1 // Copyright 2010 The Go Authors. All rights reserved.
2 // Copyright 2011 ThePiachu. All rights reserved.
3 // Copyright 2013-2014 The btcsuite developers
4 // Use of this source code is governed by an ISC
5 // license that can be found in the LICENSE file.
10 // [SECG]: Recommended Elliptic Curve Domain Parameters
11 // http://www.secg.org/sec2-v2.pdf
13 // [GECC]: Guide to Elliptic Curve Cryptography (Hankerson, Menezes, Vanstone)
15 // This package operates, internally, on Jacobian coordinates. For a given
16 // (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
17 // where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
18 // calculation can be performed within the transform (as in ScalarMult and
19 // ScalarBaseMult). But even for Add and Double, it's faster to apply and
20 // reverse the transform than to operate in affine coordinates.
29 // fieldOne is simply the integer 1 in field representation. It is
30 // used to avoid needing to create it multiple times during the internal
32 fieldOne = new(fieldVal).SetInt(1)
35 // KoblitzCurve supports a koblitz curve implementation that fits the ECC Curve
36 // interface from crypto/elliptic.
37 type KoblitzCurve struct {
40 H int // cofactor of the curve.
42 // byteSize is simply the bit size / 8 and is provided for convenience
43 // since it is calculated repeatedly.
47 bytePoints *[32][256][3]fieldVal
49 // The next 6 values are used specifically for endomorphism
50 // optimizations in ScalarMult.
52 // lambda must fulfill lambda^3 = 1 mod N where N is the order of G.
55 // beta must fulfill beta^3 = 1 mod P where P is the prime field of the
59 // See the EndomorphismVectors in gensecp256k1.go to see how these are
67 // Params returns the parameters for the curve.
68 func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
69 return curve.CurveParams
72 // bigAffineToField takes an affine point (x, y) as big integers and converts
73 // it to an affine point as field values.
74 func (curve *KoblitzCurve) bigAffineToField(x, y *big.Int) (*fieldVal, *fieldVal) {
75 x3, y3 := new(fieldVal), new(fieldVal)
76 x3.SetByteSlice(x.Bytes())
77 y3.SetByteSlice(y.Bytes())
82 // fieldJacobianToBigAffine takes a Jacobian point (x, y, z) as field values and
83 // converts it to an affine point as big integers.
84 func (curve *KoblitzCurve) fieldJacobianToBigAffine(x, y, z *fieldVal) (*big.Int, *big.Int) {
85 // Inversions are expensive and both point addition and point doubling
86 // are faster when working with points that have a z value of one. So,
87 // if the point needs to be converted to affine, go ahead and normalize
88 // the point itself at the same time as the calculation is the same.
89 var zInv, tempZ fieldVal
90 zInv.Set(z).Inverse() // zInv = Z^-1
91 tempZ.SquareVal(&zInv) // tempZ = Z^-2
92 x.Mul(&tempZ) // X = X/Z^2 (mag: 1)
93 y.Mul(tempZ.Mul(&zInv)) // Y = Y/Z^3 (mag: 1)
94 z.SetInt(1) // Z = 1 (mag: 1)
96 // Normalize the x and y values.
100 // Convert the field values for the now affine point to big.Ints.
101 x3, y3 := new(big.Int), new(big.Int)
102 x3.SetBytes(x.Bytes()[:])
103 y3.SetBytes(y.Bytes()[:])
107 // IsOnCurve returns boolean if the point (x,y) is on the curve.
108 // Part of the elliptic.Curve interface. This function differs from the
109 // crypto/elliptic algorithm since a = 0 not -3.
110 func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
111 // Convert big ints to field values for faster arithmetic.
112 fx, fy := curve.bigAffineToField(x, y)
114 // Elliptic curve equation for secp256k1 is: y^2 = x^3 + 7
115 y2 := new(fieldVal).SquareVal(fy).Normalize()
116 result := new(fieldVal).SquareVal(fx).Mul(fx).AddInt(7).Normalize()
117 return y2.Equals(result)
120 // addZ1AndZ2EqualsOne adds two Jacobian points that are already known to have
121 // z values of 1 and stores the result in (x3, y3, z3). That is to say
122 // (x1, y1, 1) + (x2, y2, 1) = (x3, y3, z3). It performs faster addition than
123 // the generic add routine since less arithmetic is needed due to the ability to
124 // avoid the z value multiplications.
125 func (curve *KoblitzCurve) addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
126 // To compute the point addition efficiently, this implementation splits
127 // the equation into intermediate elements which are used to minimize
128 // the number of field multiplications using the method shown at:
129 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
131 // In particular it performs the calculations using the following:
132 // H = X2-X1, HH = H^2, I = 4*HH, J = H*I, r = 2*(Y2-Y1), V = X1*I
133 // X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = 2*H
135 // This results in a cost of 4 field multiplications, 2 field squarings,
136 // 6 field additions, and 5 integer multiplications.
138 // When the x coordinates are the same for two points on the curve, the
139 // y coordinates either must be the same, in which case it is point
140 // doubling, or they are opposite and the result is the point at
141 // infinity per the group law for elliptic curve cryptography.
148 // Since x1 == x2 and y1 == y2, point doubling must be
149 // done, otherwise the addition would end up dividing
151 curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
155 // Since x1 == x2 and y1 == -y2, the sum is the point at
156 // infinity per the group law.
163 // Calculate X3, Y3, and Z3 according to the intermediate elements
165 var h, i, j, r, v fieldVal
166 var negJ, neg2V, negX3 fieldVal
167 h.Set(x1).Negate(1).Add(x2) // H = X2-X1 (mag: 3)
168 i.SquareVal(&h).MulInt(4) // I = 4*H^2 (mag: 4)
169 j.Mul2(&h, &i) // J = H*I (mag: 1)
170 r.Set(y1).Negate(1).Add(y2).MulInt(2) // r = 2*(Y2-Y1) (mag: 6)
171 v.Mul2(x1, &i) // V = X1*I (mag: 1)
172 negJ.Set(&j).Negate(1) // negJ = -J (mag: 2)
173 neg2V.Set(&v).MulInt(2).Negate(2) // neg2V = -(2*V) (mag: 3)
174 x3.Set(&r).Square().Add(&negJ).Add(&neg2V) // X3 = r^2-J-2*V (mag: 6)
175 negX3.Set(x3).Negate(6) // negX3 = -X3 (mag: 7)
176 j.Mul(y1).MulInt(2).Negate(2) // J = -(2*Y1*J) (mag: 3)
177 y3.Set(&v).Add(&negX3).Mul(&r).Add(&j) // Y3 = r*(V-X3)-2*Y1*J (mag: 4)
178 z3.Set(&h).MulInt(2) // Z3 = 2*H (mag: 6)
180 // Normalize the resulting field values to a magnitude of 1 as needed.
186 // addZ1EqualsZ2 adds two Jacobian points that are already known to have the
187 // same z value and stores the result in (x3, y3, z3). That is to say
188 // (x1, y1, z1) + (x2, y2, z1) = (x3, y3, z3). It performs faster addition than
189 // the generic add routine since less arithmetic is needed due to the known
191 func (curve *KoblitzCurve) addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
192 // To compute the point addition efficiently, this implementation splits
193 // the equation into intermediate elements which are used to minimize
194 // the number of field multiplications using a slightly modified version
195 // of the method shown at:
196 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
198 // In particular it performs the calculations using the following:
199 // A = X2-X1, B = A^2, C=Y2-Y1, D = C^2, E = X1*B, F = X2*B
200 // X3 = D-E-F, Y3 = C*(E-X3)-Y1*(F-E), Z3 = Z1*A
202 // This results in a cost of 5 field multiplications, 2 field squarings,
203 // 9 field additions, and 0 integer multiplications.
205 // When the x coordinates are the same for two points on the curve, the
206 // y coordinates either must be the same, in which case it is point
207 // doubling, or they are opposite and the result is the point at
208 // infinity per the group law for elliptic curve cryptography.
215 // Since x1 == x2 and y1 == y2, point doubling must be
216 // done, otherwise the addition would end up dividing
218 curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
222 // Since x1 == x2 and y1 == -y2, the sum is the point at
223 // infinity per the group law.
230 // Calculate X3, Y3, and Z3 according to the intermediate elements
232 var a, b, c, d, e, f fieldVal
233 var negX1, negY1, negE, negX3 fieldVal
234 negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
235 negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
236 a.Set(&negX1).Add(x2) // A = X2-X1 (mag: 3)
237 b.SquareVal(&a) // B = A^2 (mag: 1)
238 c.Set(&negY1).Add(y2) // C = Y2-Y1 (mag: 3)
239 d.SquareVal(&c) // D = C^2 (mag: 1)
240 e.Mul2(x1, &b) // E = X1*B (mag: 1)
241 negE.Set(&e).Negate(1) // negE = -E (mag: 2)
242 f.Mul2(x2, &b) // F = X2*B (mag: 1)
243 x3.Add2(&e, &f).Negate(3).Add(&d) // X3 = D-E-F (mag: 5)
244 negX3.Set(x3).Negate(5).Normalize() // negX3 = -X3 (mag: 1)
245 y3.Set(y1).Mul(f.Add(&negE)).Negate(3) // Y3 = -(Y1*(F-E)) (mag: 4)
246 y3.Add(e.Add(&negX3).Mul(&c)) // Y3 = C*(E-X3)+Y3 (mag: 5)
247 z3.Mul2(z1, &a) // Z3 = Z1*A (mag: 1)
249 // Normalize the resulting field values to a magnitude of 1 as needed.
254 // addZ2EqualsOne adds two Jacobian points when the second point is already
255 // known to have a z value of 1 (and the z value for the first point is not 1)
256 // and stores the result in (x3, y3, z3). That is to say (x1, y1, z1) +
257 // (x2, y2, 1) = (x3, y3, z3). It performs faster addition than the generic
258 // add routine since less arithmetic is needed due to the ability to avoid
259 // multiplications by the second point's z value.
260 func (curve *KoblitzCurve) addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
261 // To compute the point addition efficiently, this implementation splits
262 // the equation into intermediate elements which are used to minimize
263 // the number of field multiplications using the method shown at:
264 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
266 // In particular it performs the calculations using the following:
267 // Z1Z1 = Z1^2, U2 = X2*Z1Z1, S2 = Y2*Z1*Z1Z1, H = U2-X1, HH = H^2,
268 // I = 4*HH, J = H*I, r = 2*(S2-Y1), V = X1*I
269 // X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = (Z1+H)^2-Z1Z1-HH
271 // This results in a cost of 7 field multiplications, 4 field squarings,
272 // 9 field additions, and 4 integer multiplications.
274 // When the x coordinates are the same for two points on the curve, the
275 // y coordinates either must be the same, in which case it is point
276 // doubling, or they are opposite and the result is the point at
277 // infinity per the group law for elliptic curve cryptography. Since
278 // any number of Jacobian coordinates can represent the same affine
279 // point, the x and y values need to be converted to like terms. Due to
280 // the assumption made for this function that the second point has a z
281 // value of 1 (z2=1), the first point is already "converted".
282 var z1z1, u2, s2 fieldVal
285 z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
286 u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
287 s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
290 // Since x1 == x2 and y1 == y2, point doubling must be
291 // done, otherwise the addition would end up dividing
293 curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
297 // Since x1 == x2 and y1 == -y2, the sum is the point at
298 // infinity per the group law.
305 // Calculate X3, Y3, and Z3 according to the intermediate elements
307 var h, hh, i, j, r, rr, v fieldVal
308 var negX1, negY1, negX3 fieldVal
309 negX1.Set(x1).Negate(1) // negX1 = -X1 (mag: 2)
310 h.Add2(&u2, &negX1) // H = U2-X1 (mag: 3)
311 hh.SquareVal(&h) // HH = H^2 (mag: 1)
312 i.Set(&hh).MulInt(4) // I = 4 * HH (mag: 4)
313 j.Mul2(&h, &i) // J = H*I (mag: 1)
314 negY1.Set(y1).Negate(1) // negY1 = -Y1 (mag: 2)
315 r.Set(&s2).Add(&negY1).MulInt(2) // r = 2*(S2-Y1) (mag: 6)
316 rr.SquareVal(&r) // rr = r^2 (mag: 1)
317 v.Mul2(x1, &i) // V = X1*I (mag: 1)
318 x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
319 x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
320 negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
321 y3.Set(y1).Mul(&j).MulInt(2).Negate(2) // Y3 = -(2*Y1*J) (mag: 3)
322 y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
323 z3.Add2(z1, &h).Square() // Z3 = (Z1+H)^2 (mag: 1)
324 z3.Add(z1z1.Add(&hh).Negate(2)) // Z3 = Z3-(Z1Z1+HH) (mag: 4)
326 // Normalize the resulting field values to a magnitude of 1 as needed.
332 // addGeneric adds two Jacobian points (x1, y1, z1) and (x2, y2, z2) without any
333 // assumptions about the z values of the two points and stores the result in
334 // (x3, y3, z3). That is to say (x1, y1, z1) + (x2, y2, z2) = (x3, y3, z3). It
335 // is the slowest of the add routines due to requiring the most arithmetic.
336 func (curve *KoblitzCurve) addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
337 // To compute the point addition efficiently, this implementation splits
338 // the equation into intermediate elements which are used to minimize
339 // the number of field multiplications using the method shown at:
340 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
342 // In particular it performs the calculations using the following:
343 // Z1Z1 = Z1^2, Z2Z2 = Z2^2, U1 = X1*Z2Z2, U2 = X2*Z1Z1, S1 = Y1*Z2*Z2Z2
344 // S2 = Y2*Z1*Z1Z1, H = U2-U1, I = (2*H)^2, J = H*I, r = 2*(S2-S1)
346 // X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*S1*J, Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H
348 // This results in a cost of 11 field multiplications, 5 field squarings,
349 // 9 field additions, and 4 integer multiplications.
351 // When the x coordinates are the same for two points on the curve, the
352 // y coordinates either must be the same, in which case it is point
353 // doubling, or they are opposite and the result is the point at
354 // infinity. Since any number of Jacobian coordinates can represent the
355 // same affine point, the x and y values need to be converted to like
357 var z1z1, z2z2, u1, u2, s1, s2 fieldVal
358 z1z1.SquareVal(z1) // Z1Z1 = Z1^2 (mag: 1)
359 z2z2.SquareVal(z2) // Z2Z2 = Z2^2 (mag: 1)
360 u1.Set(x1).Mul(&z2z2).Normalize() // U1 = X1*Z2Z2 (mag: 1)
361 u2.Set(x2).Mul(&z1z1).Normalize() // U2 = X2*Z1Z1 (mag: 1)
362 s1.Set(y1).Mul(&z2z2).Mul(z2).Normalize() // S1 = Y1*Z2*Z2Z2 (mag: 1)
363 s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
366 // Since x1 == x2 and y1 == y2, point doubling must be
367 // done, otherwise the addition would end up dividing
369 curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
373 // Since x1 == x2 and y1 == -y2, the sum is the point at
374 // infinity per the group law.
381 // Calculate X3, Y3, and Z3 according to the intermediate elements
383 var h, i, j, r, rr, v fieldVal
384 var negU1, negS1, negX3 fieldVal
385 negU1.Set(&u1).Negate(1) // negU1 = -U1 (mag: 2)
386 h.Add2(&u2, &negU1) // H = U2-U1 (mag: 3)
387 i.Set(&h).MulInt(2).Square() // I = (2*H)^2 (mag: 2)
388 j.Mul2(&h, &i) // J = H*I (mag: 1)
389 negS1.Set(&s1).Negate(1) // negS1 = -S1 (mag: 2)
390 r.Set(&s2).Add(&negS1).MulInt(2) // r = 2*(S2-S1) (mag: 6)
391 rr.SquareVal(&r) // rr = r^2 (mag: 1)
392 v.Mul2(&u1, &i) // V = U1*I (mag: 1)
393 x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
394 x3.Add(&rr) // X3 = r^2+X3 (mag: 5)
395 negX3.Set(x3).Negate(5) // negX3 = -X3 (mag: 6)
396 y3.Mul2(&s1, &j).MulInt(2).Negate(2) // Y3 = -(2*S1*J) (mag: 3)
397 y3.Add(v.Add(&negX3).Mul(&r)) // Y3 = r*(V-X3)+Y3 (mag: 4)
398 z3.Add2(z1, z2).Square() // Z3 = (Z1+Z2)^2 (mag: 1)
399 z3.Add(z1z1.Add(&z2z2).Negate(2)) // Z3 = Z3-(Z1Z1+Z2Z2) (mag: 4)
400 z3.Mul(&h) // Z3 = Z3*H (mag: 1)
402 // Normalize the resulting field values to a magnitude of 1 as needed.
407 // addJacobian adds the passed Jacobian points (x1, y1, z1) and (x2, y2, z2)
408 // together and stores the result in (x3, y3, z3).
409 func (curve *KoblitzCurve) addJacobian(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
410 // A point at infinity is the identity according to the group law for
411 // elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
412 if (x1.IsZero() && y1.IsZero()) || z1.IsZero() {
418 if (x2.IsZero() && y2.IsZero()) || z2.IsZero() {
425 // Faster point addition can be achieved when certain assumptions are
426 // met. For example, when both points have the same z value, arithmetic
427 // on the z values can be avoided. This section thus checks for these
428 // conditions and calls an appropriate add function which is accelerated
429 // by using those assumptions.
432 isZ1One := z1.Equals(fieldOne)
433 isZ2One := z2.Equals(fieldOne)
435 case isZ1One && isZ2One:
436 curve.addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
439 curve.addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3)
442 curve.addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
446 // None of the above assumptions are true, so fall back to generic
448 curve.addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3)
451 // Add returns the sum of (x1,y1) and (x2,y2). Part of the elliptic.Curve
453 func (curve *KoblitzCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
454 // A point at infinity is the identity according to the group law for
455 // elliptic curve cryptography. Thus, ∞ + P = P and P + ∞ = P.
456 if x1.Sign() == 0 && y1.Sign() == 0 {
459 if x2.Sign() == 0 && y2.Sign() == 0 {
463 // Convert the affine coordinates from big integers to field values
464 // and do the point addition in Jacobian projective space.
465 fx1, fy1 := curve.bigAffineToField(x1, y1)
466 fx2, fy2 := curve.bigAffineToField(x2, y2)
467 fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
468 fOne := new(fieldVal).SetInt(1)
469 curve.addJacobian(fx1, fy1, fOne, fx2, fy2, fOne, fx3, fy3, fz3)
471 // Convert the Jacobian coordinate field values back to affine big
473 return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
476 // doubleZ1EqualsOne performs point doubling on the passed Jacobian point
477 // when the point is already known to have a z value of 1 and stores
478 // the result in (x3, y3, z3). That is to say (x3, y3, z3) = 2*(x1, y1, 1). It
479 // performs faster point doubling than the generic routine since less arithmetic
480 // is needed due to the ability to avoid multiplication by the z value.
481 func (curve *KoblitzCurve) doubleZ1EqualsOne(x1, y1, x3, y3, z3 *fieldVal) {
482 // This function uses the assumptions that z1 is 1, thus the point
483 // doubling formulas reduce to:
485 // X3 = (3*X1^2)^2 - 8*X1*Y1^2
486 // Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
489 // To compute the above efficiently, this implementation splits the
490 // equation into intermediate elements which are used to minimize the
491 // number of field multiplications in favor of field squarings which
492 // are roughly 35% faster than field multiplications with the current
493 // implementation at the time this was written.
495 // This uses a slightly modified version of the method shown at:
496 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-mdbl-2007-bl
498 // In particular it performs the calculations using the following:
499 // A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
500 // E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
503 // This results in a cost of 1 field multiplication, 5 field squarings,
504 // 6 field additions, and 5 integer multiplications.
505 var a, b, c, d, e, f fieldVal
506 z3.Set(y1).MulInt(2) // Z3 = 2*Y1 (mag: 2)
507 a.SquareVal(x1) // A = X1^2 (mag: 1)
508 b.SquareVal(y1) // B = Y1^2 (mag: 1)
509 c.SquareVal(&b) // C = B^2 (mag: 1)
510 b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
511 d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
512 d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
513 e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
514 f.SquareVal(&e) // F = E^2 (mag: 1)
515 x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
516 x3.Add(&f) // X3 = F+X3 (mag: 18)
517 f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
518 y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
519 y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
521 // Normalize the field values back to a magnitude of 1.
527 // doubleGeneric performs point doubling on the passed Jacobian point without
528 // any assumptions about the z value and stores the result in (x3, y3, z3).
529 // That is to say (x3, y3, z3) = 2*(x1, y1, z1). It is the slowest of the point
530 // doubling routines due to requiring the most arithmetic.
531 func (curve *KoblitzCurve) doubleGeneric(x1, y1, z1, x3, y3, z3 *fieldVal) {
532 // Point doubling formula for Jacobian coordinates for the secp256k1
534 // X3 = (3*X1^2)^2 - 8*X1*Y1^2
535 // Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
538 // To compute the above efficiently, this implementation splits the
539 // equation into intermediate elements which are used to minimize the
540 // number of field multiplications in favor of field squarings which
541 // are roughly 35% faster than field multiplications with the current
542 // implementation at the time this was written.
544 // This uses a slightly modified version of the method shown at:
545 // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
547 // In particular it performs the calculations using the following:
548 // A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
549 // E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
552 // This results in a cost of 1 field multiplication, 5 field squarings,
553 // 6 field additions, and 5 integer multiplications.
554 var a, b, c, d, e, f fieldVal
555 z3.Mul2(y1, z1).MulInt(2) // Z3 = 2*Y1*Z1 (mag: 2)
556 a.SquareVal(x1) // A = X1^2 (mag: 1)
557 b.SquareVal(y1) // B = Y1^2 (mag: 1)
558 c.SquareVal(&b) // C = B^2 (mag: 1)
559 b.Add(x1).Square() // B = (X1+B)^2 (mag: 1)
560 d.Set(&a).Add(&c).Negate(2) // D = -(A+C) (mag: 3)
561 d.Add(&b).MulInt(2) // D = 2*(B+D)(mag: 8)
562 e.Set(&a).MulInt(3) // E = 3*A (mag: 3)
563 f.SquareVal(&e) // F = E^2 (mag: 1)
564 x3.Set(&d).MulInt(2).Negate(16) // X3 = -(2*D) (mag: 17)
565 x3.Add(&f) // X3 = F+X3 (mag: 18)
566 f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
567 y3.Set(&c).MulInt(8).Negate(8) // Y3 = -(8*C) (mag: 9)
568 y3.Add(f.Mul(&e)) // Y3 = E*F+Y3 (mag: 10)
570 // Normalize the field values back to a magnitude of 1.
576 // doubleJacobian doubles the passed Jacobian point (x1, y1, z1) and stores the
577 // result in (x3, y3, z3).
578 func (curve *KoblitzCurve) doubleJacobian(x1, y1, z1, x3, y3, z3 *fieldVal) {
579 // Doubling a point at infinity is still infinity.
580 if y1.IsZero() || z1.IsZero() {
587 // Slightly faster point doubling can be achieved when the z value is 1
588 // by avoiding the multiplication on the z value. This section calls
589 // a point doubling function which is accelerated by using that
590 // assumption when possible.
591 if z1.Normalize().Equals(fieldOne) {
592 curve.doubleZ1EqualsOne(x1, y1, x3, y3, z3)
596 // Fall back to generic point doubling which works with arbitrary z
598 curve.doubleGeneric(x1, y1, z1, x3, y3, z3)
601 // Double returns 2*(x1,y1). Part of the elliptic.Curve interface.
602 func (curve *KoblitzCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
604 return new(big.Int), new(big.Int)
607 // Convert the affine coordinates from big integers to field values
608 // and do the point doubling in Jacobian projective space.
609 fx1, fy1 := curve.bigAffineToField(x1, y1)
610 fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
611 fOne := new(fieldVal).SetInt(1)
612 curve.doubleJacobian(fx1, fy1, fOne, fx3, fy3, fz3)
614 // Convert the Jacobian coordinate field values back to affine big
616 return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
619 // splitK returns a balanced length-two representation of k and their signs.
620 // This is algorithm 3.74 from [GECC].
622 // One thing of note about this algorithm is that no matter what c1 and c2 are,
623 // the final equation of k = k1 + k2 * lambda (mod n) will hold. This is
624 // provable mathematically due to how a1/b1/a2/b2 are computed.
626 // c1 and c2 are chosen to minimize the max(k1,k2).
627 func (curve *KoblitzCurve) splitK(k []byte) ([]byte, []byte, int, int) {
628 // All math here is done with big.Int, which is slow.
629 // At some point, it might be useful to write something similar to
630 // fieldVal but for N instead of P as the prime field if this ends up
631 // being a bottleneck.
632 bigIntK := new(big.Int)
633 c1, c2 := new(big.Int), new(big.Int)
634 tmp1, tmp2 := new(big.Int), new(big.Int)
635 k1, k2 := new(big.Int), new(big.Int)
638 // c1 = round(b2 * k / n) from step 4.
639 // Rounding isn't really necessary and costs too much, hence skipped
640 c1.Mul(curve.b2, bigIntK)
642 // c2 = round(b1 * k / n) from step 4 (sign reversed to optimize one step)
643 // Rounding isn't really necessary and costs too much, hence skipped
644 c2.Mul(curve.b1, bigIntK)
646 // k1 = k - c1 * a1 - c2 * a2 from step 5 (note c2's sign is reversed)
647 tmp1.Mul(c1, curve.a1)
648 tmp2.Mul(c2, curve.a2)
649 k1.Sub(bigIntK, tmp1)
651 // k2 = - c1 * b1 - c2 * b2 from step 5 (note c2's sign is reversed)
652 tmp1.Mul(c1, curve.b1)
653 tmp2.Mul(c2, curve.b2)
656 // Note Bytes() throws out the sign of k1 and k2. This matters
657 // since k1 and/or k2 can be negative. Hence, we pass that
659 return k1.Bytes(), k2.Bytes(), k1.Sign(), k2.Sign()
662 // moduloReduce reduces k from more than 32 bytes to 32 bytes and under. This
663 // is done by doing a simple modulo curve.N. We can do this since G^N = 1 and
664 // thus any other valid point on the elliptic curve has the same order.
665 func (curve *KoblitzCurve) moduloReduce(k []byte) []byte {
666 // Since the order of G is curve.N, we can use a much smaller number
667 // by doing modulo curve.N
668 if len(k) > curve.byteSize {
669 // Reduce k by performing modulo curve.N.
670 tmpK := new(big.Int).SetBytes(k)
671 tmpK.Mod(tmpK, curve.N)
678 // NAF takes a positive integer k and returns the Non-Adjacent Form (NAF) as two
679 // byte slices. The first is where 1s will be. The second is where -1s will
680 // be. NAF is convenient in that on average, only 1/3rd of its values are
681 // non-zero. This is algorithm 3.30 from [GECC].
683 // Essentially, this makes it possible to minimize the number of operations
684 // since the resulting ints returned will be at least 50% 0s.
685 func NAF(k []byte) ([]byte, []byte) {
686 // The essence of this algorithm is that whenever we have consecutive 1s
687 // in the binary, we want to put a -1 in the lowest bit and get a bunch
688 // of 0s up to the highest bit of consecutive 1s. This is due to this
690 // 2^n + 2^(n-1) + 2^(n-2) + ... + 2^(n-k) = 2^(n+1) - 2^(n-k)
692 // The algorithm thus may need to go 1 more bit than the length of the
693 // bits we actually have, hence bits being 1 bit longer than was
694 // necessary. Since we need to know whether adding will cause a carry,
695 // we go from right-to-left in this addition.
696 var carry, curIsOne, nextIsOne bool
697 // these default to zero
698 retPos := make([]byte, len(k)+1)
699 retNeg := make([]byte, len(k)+1)
700 for i := len(k) - 1; i >= 0; i-- {
702 for j := uint(0); j < 8; j++ {
703 curIsOne = curByte&1 == 1
708 nextIsOne = k[i-1]&1 == 1
711 nextIsOne = curByte&2 == 2
715 // This bit is 1, so continue to carry
716 // and don't need to do anything.
718 // We've hit a 0 after some number of
721 // Start carrying again since
722 // a new sequence of 1s is
724 retNeg[i+1] += 1 << j
726 // Stop carrying since 1s have
729 retPos[i+1] += 1 << j
734 // If this is the start of at least 2
735 // consecutive 1s, set the current one
736 // to -1 and start carrying.
737 retNeg[i+1] += 1 << j
740 // This is a singleton, not consecutive
742 retPos[i+1] += 1 << j
750 return retPos, retNeg
752 return retPos[1:], retNeg[1:]
755 // ScalarMult returns k*(Bx, By) where k is a big endian integer.
756 // Part of the elliptic.Curve interface.
757 func (curve *KoblitzCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
758 // Point Q = ∞ (point at infinity).
759 qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
761 // Decompose K into k1 and k2 in order to halve the number of EC ops.
762 // See Algorithm 3.74 in [GECC].
763 k1, k2, signK1, signK2 := curve.splitK(curve.moduloReduce(k))
765 // The main equation here to remember is:
766 // k * P = k1 * P + k2 * ϕ(P)
768 // P1 below is P in the equation, P2 below is ϕ(P) in the equation
769 p1x, p1y := curve.bigAffineToField(Bx, By)
770 p1yNeg := new(fieldVal).NegateVal(p1y, 1)
771 p1z := new(fieldVal).SetInt(1)
773 // NOTE: ϕ(x,y) = (βx,y). The Jacobian z coordinate is 1, so this math
775 p2x := new(fieldVal).Mul2(p1x, curve.beta)
776 p2y := new(fieldVal).Set(p1y)
777 p2yNeg := new(fieldVal).NegateVal(p2y, 1)
778 p2z := new(fieldVal).SetInt(1)
780 // Flip the positive and negative values of the points as needed
781 // depending on the signs of k1 and k2. As mentioned in the equation
782 // above, each of k1 and k2 are multiplied by the respective point.
783 // Since -k * P is the same thing as k * -P, and the group law for
784 // elliptic curves states that P(x, y) = -P(x, -y), it's faster and
785 // simplifies the code to just make the point negative.
787 p1y, p1yNeg = p1yNeg, p1y
790 p2y, p2yNeg = p2yNeg, p2y
793 // NAF versions of k1 and k2 should have a lot more zeros.
795 // The Pos version of the bytes contain the +1s and the Neg versions
797 k1PosNAF, k1NegNAF := NAF(k1)
798 k2PosNAF, k2NegNAF := NAF(k2)
799 k1Len := len(k1PosNAF)
800 k2Len := len(k2PosNAF)
807 // Add left-to-right using the NAF optimization. See algorithm 3.77
808 // from [GECC]. This should be faster overall since there will be a lot
809 // more instances of 0, hence reducing the number of Jacobian additions
810 // at the cost of 1 possible extra doubling.
811 var k1BytePos, k1ByteNeg, k2BytePos, k2ByteNeg byte
812 for i := 0; i < m; i++ {
813 // Since we're going left-to-right, pad the front with 0s.
818 k1BytePos = k1PosNAF[i-(m-k1Len)]
819 k1ByteNeg = k1NegNAF[i-(m-k1Len)]
825 k2BytePos = k2PosNAF[i-(m-k2Len)]
826 k2ByteNeg = k2NegNAF[i-(m-k2Len)]
829 for j := 7; j >= 0; j-- {
831 curve.doubleJacobian(qx, qy, qz, qx, qy, qz)
833 if k1BytePos&0x80 == 0x80 {
834 curve.addJacobian(qx, qy, qz, p1x, p1y, p1z,
836 } else if k1ByteNeg&0x80 == 0x80 {
837 curve.addJacobian(qx, qy, qz, p1x, p1yNeg, p1z,
841 if k2BytePos&0x80 == 0x80 {
842 curve.addJacobian(qx, qy, qz, p2x, p2y, p2z,
844 } else if k2ByteNeg&0x80 == 0x80 {
845 curve.addJacobian(qx, qy, qz, p2x, p2yNeg, p2z,
855 // Convert the Jacobian coordinate field values back to affine big.Ints.
856 return curve.fieldJacobianToBigAffine(qx, qy, qz)
859 // ScalarBaseMult returns k*G where G is the base point of the group and k is a
860 // big endian integer.
861 // Part of the elliptic.Curve interface.
862 func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
863 newK := curve.moduloReduce(k)
864 diff := len(curve.bytePoints) - len(newK)
866 // Point Q = ∞ (point at infinity).
867 qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)
869 // curve.bytePoints has all 256 byte points for each 8-bit window. The
870 // strategy is to add up the byte points. This is best understood by
871 // expressing k in base-256 which it already sort of is.
872 // Each "digit" in the 8-bit window can be looked up using bytePoints
873 // and added together.
874 for i, byteVal := range newK {
875 p := curve.bytePoints[diff+i][byteVal]
876 curve.addJacobian(qx, qy, qz, &p[0], &p[1], &p[2], qx, qy, qz)
878 return curve.fieldJacobianToBigAffine(qx, qy, qz)
881 // QPlus1Div4 returns the Q+1/4 constant for the curve for use in calculating
882 // square roots via exponention.
883 func (curve *KoblitzCurve) QPlus1Div4() *big.Int {
887 var initonce sync.Once
888 var secp256k1 KoblitzCurve
894 // fromHex converts the passed hex string into a big integer pointer and will
895 // panic is there is an error. This is only provided for the hard-coded
896 // constants so errors in the source code can bet detected. It will only (and
897 // must only) be called for initialization purposes.
898 func fromHex(s string) *big.Int {
899 r, ok := new(big.Int).SetString(s, 16)
901 panic("invalid hex in source file: " + s)
907 // Curve parameters taken from [SECG] section 2.4.1.
908 secp256k1.CurveParams = new(elliptic.CurveParams)
909 secp256k1.P = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")
910 secp256k1.N = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141")
911 secp256k1.B = fromHex("0000000000000000000000000000000000000000000000000000000000000007")
912 secp256k1.Gx = fromHex("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798")
913 secp256k1.Gy = fromHex("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8")
914 secp256k1.BitSize = 256
916 secp256k1.q = new(big.Int).Div(new(big.Int).Add(secp256k1.P,
917 big.NewInt(1)), big.NewInt(4))
919 // Provided for convenience since this gets computed repeatedly.
920 secp256k1.byteSize = secp256k1.BitSize / 8
922 // Deserialize and set the pre-computed table used to accelerate scalar
923 // base multiplication. This is hard-coded data, so any errors are
924 // panics because it means something is wrong in the source code.
925 if err := loadS256BytePoints(); err != nil {
929 // Next 6 constants are from Hal Finney's bitcointalk.org post:
930 // https://bitcointalk.org/index.php?topic=3238.msg45565#msg45565
931 // May he rest in peace.
933 // They have also been independently derived from the code in the
934 // EndomorphismVectors function in gensecp256k1.go.
935 secp256k1.lambda = fromHex("5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72")
936 secp256k1.beta = new(fieldVal).SetHex("7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE")
937 secp256k1.a1 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
938 secp256k1.b1 = fromHex("-E4437ED6010E88286F547FA90ABFE4C3")
939 secp256k1.a2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
940 secp256k1.b2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
942 // Alternatively, we can use the parameters below, however, they seem
943 // to be about 8% slower.
944 // secp256k1.lambda = fromHex("AC9C52B33FA3CF1F5AD9E3FD77ED9BA4A880B9FC8EC739C2E0CFC810B51283CE")
945 // secp256k1.beta = new(fieldVal).SetHex("851695D49A83F8EF919BB86153CBCB16630FB68AED0A766A3EC693D68E6AFA40")
946 // secp256k1.a1 = fromHex("E4437ED6010E88286F547FA90ABFE4C3")
947 // secp256k1.b1 = fromHex("-3086D221A7D46BCDE86C90E49284EB15")
948 // secp256k1.a2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
949 // secp256k1.b2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
952 // S256 returns a Curve which implements secp256k1.
953 func S256() *KoblitzCurve {