// Copyright 2016 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package edwards25519 import "encoding/binary" // This code is a port of the public domain, “ref10” implementation of ed25519 // from SUPERCOP. // FieldElement represents an element of the field GF(2^255 - 19). An element // t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 // t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on // context. type FieldElement [10]int32 var zero FieldElement func FeZero(fe *FieldElement) { copy(fe[:], zero[:]) } func FeOne(fe *FieldElement) { FeZero(fe) fe[0] = 1 } func FeAdd(dst, a, b *FieldElement) { dst[0] = a[0] + b[0] dst[1] = a[1] + b[1] dst[2] = a[2] + b[2] dst[3] = a[3] + b[3] dst[4] = a[4] + b[4] dst[5] = a[5] + b[5] dst[6] = a[6] + b[6] dst[7] = a[7] + b[7] dst[8] = a[8] + b[8] dst[9] = a[9] + b[9] } func FeSub(dst, a, b *FieldElement) { dst[0] = a[0] - b[0] dst[1] = a[1] - b[1] dst[2] = a[2] - b[2] dst[3] = a[3] - b[3] dst[4] = a[4] - b[4] dst[5] = a[5] - b[5] dst[6] = a[6] - b[6] dst[7] = a[7] - b[7] dst[8] = a[8] - b[8] dst[9] = a[9] - b[9] } func FeCopy(dst, src *FieldElement) { copy(dst[:], src[:]) } // Replace (f,g) with (g,g) if b == 1; // replace (f,g) with (f,g) if b == 0. // // Preconditions: b in {0,1}. func FeCMove(f, g *FieldElement, b int32) { b = -b f[0] ^= b & (f[0] ^ g[0]) f[1] ^= b & (f[1] ^ g[1]) f[2] ^= b & (f[2] ^ g[2]) f[3] ^= b & (f[3] ^ g[3]) f[4] ^= b & (f[4] ^ g[4]) f[5] ^= b & (f[5] ^ g[5]) f[6] ^= b & (f[6] ^ g[6]) f[7] ^= b & (f[7] ^ g[7]) f[8] ^= b & (f[8] ^ g[8]) f[9] ^= b & (f[9] ^ g[9]) } func load3(in []byte) int64 { var r int64 r = int64(in[0]) r |= int64(in[1]) << 8 r |= int64(in[2]) << 16 return r } func load4(in []byte) int64 { var r int64 r = int64(in[0]) r |= int64(in[1]) << 8 r |= int64(in[2]) << 16 r |= int64(in[3]) << 24 return r } func FeFromBytes(dst *FieldElement, src *[32]byte) { h0 := load4(src[:]) h1 := load3(src[4:]) << 6 h2 := load3(src[7:]) << 5 h3 := load3(src[10:]) << 3 h4 := load3(src[13:]) << 2 h5 := load4(src[16:]) h6 := load3(src[20:]) << 7 h7 := load3(src[23:]) << 5 h8 := load3(src[26:]) << 4 h9 := (load3(src[29:]) & 8388607) << 2 FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) } // FeToBytes marshals h to s. // Preconditions: // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. // // Write p=2^255-19; q=floor(h/p). // Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). // // Proof: // Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. // Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. // // Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). // Then 0> 25 q = (h[0] + q) >> 26 q = (h[1] + q) >> 25 q = (h[2] + q) >> 26 q = (h[3] + q) >> 25 q = (h[4] + q) >> 26 q = (h[5] + q) >> 25 q = (h[6] + q) >> 26 q = (h[7] + q) >> 25 q = (h[8] + q) >> 26 q = (h[9] + q) >> 25 // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. h[0] += 19 * q // Goal: Output h-2^255 q, which is between 0 and 2^255-20. carry[0] = h[0] >> 26 h[1] += carry[0] h[0] -= carry[0] << 26 carry[1] = h[1] >> 25 h[2] += carry[1] h[1] -= carry[1] << 25 carry[2] = h[2] >> 26 h[3] += carry[2] h[2] -= carry[2] << 26 carry[3] = h[3] >> 25 h[4] += carry[3] h[3] -= carry[3] << 25 carry[4] = h[4] >> 26 h[5] += carry[4] h[4] -= carry[4] << 26 carry[5] = h[5] >> 25 h[6] += carry[5] h[5] -= carry[5] << 25 carry[6] = h[6] >> 26 h[7] += carry[6] h[6] -= carry[6] << 26 carry[7] = h[7] >> 25 h[8] += carry[7] h[7] -= carry[7] << 25 carry[8] = h[8] >> 26 h[9] += carry[8] h[8] -= carry[8] << 26 carry[9] = h[9] >> 25 h[9] -= carry[9] << 25 // h10 = carry9 // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; // evidently 2^255 h10-2^255 q = 0. // Goal: Output h[0]+...+2^230 h[9]. s[0] = byte(h[0] >> 0) s[1] = byte(h[0] >> 8) s[2] = byte(h[0] >> 16) s[3] = byte((h[0] >> 24) | (h[1] << 2)) s[4] = byte(h[1] >> 6) s[5] = byte(h[1] >> 14) s[6] = byte((h[1] >> 22) | (h[2] << 3)) s[7] = byte(h[2] >> 5) s[8] = byte(h[2] >> 13) s[9] = byte((h[2] >> 21) | (h[3] << 5)) s[10] = byte(h[3] >> 3) s[11] = byte(h[3] >> 11) s[12] = byte((h[3] >> 19) | (h[4] << 6)) s[13] = byte(h[4] >> 2) s[14] = byte(h[4] >> 10) s[15] = byte(h[4] >> 18) s[16] = byte(h[5] >> 0) s[17] = byte(h[5] >> 8) s[18] = byte(h[5] >> 16) s[19] = byte((h[5] >> 24) | (h[6] << 1)) s[20] = byte(h[6] >> 7) s[21] = byte(h[6] >> 15) s[22] = byte((h[6] >> 23) | (h[7] << 3)) s[23] = byte(h[7] >> 5) s[24] = byte(h[7] >> 13) s[25] = byte((h[7] >> 21) | (h[8] << 4)) s[26] = byte(h[8] >> 4) s[27] = byte(h[8] >> 12) s[28] = byte((h[8] >> 20) | (h[9] << 6)) s[29] = byte(h[9] >> 2) s[30] = byte(h[9] >> 10) s[31] = byte(h[9] >> 18) } func FeIsNegative(f *FieldElement) byte { var s [32]byte FeToBytes(&s, f) return s[0] & 1 } func FeIsNonZero(f *FieldElement) int32 { var s [32]byte FeToBytes(&s, f) var x uint8 for _, b := range s { x |= b } x |= x >> 4 x |= x >> 2 x |= x >> 1 return int32(x & 1) } // FeNeg sets h = -f // // Preconditions: // |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. // // Postconditions: // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. func FeNeg(h, f *FieldElement) { h[0] = -f[0] h[1] = -f[1] h[2] = -f[2] h[3] = -f[3] h[4] = -f[4] h[5] = -f[5] h[6] = -f[6] h[7] = -f[7] h[8] = -f[8] h[9] = -f[9] } func FeCombine(h *FieldElement, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { var c0, c1, c2, c3, c4, c5, c6, c7, c8, c9 int64 /* |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8 |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9 */ c0 = (h0 + (1 << 25)) >> 26 h1 += c0 h0 -= c0 << 26 c4 = (h4 + (1 << 25)) >> 26 h5 += c4 h4 -= c4 << 26 /* |h0| <= 2^25 */ /* |h4| <= 2^25 */ /* |h1| <= 1.51*2^58 */ /* |h5| <= 1.51*2^58 */ c1 = (h1 + (1 << 24)) >> 25 h2 += c1 h1 -= c1 << 25 c5 = (h5 + (1 << 24)) >> 25 h6 += c5 h5 -= c5 << 25 /* |h1| <= 2^24; from now on fits into int32 */ /* |h5| <= 2^24; from now on fits into int32 */ /* |h2| <= 1.21*2^59 */ /* |h6| <= 1.21*2^59 */ c2 = (h2 + (1 << 25)) >> 26 h3 += c2 h2 -= c2 << 26 c6 = (h6 + (1 << 25)) >> 26 h7 += c6 h6 -= c6 << 26 /* |h2| <= 2^25; from now on fits into int32 unchanged */ /* |h6| <= 2^25; from now on fits into int32 unchanged */ /* |h3| <= 1.51*2^58 */ /* |h7| <= 1.51*2^58 */ c3 = (h3 + (1 << 24)) >> 25 h4 += c3 h3 -= c3 << 25 c7 = (h7 + (1 << 24)) >> 25 h8 += c7 h7 -= c7 << 25 /* |h3| <= 2^24; from now on fits into int32 unchanged */ /* |h7| <= 2^24; from now on fits into int32 unchanged */ /* |h4| <= 1.52*2^33 */ /* |h8| <= 1.52*2^33 */ c4 = (h4 + (1 << 25)) >> 26 h5 += c4 h4 -= c4 << 26 c8 = (h8 + (1 << 25)) >> 26 h9 += c8 h8 -= c8 << 26 /* |h4| <= 2^25; from now on fits into int32 unchanged */ /* |h8| <= 2^25; from now on fits into int32 unchanged */ /* |h5| <= 1.01*2^24 */ /* |h9| <= 1.51*2^58 */ c9 = (h9 + (1 << 24)) >> 25 h0 += c9 * 19 h9 -= c9 << 25 /* |h9| <= 2^24; from now on fits into int32 unchanged */ /* |h0| <= 1.8*2^37 */ c0 = (h0 + (1 << 25)) >> 26 h1 += c0 h0 -= c0 << 26 /* |h0| <= 2^25; from now on fits into int32 unchanged */ /* |h1| <= 1.01*2^24 */ h[0] = int32(h0) h[1] = int32(h1) h[2] = int32(h2) h[3] = int32(h3) h[4] = int32(h4) h[5] = int32(h5) h[6] = int32(h6) h[7] = int32(h7) h[8] = int32(h8) h[9] = int32(h9) } // FeMul calculates h = f * g // Can overlap h with f or g. // // Preconditions: // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. // |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. // // Postconditions: // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. // // Notes on implementation strategy: // // Using schoolbook multiplication. // Karatsuba would save a little in some cost models. // // Most multiplications by 2 and 19 are 32-bit precomputations; // cheaper than 64-bit postcomputations. // // There is one remaining multiplication by 19 in the carry chain; // one *19 precomputation can be merged into this, // but the resulting data flow is considerably less clean. // // There are 12 carries below. // 10 of them are 2-way parallelizable and vectorizable. // Can get away with 11 carries, but then data flow is much deeper. // // With tighter constraints on inputs, can squeeze carries into int32. func FeMul(h, f, g *FieldElement) { f0 := int64(f[0]) f1 := int64(f[1]) f2 := int64(f[2]) f3 := int64(f[3]) f4 := int64(f[4]) f5 := int64(f[5]) f6 := int64(f[6]) f7 := int64(f[7]) f8 := int64(f[8]) f9 := int64(f[9]) f1_2 := int64(2 * f[1]) f3_2 := int64(2 * f[3]) f5_2 := int64(2 * f[5]) f7_2 := int64(2 * f[7]) f9_2 := int64(2 * f[9]) g0 := int64(g[0]) g1 := int64(g[1]) g2 := int64(g[2]) g3 := int64(g[3]) g4 := int64(g[4]) g5 := int64(g[5]) g6 := int64(g[6]) g7 := int64(g[7]) g8 := int64(g[8]) g9 := int64(g[9]) g1_19 := int64(19 * g[1]) /* 1.4*2^29 */ g2_19 := int64(19 * g[2]) /* 1.4*2^30; still ok */ g3_19 := int64(19 * g[3]) g4_19 := int64(19 * g[4]) g5_19 := int64(19 * g[5]) g6_19 := int64(19 * g[6]) g7_19 := int64(19 * g[7]) g8_19 := int64(19 * g[8]) g9_19 := int64(19 * g[9]) h0 := f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19 h1 := f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19 h2 := f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19 h3 := f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19 h4 := f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19 h5 := f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19 h6 := f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19 h7 := f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19 h8 := f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19 h9 := f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0 FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) } func feSquare(f *FieldElement) (h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { f0 := int64(f[0]) f1 := int64(f[1]) f2 := int64(f[2]) f3 := int64(f[3]) f4 := int64(f[4]) f5 := int64(f[5]) f6 := int64(f[6]) f7 := int64(f[7]) f8 := int64(f[8]) f9 := int64(f[9]) f0_2 := int64(2 * f[0]) f1_2 := int64(2 * f[1]) f2_2 := int64(2 * f[2]) f3_2 := int64(2 * f[3]) f4_2 := int64(2 * f[4]) f5_2 := int64(2 * f[5]) f6_2 := int64(2 * f[6]) f7_2 := int64(2 * f[7]) f5_38 := 38 * f5 // 1.31*2^30 f6_19 := 19 * f6 // 1.31*2^30 f7_38 := 38 * f7 // 1.31*2^30 f8_19 := 19 * f8 // 1.31*2^30 f9_38 := 38 * f9 // 1.31*2^30 h0 = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38 h1 = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19 h2 = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19 h3 = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38 h4 = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38 h5 = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19 h6 = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19 h7 = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38 h8 = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38 h9 = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5 return } // FeSquare calculates h = f*f. Can overlap h with f. // // Preconditions: // |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. // // Postconditions: // |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. func FeSquare(h, f *FieldElement) { h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) } // FeSquare2 sets h = 2 * f * f // // Can overlap h with f. // // Preconditions: // |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. // // Postconditions: // |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. // See fe_mul.c for discussion of implementation strategy. func FeSquare2(h, f *FieldElement) { h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) h0 += h0 h1 += h1 h2 += h2 h3 += h3 h4 += h4 h5 += h5 h6 += h6 h7 += h7 h8 += h8 h9 += h9 FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) } func FeInvert(out, z *FieldElement) { var t0, t1, t2, t3 FieldElement var i int FeSquare(&t0, z) // 2^1 FeSquare(&t1, &t0) // 2^2 for i = 1; i < 2; i++ { // 2^3 FeSquare(&t1, &t1) } FeMul(&t1, z, &t1) // 2^3 + 2^0 FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0 FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1 FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0 FeSquare(&t2, &t1) // 5,4,3,2,1 for i = 1; i < 5; i++ { // 9,8,7,6,5 FeSquare(&t2, &t2) } FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 FeSquare(&t2, &t1) // 10..1 for i = 1; i < 10; i++ { // 19..10 FeSquare(&t2, &t2) } FeMul(&t2, &t2, &t1) // 19..0 FeSquare(&t3, &t2) // 20..1 for i = 1; i < 20; i++ { // 39..20 FeSquare(&t3, &t3) } FeMul(&t2, &t3, &t2) // 39..0 FeSquare(&t2, &t2) // 40..1 for i = 1; i < 10; i++ { // 49..10 FeSquare(&t2, &t2) } FeMul(&t1, &t2, &t1) // 49..0 FeSquare(&t2, &t1) // 50..1 for i = 1; i < 50; i++ { // 99..50 FeSquare(&t2, &t2) } FeMul(&t2, &t2, &t1) // 99..0 FeSquare(&t3, &t2) // 100..1 for i = 1; i < 100; i++ { // 199..100 FeSquare(&t3, &t3) } FeMul(&t2, &t3, &t2) // 199..0 FeSquare(&t2, &t2) // 200..1 for i = 1; i < 50; i++ { // 249..50 FeSquare(&t2, &t2) } FeMul(&t1, &t2, &t1) // 249..0 FeSquare(&t1, &t1) // 250..1 for i = 1; i < 5; i++ { // 254..5 FeSquare(&t1, &t1) } FeMul(out, &t1, &t0) // 254..5,3,1,0 } func fePow22523(out, z *FieldElement) { var t0, t1, t2 FieldElement var i int FeSquare(&t0, z) for i = 1; i < 1; i++ { FeSquare(&t0, &t0) } FeSquare(&t1, &t0) for i = 1; i < 2; i++ { FeSquare(&t1, &t1) } FeMul(&t1, z, &t1) FeMul(&t0, &t0, &t1) FeSquare(&t0, &t0) for i = 1; i < 1; i++ { FeSquare(&t0, &t0) } FeMul(&t0, &t1, &t0) FeSquare(&t1, &t0) for i = 1; i < 5; i++ { FeSquare(&t1, &t1) } FeMul(&t0, &t1, &t0) FeSquare(&t1, &t0) for i = 1; i < 10; i++ { FeSquare(&t1, &t1) } FeMul(&t1, &t1, &t0) FeSquare(&t2, &t1) for i = 1; i < 20; i++ { FeSquare(&t2, &t2) } FeMul(&t1, &t2, &t1) FeSquare(&t1, &t1) for i = 1; i < 10; i++ { FeSquare(&t1, &t1) } FeMul(&t0, &t1, &t0) FeSquare(&t1, &t0) for i = 1; i < 50; i++ { FeSquare(&t1, &t1) } FeMul(&t1, &t1, &t0) FeSquare(&t2, &t1) for i = 1; i < 100; i++ { FeSquare(&t2, &t2) } FeMul(&t1, &t2, &t1) FeSquare(&t1, &t1) for i = 1; i < 50; i++ { FeSquare(&t1, &t1) } FeMul(&t0, &t1, &t0) FeSquare(&t0, &t0) for i = 1; i < 2; i++ { FeSquare(&t0, &t0) } FeMul(out, &t0, z) } // Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 * // y^2 where d = -121665/121666. // // Several representations are used: // ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z // ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT // CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T // PreComputedGroupElement: (y+x,y-x,2dxy) type ProjectiveGroupElement struct { X, Y, Z FieldElement } type ExtendedGroupElement struct { X, Y, Z, T FieldElement } type CompletedGroupElement struct { X, Y, Z, T FieldElement } type PreComputedGroupElement struct { yPlusX, yMinusX, xy2d FieldElement } type CachedGroupElement struct { yPlusX, yMinusX, Z, T2d FieldElement } func (p *ProjectiveGroupElement) Zero() { FeZero(&p.X) FeOne(&p.Y) FeOne(&p.Z) } func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) { var t0 FieldElement FeSquare(&r.X, &p.X) FeSquare(&r.Z, &p.Y) FeSquare2(&r.T, &p.Z) FeAdd(&r.Y, &p.X, &p.Y) FeSquare(&t0, &r.Y) FeAdd(&r.Y, &r.Z, &r.X) FeSub(&r.Z, &r.Z, &r.X) FeSub(&r.X, &t0, &r.Y) FeSub(&r.T, &r.T, &r.Z) } func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) { var recip, x, y FieldElement FeInvert(&recip, &p.Z) FeMul(&x, &p.X, &recip) FeMul(&y, &p.Y, &recip) FeToBytes(s, &y) s[31] ^= FeIsNegative(&x) << 7 } func (p *ExtendedGroupElement) Zero() { FeZero(&p.X) FeOne(&p.Y) FeOne(&p.Z) FeZero(&p.T) } func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) { var q ProjectiveGroupElement p.ToProjective(&q) q.Double(r) } func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) { FeAdd(&r.yPlusX, &p.Y, &p.X) FeSub(&r.yMinusX, &p.Y, &p.X) FeCopy(&r.Z, &p.Z) FeMul(&r.T2d, &p.T, &d2) } func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) { FeCopy(&r.X, &p.X) FeCopy(&r.Y, &p.Y) FeCopy(&r.Z, &p.Z) } func (p *ExtendedGroupElement) ToBytes(s *[32]byte) { var recip, x, y FieldElement FeInvert(&recip, &p.Z) FeMul(&x, &p.X, &recip) FeMul(&y, &p.Y, &recip) FeToBytes(s, &y) s[31] ^= FeIsNegative(&x) << 7 } func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool { var u, v, v3, vxx, check FieldElement FeFromBytes(&p.Y, s) FeOne(&p.Z) FeSquare(&u, &p.Y) FeMul(&v, &u, &d) FeSub(&u, &u, &p.Z) // y = y^2-1 FeAdd(&v, &v, &p.Z) // v = dy^2+1 FeSquare(&v3, &v) FeMul(&v3, &v3, &v) // v3 = v^3 FeSquare(&p.X, &v3) FeMul(&p.X, &p.X, &v) FeMul(&p.X, &p.X, &u) // x = uv^7 fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8) FeMul(&p.X, &p.X, &v3) FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8) var tmpX, tmp2 [32]byte FeSquare(&vxx, &p.X) FeMul(&vxx, &vxx, &v) FeSub(&check, &vxx, &u) // vx^2-u if FeIsNonZero(&check) == 1 { FeAdd(&check, &vxx, &u) // vx^2+u if FeIsNonZero(&check) == 1 { return false } FeMul(&p.X, &p.X, &SqrtM1) FeToBytes(&tmpX, &p.X) for i, v := range tmpX { tmp2[31-i] = v } } if FeIsNegative(&p.X) != (s[31] >> 7) { FeNeg(&p.X, &p.X) } FeMul(&p.T, &p.X, &p.Y) return true } func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) { FeMul(&r.X, &p.X, &p.T) FeMul(&r.Y, &p.Y, &p.Z) FeMul(&r.Z, &p.Z, &p.T) } func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) { FeMul(&r.X, &p.X, &p.T) FeMul(&r.Y, &p.Y, &p.Z) FeMul(&r.Z, &p.Z, &p.T) FeMul(&r.T, &p.X, &p.Y) } func (p *PreComputedGroupElement) Zero() { FeOne(&p.yPlusX) FeOne(&p.yMinusX) FeZero(&p.xy2d) } func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { var t0 FieldElement FeAdd(&r.X, &p.Y, &p.X) FeSub(&r.Y, &p.Y, &p.X) FeMul(&r.Z, &r.X, &q.yPlusX) FeMul(&r.Y, &r.Y, &q.yMinusX) FeMul(&r.T, &q.T2d, &p.T) FeMul(&r.X, &p.Z, &q.Z) FeAdd(&t0, &r.X, &r.X) FeSub(&r.X, &r.Z, &r.Y) FeAdd(&r.Y, &r.Z, &r.Y) FeAdd(&r.Z, &t0, &r.T) FeSub(&r.T, &t0, &r.T) } func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { var t0 FieldElement FeAdd(&r.X, &p.Y, &p.X) FeSub(&r.Y, &p.Y, &p.X) FeMul(&r.Z, &r.X, &q.yMinusX) FeMul(&r.Y, &r.Y, &q.yPlusX) FeMul(&r.T, &q.T2d, &p.T) FeMul(&r.X, &p.Z, &q.Z) FeAdd(&t0, &r.X, &r.X) FeSub(&r.X, &r.Z, &r.Y) FeAdd(&r.Y, &r.Z, &r.Y) FeSub(&r.Z, &t0, &r.T) FeAdd(&r.T, &t0, &r.T) } func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { var t0 FieldElement FeAdd(&r.X, &p.Y, &p.X) FeSub(&r.Y, &p.Y, &p.X) FeMul(&r.Z, &r.X, &q.yPlusX) FeMul(&r.Y, &r.Y, &q.yMinusX) FeMul(&r.T, &q.xy2d, &p.T) FeAdd(&t0, &p.Z, &p.Z) FeSub(&r.X, &r.Z, &r.Y) FeAdd(&r.Y, &r.Z, &r.Y) FeAdd(&r.Z, &t0, &r.T) FeSub(&r.T, &t0, &r.T) } func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { var t0 FieldElement FeAdd(&r.X, &p.Y, &p.X) FeSub(&r.Y, &p.Y, &p.X) FeMul(&r.Z, &r.X, &q.yMinusX) FeMul(&r.Y, &r.Y, &q.yPlusX) FeMul(&r.T, &q.xy2d, &p.T) FeAdd(&t0, &p.Z, &p.Z) FeSub(&r.X, &r.Z, &r.Y) FeAdd(&r.Y, &r.Z, &r.Y) FeSub(&r.Z, &t0, &r.T) FeAdd(&r.T, &t0, &r.T) } func slide(r *[256]int8, a *[32]byte) { for i := range r { r[i] = int8(1 & (a[i>>3] >> uint(i&7))) } for i := range r { if r[i] != 0 { for b := 1; b <= 6 && i+b < 256; b++ { if r[i+b] != 0 { if r[i]+(r[i+b]<= -15 { r[i] -= r[i+b] << uint(b) for k := i + b; k < 256; k++ { if r[k] == 0 { r[k] = 1 break } r[k] = 0 } } else { break } } } } } } // GeDoubleScalarMultVartime sets r = a*A + b*B // where a = a[0]+256*a[1]+...+256^31 a[31]. // and b = b[0]+256*b[1]+...+256^31 b[31]. // B is the Ed25519 base point (x,4/5) with x positive. func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { var aSlide, bSlide [256]int8 var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A var t CompletedGroupElement var u, A2 ExtendedGroupElement var i int slide(&aSlide, a) slide(&bSlide, b) A.ToCached(&Ai[0]) A.Double(&t) t.ToExtended(&A2) for i := 0; i < 7; i++ { geAdd(&t, &A2, &Ai[i]) t.ToExtended(&u) u.ToCached(&Ai[i+1]) } r.Zero() for i = 255; i >= 0; i-- { if aSlide[i] != 0 || bSlide[i] != 0 { break } } for ; i >= 0; i-- { r.Double(&t) if aSlide[i] > 0 { t.ToExtended(&u) geAdd(&t, &u, &Ai[aSlide[i]/2]) } else if aSlide[i] < 0 { t.ToExtended(&u) geSub(&t, &u, &Ai[(-aSlide[i])/2]) } if bSlide[i] > 0 { t.ToExtended(&u) geMixedAdd(&t, &u, &bi[bSlide[i]/2]) } else if bSlide[i] < 0 { t.ToExtended(&u) geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) } t.ToProjective(r) } } // equal returns 1 if b == c and 0 otherwise, assuming that b and c are // non-negative. func equal(b, c int32) int32 { x := uint32(b ^ c) x-- return int32(x >> 31) } // negative returns 1 if b < 0 and 0 otherwise. func negative(b int32) int32 { return (b >> 31) & 1 } func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { FeCMove(&t.yPlusX, &u.yPlusX, b) FeCMove(&t.yMinusX, &u.yMinusX, b) FeCMove(&t.xy2d, &u.xy2d, b) } func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { var minusT PreComputedGroupElement bNegative := negative(b) bAbs := b - (((-bNegative) & b) << 1) t.Zero() for i := int32(0); i < 8; i++ { PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) } FeCopy(&minusT.yPlusX, &t.yMinusX) FeCopy(&minusT.yMinusX, &t.yPlusX) FeNeg(&minusT.xy2d, &t.xy2d) PreComputedGroupElementCMove(t, &minusT, bNegative) } // GeScalarMultBase computes h = a*B, where // a = a[0]+256*a[1]+...+256^31 a[31] // B is the Ed25519 base point (x,4/5) with x positive. // // Preconditions: // a[31] <= 127 func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { var e [64]int8 for i, v := range a { e[2*i] = int8(v & 15) e[2*i+1] = int8((v >> 4) & 15) } // each e[i] is between 0 and 15 and e[63] is between 0 and 7. carry := int8(0) for i := 0; i < 63; i++ { e[i] += carry carry = (e[i] + 8) >> 4 e[i] -= carry << 4 } e[63] += carry // each e[i] is between -8 and 8. h.Zero() var t PreComputedGroupElement var r CompletedGroupElement for i := int32(1); i < 64; i += 2 { selectPoint(&t, i/2, int32(e[i])) geMixedAdd(&r, h, &t) r.ToExtended(h) } var s ProjectiveGroupElement h.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToProjective(&s) s.Double(&r) r.ToExtended(h) for i := int32(0); i < 64; i += 2 { selectPoint(&t, i/2, int32(e[i])) geMixedAdd(&r, h, &t) r.ToExtended(h) } } // The scalars are GF(2^252 + 27742317777372353535851937790883648493). // Input: // a[0]+256*a[1]+...+256^31*a[31] = a // b[0]+256*b[1]+...+256^31*b[31] = b // c[0]+256*c[1]+...+256^31*c[31] = c // // Output: // s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l // where l = 2^252 + 27742317777372353535851937790883648493. func ScMulAdd(s, a, b, c *[32]byte) { a0 := 2097151 & load3(a[:]) a1 := 2097151 & (load4(a[2:]) >> 5) a2 := 2097151 & (load3(a[5:]) >> 2) a3 := 2097151 & (load4(a[7:]) >> 7) a4 := 2097151 & (load4(a[10:]) >> 4) a5 := 2097151 & (load3(a[13:]) >> 1) a6 := 2097151 & (load4(a[15:]) >> 6) a7 := 2097151 & (load3(a[18:]) >> 3) a8 := 2097151 & load3(a[21:]) a9 := 2097151 & (load4(a[23:]) >> 5) a10 := 2097151 & (load3(a[26:]) >> 2) a11 := (load4(a[28:]) >> 7) b0 := 2097151 & load3(b[:]) b1 := 2097151 & (load4(b[2:]) >> 5) b2 := 2097151 & (load3(b[5:]) >> 2) b3 := 2097151 & (load4(b[7:]) >> 7) b4 := 2097151 & (load4(b[10:]) >> 4) b5 := 2097151 & (load3(b[13:]) >> 1) b6 := 2097151 & (load4(b[15:]) >> 6) b7 := 2097151 & (load3(b[18:]) >> 3) b8 := 2097151 & load3(b[21:]) b9 := 2097151 & (load4(b[23:]) >> 5) b10 := 2097151 & (load3(b[26:]) >> 2) b11 := (load4(b[28:]) >> 7) c0 := 2097151 & load3(c[:]) c1 := 2097151 & (load4(c[2:]) >> 5) c2 := 2097151 & (load3(c[5:]) >> 2) c3 := 2097151 & (load4(c[7:]) >> 7) c4 := 2097151 & (load4(c[10:]) >> 4) c5 := 2097151 & (load3(c[13:]) >> 1) c6 := 2097151 & (load4(c[15:]) >> 6) c7 := 2097151 & (load3(c[18:]) >> 3) c8 := 2097151 & load3(c[21:]) c9 := 2097151 & (load4(c[23:]) >> 5) c10 := 2097151 & (load3(c[26:]) >> 2) c11 := (load4(c[28:]) >> 7) var carry [23]int64 s0 := c0 + a0*b0 s1 := c1 + a0*b1 + a1*b0 s2 := c2 + a0*b2 + a1*b1 + a2*b0 s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0 s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 s20 := a9*b11 + a10*b10 + a11*b9 s21 := a10*b11 + a11*b10 s22 := a11 * b11 s23 := int64(0) carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[18] = (s18 + (1 << 20)) >> 21 s19 += carry[18] s18 -= carry[18] << 21 carry[20] = (s20 + (1 << 20)) >> 21 s21 += carry[20] s20 -= carry[20] << 21 carry[22] = (s22 + (1 << 20)) >> 21 s23 += carry[22] s22 -= carry[22] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 carry[17] = (s17 + (1 << 20)) >> 21 s18 += carry[17] s17 -= carry[17] << 21 carry[19] = (s19 + (1 << 20)) >> 21 s20 += carry[19] s19 -= carry[19] << 21 carry[21] = (s21 + (1 << 20)) >> 21 s22 += carry[21] s21 -= carry[21] << 21 s11 += s23 * 666643 s12 += s23 * 470296 s13 += s23 * 654183 s14 -= s23 * 997805 s15 += s23 * 136657 s16 -= s23 * 683901 s23 = 0 s10 += s22 * 666643 s11 += s22 * 470296 s12 += s22 * 654183 s13 -= s22 * 997805 s14 += s22 * 136657 s15 -= s22 * 683901 s22 = 0 s9 += s21 * 666643 s10 += s21 * 470296 s11 += s21 * 654183 s12 -= s21 * 997805 s13 += s21 * 136657 s14 -= s21 * 683901 s21 = 0 s8 += s20 * 666643 s9 += s20 * 470296 s10 += s20 * 654183 s11 -= s20 * 997805 s12 += s20 * 136657 s13 -= s20 * 683901 s20 = 0 s7 += s19 * 666643 s8 += s19 * 470296 s9 += s19 * 654183 s10 -= s19 * 997805 s11 += s19 * 136657 s12 -= s19 * 683901 s19 = 0 s6 += s18 * 666643 s7 += s18 * 470296 s8 += s18 * 654183 s9 -= s18 * 997805 s10 += s18 * 136657 s11 -= s18 * 683901 s18 = 0 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 s5 += s17 * 666643 s6 += s17 * 470296 s7 += s17 * 654183 s8 -= s17 * 997805 s9 += s17 * 136657 s10 -= s17 * 683901 s17 = 0 s4 += s16 * 666643 s5 += s16 * 470296 s6 += s16 * 654183 s7 -= s16 * 997805 s8 += s16 * 136657 s9 -= s16 * 683901 s16 = 0 s3 += s15 * 666643 s4 += s15 * 470296 s5 += s15 * 654183 s6 -= s15 * 997805 s7 += s15 * 136657 s8 -= s15 * 683901 s15 = 0 s2 += s14 * 666643 s3 += s14 * 470296 s4 += s14 * 654183 s5 -= s14 * 997805 s6 += s14 * 136657 s7 -= s14 * 683901 s14 = 0 s1 += s13 * 666643 s2 += s13 * 470296 s3 += s13 * 654183 s4 -= s13 * 997805 s5 += s13 * 136657 s6 -= s13 * 683901 s13 = 0 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[11] = s11 >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 s[0] = byte(s0 >> 0) s[1] = byte(s0 >> 8) s[2] = byte((s0 >> 16) | (s1 << 5)) s[3] = byte(s1 >> 3) s[4] = byte(s1 >> 11) s[5] = byte((s1 >> 19) | (s2 << 2)) s[6] = byte(s2 >> 6) s[7] = byte((s2 >> 14) | (s3 << 7)) s[8] = byte(s3 >> 1) s[9] = byte(s3 >> 9) s[10] = byte((s3 >> 17) | (s4 << 4)) s[11] = byte(s4 >> 4) s[12] = byte(s4 >> 12) s[13] = byte((s4 >> 20) | (s5 << 1)) s[14] = byte(s5 >> 7) s[15] = byte((s5 >> 15) | (s6 << 6)) s[16] = byte(s6 >> 2) s[17] = byte(s6 >> 10) s[18] = byte((s6 >> 18) | (s7 << 3)) s[19] = byte(s7 >> 5) s[20] = byte(s7 >> 13) s[21] = byte(s8 >> 0) s[22] = byte(s8 >> 8) s[23] = byte((s8 >> 16) | (s9 << 5)) s[24] = byte(s9 >> 3) s[25] = byte(s9 >> 11) s[26] = byte((s9 >> 19) | (s10 << 2)) s[27] = byte(s10 >> 6) s[28] = byte((s10 >> 14) | (s11 << 7)) s[29] = byte(s11 >> 1) s[30] = byte(s11 >> 9) s[31] = byte(s11 >> 17) } // Input: // s[0]+256*s[1]+...+256^63*s[63] = s // // Output: // s[0]+256*s[1]+...+256^31*s[31] = s mod l // where l = 2^252 + 27742317777372353535851937790883648493. func ScReduce(out *[32]byte, s *[64]byte) { s0 := 2097151 & load3(s[:]) s1 := 2097151 & (load4(s[2:]) >> 5) s2 := 2097151 & (load3(s[5:]) >> 2) s3 := 2097151 & (load4(s[7:]) >> 7) s4 := 2097151 & (load4(s[10:]) >> 4) s5 := 2097151 & (load3(s[13:]) >> 1) s6 := 2097151 & (load4(s[15:]) >> 6) s7 := 2097151 & (load3(s[18:]) >> 3) s8 := 2097151 & load3(s[21:]) s9 := 2097151 & (load4(s[23:]) >> 5) s10 := 2097151 & (load3(s[26:]) >> 2) s11 := 2097151 & (load4(s[28:]) >> 7) s12 := 2097151 & (load4(s[31:]) >> 4) s13 := 2097151 & (load3(s[34:]) >> 1) s14 := 2097151 & (load4(s[36:]) >> 6) s15 := 2097151 & (load3(s[39:]) >> 3) s16 := 2097151 & load3(s[42:]) s17 := 2097151 & (load4(s[44:]) >> 5) s18 := 2097151 & (load3(s[47:]) >> 2) s19 := 2097151 & (load4(s[49:]) >> 7) s20 := 2097151 & (load4(s[52:]) >> 4) s21 := 2097151 & (load3(s[55:]) >> 1) s22 := 2097151 & (load4(s[57:]) >> 6) s23 := (load4(s[60:]) >> 3) s11 += s23 * 666643 s12 += s23 * 470296 s13 += s23 * 654183 s14 -= s23 * 997805 s15 += s23 * 136657 s16 -= s23 * 683901 s23 = 0 s10 += s22 * 666643 s11 += s22 * 470296 s12 += s22 * 654183 s13 -= s22 * 997805 s14 += s22 * 136657 s15 -= s22 * 683901 s22 = 0 s9 += s21 * 666643 s10 += s21 * 470296 s11 += s21 * 654183 s12 -= s21 * 997805 s13 += s21 * 136657 s14 -= s21 * 683901 s21 = 0 s8 += s20 * 666643 s9 += s20 * 470296 s10 += s20 * 654183 s11 -= s20 * 997805 s12 += s20 * 136657 s13 -= s20 * 683901 s20 = 0 s7 += s19 * 666643 s8 += s19 * 470296 s9 += s19 * 654183 s10 -= s19 * 997805 s11 += s19 * 136657 s12 -= s19 * 683901 s19 = 0 s6 += s18 * 666643 s7 += s18 * 470296 s8 += s18 * 654183 s9 -= s18 * 997805 s10 += s18 * 136657 s11 -= s18 * 683901 s18 = 0 var carry [17]int64 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[12] = (s12 + (1 << 20)) >> 21 s13 += carry[12] s12 -= carry[12] << 21 carry[14] = (s14 + (1 << 20)) >> 21 s15 += carry[14] s14 -= carry[14] << 21 carry[16] = (s16 + (1 << 20)) >> 21 s17 += carry[16] s16 -= carry[16] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 carry[13] = (s13 + (1 << 20)) >> 21 s14 += carry[13] s13 -= carry[13] << 21 carry[15] = (s15 + (1 << 20)) >> 21 s16 += carry[15] s15 -= carry[15] << 21 s5 += s17 * 666643 s6 += s17 * 470296 s7 += s17 * 654183 s8 -= s17 * 997805 s9 += s17 * 136657 s10 -= s17 * 683901 s17 = 0 s4 += s16 * 666643 s5 += s16 * 470296 s6 += s16 * 654183 s7 -= s16 * 997805 s8 += s16 * 136657 s9 -= s16 * 683901 s16 = 0 s3 += s15 * 666643 s4 += s15 * 470296 s5 += s15 * 654183 s6 -= s15 * 997805 s7 += s15 * 136657 s8 -= s15 * 683901 s15 = 0 s2 += s14 * 666643 s3 += s14 * 470296 s4 += s14 * 654183 s5 -= s14 * 997805 s6 += s14 * 136657 s7 -= s14 * 683901 s14 = 0 s1 += s13 * 666643 s2 += s13 * 470296 s3 += s13 * 654183 s4 -= s13 * 997805 s5 += s13 * 136657 s6 -= s13 * 683901 s13 = 0 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = (s0 + (1 << 20)) >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[2] = (s2 + (1 << 20)) >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[4] = (s4 + (1 << 20)) >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[6] = (s6 + (1 << 20)) >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[8] = (s8 + (1 << 20)) >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[10] = (s10 + (1 << 20)) >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[1] = (s1 + (1 << 20)) >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[3] = (s3 + (1 << 20)) >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[5] = (s5 + (1 << 20)) >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[7] = (s7 + (1 << 20)) >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[9] = (s9 + (1 << 20)) >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[11] = (s11 + (1 << 20)) >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 carry[11] = s11 >> 21 s12 += carry[11] s11 -= carry[11] << 21 s0 += s12 * 666643 s1 += s12 * 470296 s2 += s12 * 654183 s3 -= s12 * 997805 s4 += s12 * 136657 s5 -= s12 * 683901 s12 = 0 carry[0] = s0 >> 21 s1 += carry[0] s0 -= carry[0] << 21 carry[1] = s1 >> 21 s2 += carry[1] s1 -= carry[1] << 21 carry[2] = s2 >> 21 s3 += carry[2] s2 -= carry[2] << 21 carry[3] = s3 >> 21 s4 += carry[3] s3 -= carry[3] << 21 carry[4] = s4 >> 21 s5 += carry[4] s4 -= carry[4] << 21 carry[5] = s5 >> 21 s6 += carry[5] s5 -= carry[5] << 21 carry[6] = s6 >> 21 s7 += carry[6] s6 -= carry[6] << 21 carry[7] = s7 >> 21 s8 += carry[7] s7 -= carry[7] << 21 carry[8] = s8 >> 21 s9 += carry[8] s8 -= carry[8] << 21 carry[9] = s9 >> 21 s10 += carry[9] s9 -= carry[9] << 21 carry[10] = s10 >> 21 s11 += carry[10] s10 -= carry[10] << 21 out[0] = byte(s0 >> 0) out[1] = byte(s0 >> 8) out[2] = byte((s0 >> 16) | (s1 << 5)) out[3] = byte(s1 >> 3) out[4] = byte(s1 >> 11) out[5] = byte((s1 >> 19) | (s2 << 2)) out[6] = byte(s2 >> 6) out[7] = byte((s2 >> 14) | (s3 << 7)) out[8] = byte(s3 >> 1) out[9] = byte(s3 >> 9) out[10] = byte((s3 >> 17) | (s4 << 4)) out[11] = byte(s4 >> 4) out[12] = byte(s4 >> 12) out[13] = byte((s4 >> 20) | (s5 << 1)) out[14] = byte(s5 >> 7) out[15] = byte((s5 >> 15) | (s6 << 6)) out[16] = byte(s6 >> 2) out[17] = byte(s6 >> 10) out[18] = byte((s6 >> 18) | (s7 << 3)) out[19] = byte(s7 >> 5) out[20] = byte(s7 >> 13) out[21] = byte(s8 >> 0) out[22] = byte(s8 >> 8) out[23] = byte((s8 >> 16) | (s9 << 5)) out[24] = byte(s9 >> 3) out[25] = byte(s9 >> 11) out[26] = byte((s9 >> 19) | (s10 << 2)) out[27] = byte(s10 >> 6) out[28] = byte((s10 >> 14) | (s11 << 7)) out[29] = byte(s11 >> 1) out[30] = byte(s11 >> 9) out[31] = byte(s11 >> 17) } // order is the order of Curve25519 in little-endian form. var order = [4]uint64{0x5812631a5cf5d3ed, 0x14def9dea2f79cd6, 0, 0x1000000000000000} // ScMinimal returns true if the given scalar is less than the order of the // curve. func ScMinimal(scalar *[32]byte) bool { for i := 3; ; i-- { v := binary.LittleEndian.Uint64(scalar[i*8:]) if v > order[i] { return false } else if v < order[i] { break } else if i == 0 { return false } } return true }