// Copyright 2013 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. // We have a implementation in amd64 assembly so this code is only run on // non-amd64 platforms. The amd64 assembly does not support gccgo. // +build !amd64 gccgo appengine package curve25519 import ( "encoding/binary" ) // This code is a port of the public domain, "ref10" implementation of // curve25519 from SUPERCOP 20130419 by D. J. Bernstein. // 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 func feZero(fe *fieldElement) { for i := range fe { fe[i] = 0 } } func feOne(fe *fieldElement) { feZero(fe) fe[0] = 1 } func feAdd(dst, a, b *fieldElement) { for i := range dst { dst[i] = a[i] + b[i] } } func feSub(dst, a, b *fieldElement) { for i := range dst { dst[i] = a[i] - b[i] } } func feCopy(dst, src *fieldElement) { for i := range dst { dst[i] = src[i] } } // feCSwap replaces (f,g) with (g,f) if b == 1; replaces (f,g) with (f,g) if b == 0. // // Preconditions: b in {0,1}. func feCSwap(f, g *fieldElement, b int32) { b = -b for i := range f { t := b & (f[i] ^ g[i]) f[i] ^= t g[i] ^= t } } // load3 reads a 24-bit, little-endian value from in. func load3(in []byte) int64 { var r int64 r = int64(in[0]) r |= int64(in[1]) << 8 r |= int64(in[2]) << 16 return r } // load4 reads a 32-bit, little-endian value from in. func load4(in []byte) int64 { return int64(binary.LittleEndian.Uint32(in)) } 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:]) << 2 var carry [10]int64 carry[9] = (h9 + 1<<24) >> 25 h0 += carry[9] * 19 h9 -= carry[9] << 25 carry[1] = (h1 + 1<<24) >> 25 h2 += carry[1] h1 -= carry[1] << 25 carry[3] = (h3 + 1<<24) >> 25 h4 += carry[3] h3 -= carry[3] << 25 carry[5] = (h5 + 1<<24) >> 25 h6 += carry[5] h5 -= carry[5] << 25 carry[7] = (h7 + 1<<24) >> 25 h8 += carry[7] h7 -= carry[7] << 25 carry[0] = (h0 + 1<<25) >> 26 h1 += carry[0] h0 -= carry[0] << 26 carry[2] = (h2 + 1<<25) >> 26 h3 += carry[2] h2 -= carry[2] << 26 carry[4] = (h4 + 1<<25) >> 26 h5 += carry[4] h4 -= carry[4] << 26 carry[6] = (h6 + 1<<25) >> 26 h7 += carry[6] h6 -= carry[6] << 26 carry[8] = (h8 + 1<<25) >> 26 h9 += carry[8] h8 -= carry[8] << 26 dst[0] = int32(h0) dst[1] = int32(h1) dst[2] = int32(h2) dst[3] = int32(h3) dst[4] = int32(h4) dst[5] = int32(h5) dst[6] = int32(h6) dst[7] = int32(h7) dst[8] = int32(h8) dst[9] = int32(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<y<1. // // Write r=h-pq. // Have 0<=r<=p-1=2^255-20. // Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1. // // Write x=r+19(2^-255)r+y. // Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q. // // Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1)) // so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. func feToBytes(s *[32]byte, h *fieldElement) { var carry [10]int32 q := (19*h[9] + (1 << 24)) >> 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) } // 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 := f[0] f1 := f[1] f2 := f[2] f3 := f[3] f4 := f[4] f5 := f[5] f6 := f[6] f7 := f[7] f8 := f[8] f9 := f[9] g0 := g[0] g1 := g[1] g2 := g[2] g3 := g[3] g4 := g[4] g5 := g[5] g6 := g[6] g7 := g[7] g8 := g[8] g9 := g[9] g1_19 := 19 * g1 // 1.4*2^29 g2_19 := 19 * g2 // 1.4*2^30; still ok g3_19 := 19 * g3 g4_19 := 19 * g4 g5_19 := 19 * g5 g6_19 := 19 * g6 g7_19 := 19 * g7 g8_19 := 19 * g8 g9_19 := 19 * g9 f1_2 := 2 * f1 f3_2 := 2 * f3 f5_2 := 2 * f5 f7_2 := 2 * f7 f9_2 := 2 * f9 f0g0 := int64(f0) * int64(g0) f0g1 := int64(f0) * int64(g1) f0g2 := int64(f0) * int64(g2) f0g3 := int64(f0) * int64(g3) f0g4 := int64(f0) * int64(g4) f0g5 := int64(f0) * int64(g5) f0g6 := int64(f0) * int64(g6) f0g7 := int64(f0) * int64(g7) f0g8 := int64(f0) * int64(g8) f0g9 := int64(f0) * int64(g9) f1g0 := int64(f1) * int64(g0) f1g1_2 := int64(f1_2) * int64(g1) f1g2 := int64(f1) * int64(g2) f1g3_2 := int64(f1_2) * int64(g3) f1g4 := int64(f1) * int64(g4) f1g5_2 := int64(f1_2) * int64(g5) f1g6 := int64(f1) * int64(g6) f1g7_2 := int64(f1_2) * int64(g7) f1g8 := int64(f1) * int64(g8) f1g9_38 := int64(f1_2) * int64(g9_19) f2g0 := int64(f2) * int64(g0) f2g1 := int64(f2) * int64(g1) f2g2 := int64(f2) * int64(g2) f2g3 := int64(f2) * int64(g3) f2g4 := int64(f2) * int64(g4) f2g5 := int64(f2) * int64(g5) f2g6 := int64(f2) * int64(g6) f2g7 := int64(f2) * int64(g7) f2g8_19 := int64(f2) * int64(g8_19) f2g9_19 := int64(f2) * int64(g9_19) f3g0 := int64(f3) * int64(g0) f3g1_2 := int64(f3_2) * int64(g1) f3g2 := int64(f3) * int64(g2) f3g3_2 := int64(f3_2) * int64(g3) f3g4 := int64(f3) * int64(g4) f3g5_2 := int64(f3_2) * int64(g5) f3g6 := int64(f3) * int64(g6) f3g7_38 := int64(f3_2) * int64(g7_19) f3g8_19 := int64(f3) * int64(g8_19) f3g9_38 := int64(f3_2) * int64(g9_19) f4g0 := int64(f4) * int64(g0) f4g1 := int64(f4) * int64(g1) f4g2 := int64(f4) * int64(g2) f4g3 := int64(f4) * int64(g3) f4g4 := int64(f4) * int64(g4) f4g5 := int64(f4) * int64(g5) f4g6_19 := int64(f4) * int64(g6_19) f4g7_19 := int64(f4) * int64(g7_19) f4g8_19 := int64(f4) * int64(g8_19) f4g9_19 := int64(f4) * int64(g9_19) f5g0 := int64(f5) * int64(g0) f5g1_2 := int64(f5_2) * int64(g1) f5g2 := int64(f5) * int64(g2) f5g3_2 := int64(f5_2) * int64(g3) f5g4 := int64(f5) * int64(g4) f5g5_38 := int64(f5_2) * int64(g5_19) f5g6_19 := int64(f5) * int64(g6_19) f5g7_38 := int64(f5_2) * int64(g7_19) f5g8_19 := int64(f5) * int64(g8_19) f5g9_38 := int64(f5_2) * int64(g9_19) f6g0 := int64(f6) * int64(g0) f6g1 := int64(f6) * int64(g1) f6g2 := int64(f6) * int64(g2) f6g3 := int64(f6) * int64(g3) f6g4_19 := int64(f6) * int64(g4_19) f6g5_19 := int64(f6) * int64(g5_19) f6g6_19 := int64(f6) * int64(g6_19) f6g7_19 := int64(f6) * int64(g7_19) f6g8_19 := int64(f6) * int64(g8_19) f6g9_19 := int64(f6) * int64(g9_19) f7g0 := int64(f7) * int64(g0) f7g1_2 := int64(f7_2) * int64(g1) f7g2 := int64(f7) * int64(g2) f7g3_38 := int64(f7_2) * int64(g3_19) f7g4_19 := int64(f7) * int64(g4_19) f7g5_38 := int64(f7_2) * int64(g5_19) f7g6_19 := int64(f7) * int64(g6_19) f7g7_38 := int64(f7_2) * int64(g7_19) f7g8_19 := int64(f7) * int64(g8_19) f7g9_38 := int64(f7_2) * int64(g9_19) f8g0 := int64(f8) * int64(g0) f8g1 := int64(f8) * int64(g1) f8g2_19 := int64(f8) * int64(g2_19) f8g3_19 := int64(f8) * int64(g3_19) f8g4_19 := int64(f8) * int64(g4_19) f8g5_19 := int64(f8) * int64(g5_19) f8g6_19 := int64(f8) * int64(g6_19) f8g7_19 := int64(f8) * int64(g7_19) f8g8_19 := int64(f8) * int64(g8_19) f8g9_19 := int64(f8) * int64(g9_19) f9g0 := int64(f9) * int64(g0) f9g1_38 := int64(f9_2) * int64(g1_19) f9g2_19 := int64(f9) * int64(g2_19) f9g3_38 := int64(f9_2) * int64(g3_19) f9g4_19 := int64(f9) * int64(g4_19) f9g5_38 := int64(f9_2) * int64(g5_19) f9g6_19 := int64(f9) * int64(g6_19) f9g7_38 := int64(f9_2) * int64(g7_19) f9g8_19 := int64(f9) * int64(g8_19) f9g9_38 := int64(f9_2) * int64(g9_19) h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38 h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19 h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38 h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19 h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38 h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19 h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38 h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19 h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38 h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 var carry [10]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 carry[0] = (h0 + (1 << 25)) >> 26 h1 += carry[0] h0 -= carry[0] << 26 carry[4] = (h4 + (1 << 25)) >> 26 h5 += carry[4] h4 -= carry[4] << 26 // |h0| <= 2^25 // |h4| <= 2^25 // |h1| <= 1.51*2^58 // |h5| <= 1.51*2^58 carry[1] = (h1 + (1 << 24)) >> 25 h2 += carry[1] h1 -= carry[1] << 25 carry[5] = (h5 + (1 << 24)) >> 25 h6 += carry[5] h5 -= carry[5] << 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 carry[2] = (h2 + (1 << 25)) >> 26 h3 += carry[2] h2 -= carry[2] << 26 carry[6] = (h6 + (1 << 25)) >> 26 h7 += carry[6] h6 -= carry[6] << 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 carry[3] = (h3 + (1 << 24)) >> 25 h4 += carry[3] h3 -= carry[3] << 25 carry[7] = (h7 + (1 << 24)) >> 25 h8 += carry[7] h7 -= carry[7] << 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 carry[4] = (h4 + (1 << 25)) >> 26 h5 += carry[4] h4 -= carry[4] << 26 carry[8] = (h8 + (1 << 25)) >> 26 h9 += carry[8] h8 -= carry[8] << 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 carry[9] = (h9 + (1 << 24)) >> 25 h0 += carry[9] * 19 h9 -= carry[9] << 25 // |h9| <= 2^24; from now on fits into int32 unchanged // |h0| <= 1.8*2^37 carry[0] = (h0 + (1 << 25)) >> 26 h1 += carry[0] h0 -= carry[0] << 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) } // 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) { f0 := f[0] f1 := f[1] f2 := f[2] f3 := f[3] f4 := f[4] f5 := f[5] f6 := f[6] f7 := f[7] f8 := f[8] f9 := f[9] f0_2 := 2 * f0 f1_2 := 2 * f1 f2_2 := 2 * f2 f3_2 := 2 * f3 f4_2 := 2 * f4 f5_2 := 2 * f5 f6_2 := 2 * f6 f7_2 := 2 * f7 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 f0f0 := int64(f0) * int64(f0) f0f1_2 := int64(f0_2) * int64(f1) f0f2_2 := int64(f0_2) * int64(f2) f0f3_2 := int64(f0_2) * int64(f3) f0f4_2 := int64(f0_2) * int64(f4) f0f5_2 := int64(f0_2) * int64(f5) f0f6_2 := int64(f0_2) * int64(f6) f0f7_2 := int64(f0_2) * int64(f7) f0f8_2 := int64(f0_2) * int64(f8) f0f9_2 := int64(f0_2) * int64(f9) f1f1_2 := int64(f1_2) * int64(f1) f1f2_2 := int64(f1_2) * int64(f2) f1f3_4 := int64(f1_2) * int64(f3_2) f1f4_2 := int64(f1_2) * int64(f4) f1f5_4 := int64(f1_2) * int64(f5_2) f1f6_2 := int64(f1_2) * int64(f6) f1f7_4 := int64(f1_2) * int64(f7_2) f1f8_2 := int64(f1_2) * int64(f8) f1f9_76 := int64(f1_2) * int64(f9_38) f2f2 := int64(f2) * int64(f2) f2f3_2 := int64(f2_2) * int64(f3) f2f4_2 := int64(f2_2) * int64(f4) f2f5_2 := int64(f2_2) * int64(f5) f2f6_2 := int64(f2_2) * int64(f6) f2f7_2 := int64(f2_2) * int64(f7) f2f8_38 := int64(f2_2) * int64(f8_19) f2f9_38 := int64(f2) * int64(f9_38) f3f3_2 := int64(f3_2) * int64(f3) f3f4_2 := int64(f3_2) * int64(f4) f3f5_4 := int64(f3_2) * int64(f5_2) f3f6_2 := int64(f3_2) * int64(f6) f3f7_76 := int64(f3_2) * int64(f7_38) f3f8_38 := int64(f3_2) * int64(f8_19) f3f9_76 := int64(f3_2) * int64(f9_38) f4f4 := int64(f4) * int64(f4) f4f5_2 := int64(f4_2) * int64(f5) f4f6_38 := int64(f4_2) * int64(f6_19) f4f7_38 := int64(f4) * int64(f7_38) f4f8_38 := int64(f4_2) * int64(f8_19) f4f9_38 := int64(f4) * int64(f9_38) f5f5_38 := int64(f5) * int64(f5_38) f5f6_38 := int64(f5_2) * int64(f6_19) f5f7_76 := int64(f5_2) * int64(f7_38) f5f8_38 := int64(f5_2) * int64(f8_19) f5f9_76 := int64(f5_2) * int64(f9_38) f6f6_19 := int64(f6) * int64(f6_19) f6f7_38 := int64(f6) * int64(f7_38) f6f8_38 := int64(f6_2) * int64(f8_19) f6f9_38 := int64(f6) * int64(f9_38) f7f7_38 := int64(f7) * int64(f7_38) f7f8_38 := int64(f7_2) * int64(f8_19) f7f9_76 := int64(f7_2) * int64(f9_38) f8f8_19 := int64(f8) * int64(f8_19) f8f9_38 := int64(f8) * int64(f9_38) f9f9_38 := int64(f9) * int64(f9_38) h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38 h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38 h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19 h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38 h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38 h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38 h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19 h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38 h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38 h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2 var carry [10]int64 carry[0] = (h0 + (1 << 25)) >> 26 h1 += carry[0] h0 -= carry[0] << 26 carry[4] = (h4 + (1 << 25)) >> 26 h5 += carry[4] h4 -= carry[4] << 26 carry[1] = (h1 + (1 << 24)) >> 25 h2 += carry[1] h1 -= carry[1] << 25 carry[5] = (h5 + (1 << 24)) >> 25 h6 += carry[5] h5 -= carry[5] << 25 carry[2] = (h2 + (1 << 25)) >> 26 h3 += carry[2] h2 -= carry[2] << 26 carry[6] = (h6 + (1 << 25)) >> 26 h7 += carry[6] h6 -= carry[6] << 26 carry[3] = (h3 + (1 << 24)) >> 25 h4 += carry[3] h3 -= carry[3] << 25 carry[7] = (h7 + (1 << 24)) >> 25 h8 += carry[7] h7 -= carry[7] << 25 carry[4] = (h4 + (1 << 25)) >> 26 h5 += carry[4] h4 -= carry[4] << 26 carry[8] = (h8 + (1 << 25)) >> 26 h9 += carry[8] h8 -= carry[8] << 26 carry[9] = (h9 + (1 << 24)) >> 25 h0 += carry[9] * 19 h9 -= carry[9] << 25 carry[0] = (h0 + (1 << 25)) >> 26 h1 += carry[0] h0 -= carry[0] << 26 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) } // feMul121666 calculates h = f * 121666. 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 feMul121666(h, f *fieldElement) { h0 := int64(f[0]) * 121666 h1 := int64(f[1]) * 121666 h2 := int64(f[2]) * 121666 h3 := int64(f[3]) * 121666 h4 := int64(f[4]) * 121666 h5 := int64(f[5]) * 121666 h6 := int64(f[6]) * 121666 h7 := int64(f[7]) * 121666 h8 := int64(f[8]) * 121666 h9 := int64(f[9]) * 121666 var carry [10]int64 carry[9] = (h9 + (1 << 24)) >> 25 h0 += carry[9] * 19 h9 -= carry[9] << 25 carry[1] = (h1 + (1 << 24)) >> 25 h2 += carry[1] h1 -= carry[1] << 25 carry[3] = (h3 + (1 << 24)) >> 25 h4 += carry[3] h3 -= carry[3] << 25 carry[5] = (h5 + (1 << 24)) >> 25 h6 += carry[5] h5 -= carry[5] << 25 carry[7] = (h7 + (1 << 24)) >> 25 h8 += carry[7] h7 -= carry[7] << 25 carry[0] = (h0 + (1 << 25)) >> 26 h1 += carry[0] h0 -= carry[0] << 26 carry[2] = (h2 + (1 << 25)) >> 26 h3 += carry[2] h2 -= carry[2] << 26 carry[4] = (h4 + (1 << 25)) >> 26 h5 += carry[4] h4 -= carry[4] << 26 carry[6] = (h6 + (1 << 25)) >> 26 h7 += carry[6] h6 -= carry[6] << 26 carry[8] = (h8 + (1 << 25)) >> 26 h9 += carry[8] h8 -= carry[8] << 26 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) } // feInvert sets out = z^-1. func feInvert(out, z *fieldElement) { var t0, t1, t2, t3 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(&t2, &t0) for i = 1; i < 1; i++ { feSquare(&t2, &t2) } feMul(&t1, &t1, &t2) feSquare(&t2, &t1) for i = 1; i < 5; i++ { feSquare(&t2, &t2) } feMul(&t1, &t2, &t1) feSquare(&t2, &t1) for i = 1; i < 10; i++ { feSquare(&t2, &t2) } feMul(&t2, &t2, &t1) feSquare(&t3, &t2) for i = 1; i < 20; i++ { feSquare(&t3, &t3) } feMul(&t2, &t3, &t2) feSquare(&t2, &t2) for i = 1; i < 10; i++ { feSquare(&t2, &t2) } feMul(&t1, &t2, &t1) feSquare(&t2, &t1) for i = 1; i < 50; i++ { feSquare(&t2, &t2) } feMul(&t2, &t2, &t1) feSquare(&t3, &t2) for i = 1; i < 100; i++ { feSquare(&t3, &t3) } feMul(&t2, &t3, &t2) feSquare(&t2, &t2) for i = 1; i < 50; i++ { feSquare(&t2, &t2) } feMul(&t1, &t2, &t1) feSquare(&t1, &t1) for i = 1; i < 5; i++ { feSquare(&t1, &t1) } feMul(out, &t1, &t0) } func scalarMult(out, in, base *[32]byte) { var e [32]byte copy(e[:], in[:]) e[0] &= 248 e[31] &= 127 e[31] |= 64 var x1, x2, z2, x3, z3, tmp0, tmp1 fieldElement feFromBytes(&x1, base) feOne(&x2) feCopy(&x3, &x1) feOne(&z3) swap := int32(0) for pos := 254; pos >= 0; pos-- { b := e[pos/8] >> uint(pos&7) b &= 1 swap ^= int32(b) feCSwap(&x2, &x3, swap) feCSwap(&z2, &z3, swap) swap = int32(b) feSub(&tmp0, &x3, &z3) feSub(&tmp1, &x2, &z2) feAdd(&x2, &x2, &z2) feAdd(&z2, &x3, &z3) feMul(&z3, &tmp0, &x2) feMul(&z2, &z2, &tmp1) feSquare(&tmp0, &tmp1) feSquare(&tmp1, &x2) feAdd(&x3, &z3, &z2) feSub(&z2, &z3, &z2) feMul(&x2, &tmp1, &tmp0) feSub(&tmp1, &tmp1, &tmp0) feSquare(&z2, &z2) feMul121666(&z3, &tmp1) feSquare(&x3, &x3) feAdd(&tmp0, &tmp0, &z3) feMul(&z3, &x1, &z2) feMul(&z2, &tmp1, &tmp0) } feCSwap(&x2, &x3, swap) feCSwap(&z2, &z3, swap) feInvert(&z2, &z2) feMul(&x2, &x2, &z2) feToBytes(out, &x2) }