on average, only 1/3rd of its values are
// non-zero. This is algorithm 3.30 from [GECC].
//
// Essentially, this makes it possible to minimize the number of operations
// since the resulting ints returned will be at least 50% 0s.
func NAF(k []byte) ([]byte, []byte) {
// The essence of this algorithm is that whenever we have consecutive 1s
// in the binary, we want to put a -1 in the lowest bit and get a bunch
// of 0s up to the highest bit of consecutive 1s. This is due to this
// identity:
// 2^n + 2^(n-1) + 2^(n-2) + ... + 2^(n-k) = 2^(n+1) - 2^(n-k)
//
// The algorithm thus may need to go 1 more bit than the length of the
// bits we actually have, hence bits being 1 bit longer than was
// necessary. Since we need to know whether adding will cause a carry,
// we go from right-to-left in this addition.
var carry, curIsOne, nextIsOne bool
// these default to zero
retPos := make([]byte, len(k)+1)
retNeg := make([]byte, len(k)+1)
for i := len(k) - 1; i >= 0; i-- {
curByte := k[i]
for j := uint(0); j < 8; j++ {
curIsOne = curByte&1 == 1
if j == 7 {
if i == 0 {
nextIsOne = false
} else {
nextIsOne = k[i-1]&1 == 1
}
} else {
nextIsOne = curByte&2 == 2
}
if carry {
if curIsOne {
// This bit is 1, so continue to carry
// and don't need to do anything.
} else {
// We've hit a 0 after some number of
// 1s.
if nextIsOne {
// Start carrying again since
// a new sequence of 1s is
// starting.
retNeg[i+1] += 1 << j
} else {
// Stop carrying since 1s have
// stopped.
carry = false
retPos[i+1] += 1 << j
}
}
} else if curIsOne {
if nextIsOne {
// If this is the start of at least 2
// consecutive 1s, set the current one
// to -1 and start carrying.
retNeg[i+1] += 1 << j
carry = true
} else {
// This is a singleton, not consecutive
// 1s.
retPos[i+1] += 1 << j
}
}
curByte >>= 1
}
}
if carry {
retPos[0] = 1
}
return retPos, retNeg
}
// ScalarMult returns k*(Bx, By) where k is a big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
// Point Q = ∞ (point at infinity).
qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
// Decompose K into k1 and k2 in order to halve the number of EC ops.
// See Algorithm 3.74 in [GECC].
k1, k2, signK1, signK2 := curve.splitK(curve.moduloReduce(k))
// The main equation here to remember is:
// k * P = k1 * P + k2 * ϕ(P)
//
// P1 below is P in the equation, P2 below is ϕ(P) in the equation
p1x, p1y := curve.bigAffineToField(Bx, By)
p1yNeg := new(fieldVal).NegateVal(p1y, 1)
p1z := new(fieldVal).SetInt(1)
// NOTE: ϕ(x,y) = (βx,y). The Jacobian z coordinate is 1, so this math
// goes through.
p2x := new(fieldVal).Mul2(p1x, curve.beta)
p2y := new(fieldVal).Set(p1y)
p2yNeg := new(fieldVal).NegateVal(p2y, 1)
p2z := new(fieldVal).SetInt(1)
// Flip the positive and negative values of the points as needed
// depending on the signs of k1 and k2. As mentioned in the equation
// above, each of k1 and k2 are multiplied by the respective point.
// Since -k * P is the same thing as k * -P, and the group law for
// elliptic curves states that P(x, y) = -P(x, -y), it's faster and
// simplifies the code to just make the point negative.
if signK1 == -1 {
p1y, p1yNeg = p1yNeg, p1y
}
if signK2 == -1 {
p2y, p2yNeg = p2yNeg, p2y
}
// NAF versions of k1 and k2 should have a lot more zeros.
//
// The Pos version of the bytes contain the +1s and the Neg versions
// contain the -1s.
k1PosNAF, k1NegNAF := NAF(k1)
k2PosNAF, k2NegNAF := NAF(k2)
k1Len := len(k1PosNAF)
k2Len := len(k2PosNAF)
m := k1Len
if m < k2Len {
m = k2Len
}
// Add left-to-right using the NAF optimization. See algorithm 3.77
// from [GECC]. This should be faster overall since there will be a lot
// more instances of 0, hence reducing the number of Jacobian additions
// at the cost of 1 possible extra doubling.
var k1BytePos, k1ByteNeg, k2BytePos, k2ByteNeg byte
for i := 0; i < m; i++ {
// Since we're going left-to-right, pad the front with 0s.
if i < m-k1Len {
k1BytePos = 0
k1ByteNeg = 0
} else {
k1BytePos = k1PosNAF[i-(m-k1Len)]
k1ByteNeg = k1NegNAF[i-(m-k1Len)]
}
if i < m-k2Len {
k2BytePos = 0
k2ByteNeg = 0
} else {
k2BytePos = k2PosNAF[i-(m-k2Len)]
k2ByteNeg = k2NegNAF[i-(m-k2Len)]
}
for j := 7; j >= 0; j-- {
// Q = 2 * Q
curve.doubleJacobian(qx, qy, qz, qx, qy, qz)
if k1BytePos&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p1x, p1y, p1z,
qx, qy, qz)
} else if k1ByteNeg&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p1x, p1yNeg, p1z,
qx, qy, qz)
}
if k2BytePos&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p2x, p2y, p2z,
qx, qy, qz)
} else if k2ByteNeg&0x80 == 0x80 {
curve.addJacobian(qx, qy, qz, p2x, p2yNeg, p2z,
qx, qy, qz)
}
k1BytePos <<= 1
k1ByteNeg <<= 1
k2BytePos <<= 1
k2ByteNeg <<= 1
}
}
// Convert the Jacobian coordinate field values back to affine big.Ints.
return curve.fieldJacobianToBigAffine(qx, qy, qz)
}
// ScalarBaseMult returns k*G where G is the base point of the group and k is a
// big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
newK := curve.moduloReduce(k)
diff := len(curve.bytePoints) - len(newK)
// Point Q = ∞ (point at infinity).
qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
// curve.bytePoints has all 256 byte points for each 8-bit window. The
// strategy is to add up the byte points. This is best understood by
// expressing k in base-256 which it already sort of is.
// Each "digit" in the 8-bit window can be looked up using bytePoints
// and added together.
for i, byteVal := range newK {
p := curve.bytePoints[diff+i][byteVal]
curve.addJacobian(qx, qy, qz, &p[0], &p[1], &p[2], qx, qy, qz)
}
return curve.fieldJacobianToBigAffine(qx, qy, qz)
}
// QPlus1Div4 returns the Q+1/4 constant for the curve for use in calculating
// square roots via exponention.
func (curve *KoblitzCurve) QPlus1Div4() *big.Int {
return curve.q
}
var initonce sync.Once
var secp256k1 KoblitzCurve
func initAll() {
initS256()
}
// fromHex converts the passed hex string into a big integer pointer and will
// panic is there is an error. This is only provided for the hard-coded
// constants so errors in the source code can bet detected. It will only (and
// must only) be called for initialization purposes.
func fromHex(s string) *big.Int {
r, ok := new(big.Int).SetString(s, 16)
if !ok {
panic("invalid hex in source file: " + s)
}
return r
}
func initS256() {
// Curve parameters taken from [SECG] section 2.4.1.
secp256k1.CurveParams = new(elliptic.CurveParams)
secp256k1.P = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")
secp256k1.N = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141")
secp256k1.B = fromHex("0000000000000000000000000000000000000000000000000000000000000007")
secp256k1.Gx = fromHex("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798")
secp256k1.Gy = fromHex("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8")
secp256k1.BitSize = 256
secp256k1.H = 1
secp256k1.q = new(big.Int).Div(new(big.Int).Add(secp256k1.P,
big.NewInt(1)), big.NewInt(4))
// Provided for convenience since this gets computed repeatedly.
secp256k1.byteSize = secp256k1.BitSize / 8
// Deserialize and set the pre-computed table used to accelerate scalar
// base multiplication. This is hard-coded data, so any errors are
// panics because it means something is wrong in the source code.
if err := loadS256BytePoints(); err != nil {
panic(err)
}
// Next 6 constants are from Hal Finney's bitcointalk.org post:
// https://bitcointalk.org/index.php?topic=3238.msg45565#msg45565
// May he rest in peace.
//
// They have also been independently derived from the code in the
// EndomorphismVectors function in gensecp256k1.go.
secp256k1.lambda = fromHex("5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72")
secp256k1.beta = new(fieldVal).SetHex("7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE")
secp256k1.a1 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
secp256k1.b1 = fromHex("-E4437ED6010E88286F547FA90ABFE4C3")
secp256k1.a2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
secp256k1.b2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
// Alternatively, we can use the parameters below, however, they seem
// to be about 8% slower.
// secp256k1.lambda = fromHex("AC9C52B33FA3CF1F5AD9E3FD77ED9BA4A880B9FC8EC739C2E0CFC810B51283CE")
// secp256k1.beta = new(fieldVal).SetHex("851695D49A83F8EF919BB86153CBCB16630FB68AED0A766A3EC693D68E6AFA40")
// secp256k1.a1 = fromHex("E4437ED6010E88286F547FA90ABFE4C3")
// secp256k1.b1 = fromHex("-3086D221A7D46BCDE86C90E49284EB15")
// secp256k1.a2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
// secp256k1.b2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
}
// S256 returns a Curve which implements secp256k1.
func S256() *KoblitzCurve {
initonce.Do(initAll)
return &secp256k1
}
|