/* SPDX-License-Identifier: GPL-2.0 * * Copyright (C) 2008 Google Inc. All Rights Reserved. * Copyright (C) 2015-2017 Jason A. Donenfeld . All Rights Reserved. * * Original author: Adam Langley */ #include "curve25519.h" #include #include #ifndef __always_inline #define __always_inline __inline __attribute__((__always_inline__)) #endif #ifdef __SIZEOF_INT128__ typedef uint64_t limb; typedef limb felem[5]; typedef __uint128_t uint128_t; /* Sum two numbers: output += in */ static __always_inline void fsum(limb *output, const limb *in) { output[0] += in[0]; output[1] += in[1]; output[2] += in[2]; output[3] += in[3]; output[4] += in[4]; } /* Find the difference of two numbers: output = in - output * (note the order of the arguments!) * * Assumes that out[i] < 2**52 * On return, out[i] < 2**55 */ static __always_inline void fdifference_backwards(felem out, const felem in) { /* 152 is 19 << 3 */ static const limb two54m152 = (((limb)1) << 54) - 152; static const limb two54m8 = (((limb)1) << 54) - 8; out[0] = in[0] + two54m152 - out[0]; out[1] = in[1] + two54m8 - out[1]; out[2] = in[2] + two54m8 - out[2]; out[3] = in[3] + two54m8 - out[3]; out[4] = in[4] + two54m8 - out[4]; } /* Multiply a number by a scalar: output = in * scalar */ static __always_inline void fscalar_product(felem output, const felem in, const limb scalar) { uint128_t a; a = ((uint128_t) in[0]) * scalar; output[0] = ((limb)a) & 0x7ffffffffffffUL; a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51)); output[1] = ((limb)a) & 0x7ffffffffffffUL; a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51)); output[2] = ((limb)a) & 0x7ffffffffffffUL; a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51)); output[3] = ((limb)a) & 0x7ffffffffffffUL; a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51)); output[4] = ((limb)a) & 0x7ffffffffffffUL; output[0] += (a >> 51) * 19; } /* Multiply two numbers: output = in2 * in * * output must be distinct to both inputs. The inputs are reduced coefficient * form, the output is not. * * Assumes that in[i] < 2**55 and likewise for in2. * On return, output[i] < 2**52 */ static __always_inline void fmul(felem output, const felem in2, const felem in) { uint128_t t[5]; limb r0, r1, r2, r3, r4, s0, s1, s2, s3, s4, c; r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; s0 = in2[0]; s1 = in2[1]; s2 = in2[2]; s3 = in2[3]; s4 = in2[4]; t[0] = ((uint128_t) r0) * s0; t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0; t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1; t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1; t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2; r4 *= 19; r1 *= 19; r2 *= 19; r3 *= 19; t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2; t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3; t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4; t[3] += ((uint128_t) r4) * s4; r0 = (limb)t[0] & 0x7ffffffffffffUL; c = (limb)(t[0] >> 51); t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffffUL; c = (limb)(t[1] >> 51); t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffffUL; c = (limb)(t[2] >> 51); t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffffUL; c = (limb)(t[3] >> 51); t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffffUL; c = (limb)(t[4] >> 51); r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffffUL; r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffffUL; r2 += c; output[0] = r0; output[1] = r1; output[2] = r2; output[3] = r3; output[4] = r4; } static __always_inline void fsquare_times(felem output, const felem in, limb count) { uint128_t t[5]; limb r0, r1, r2, r3, r4, c; limb d0, d1, d2, d4, d419; r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; do { d0 = r0 * 2; d1 = r1 * 2; d2 = r2 * 2 * 19; d419 = r4 * 19; d4 = d419 * 2; t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 )); t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19)); t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 )); t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 )); t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 )); r0 = (limb)t[0] & 0x7ffffffffffffUL; c = (limb)(t[0] >> 51); t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffffUL; c = (limb)(t[1] >> 51); t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffffUL; c = (limb)(t[2] >> 51); t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffffUL; c = (limb)(t[3] >> 51); t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffffUL; c = (limb)(t[4] >> 51); r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffffUL; r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffffUL; r2 += c; } while (--count); output[0] = r0; output[1] = r1; output[2] = r2; output[3] = r3; output[4] = r4; } /* Load a little-endian 64-bit number */ static limb load_limb(const uint8_t *in) { return ((limb)in[0]) | (((limb)in[1]) << 8) | (((limb)in[2]) << 16) | (((limb)in[3]) << 24) | (((limb)in[4]) << 32) | (((limb)in[5]) << 40) | (((limb)in[6]) << 48) | (((limb)in[7]) << 56); } static void store_limb(uint8_t *out, limb in) { out[0] = in & 0xff; out[1] = (in >> 8) & 0xff; out[2] = (in >> 16) & 0xff; out[3] = (in >> 24) & 0xff; out[4] = (in >> 32) & 0xff; out[5] = (in >> 40) & 0xff; out[6] = (in >> 48) & 0xff; out[7] = (in >> 56) & 0xff; } /* Take a little-endian, 32-byte number and expand it into polynomial form */ static void fexpand(limb *output, const uint8_t *in) { output[0] = load_limb(in) & 0x7ffffffffffffUL; output[1] = (load_limb(in+6) >> 3) & 0x7ffffffffffffUL; output[2] = (load_limb(in+12) >> 6) & 0x7ffffffffffffUL; output[3] = (load_limb(in+19) >> 1) & 0x7ffffffffffffUL; output[4] = (load_limb(in+24) >> 12) & 0x7ffffffffffffUL; } /* Take a fully reduced polynomial form number and contract it into a * little-endian, 32-byte array */ static void fcontract(uint8_t *output, const felem input) { uint128_t t[5]; t[0] = input[0]; t[1] = input[1]; t[2] = input[2]; t[3] = input[3]; t[4] = input[4]; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL; /* now t is between 0 and 2^255-1, properly carried. */ /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */ t[0] += 19; t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL; t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL; /* now between 19 and 2^255-1 in both cases, and offset by 19. */ t[0] += 0x8000000000000UL - 19; t[1] += 0x8000000000000UL - 1; t[2] += 0x8000000000000UL - 1; t[3] += 0x8000000000000UL - 1; t[4] += 0x8000000000000UL - 1; /* now between 2^255 and 2^256-20, and offset by 2^255. */ t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL; t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL; t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL; t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL; t[4] &= 0x7ffffffffffffUL; store_limb(output, t[0] | (t[1] << 51)); store_limb(output+8, (t[1] >> 13) | (t[2] << 38)); store_limb(output+16, (t[2] >> 26) | (t[3] << 25)); store_limb(output+24, (t[3] >> 39) | (t[4] << 12)); } /* Input: Q, Q', Q-Q' * Output: 2Q, Q+Q' * * x2 z3: long form * x3 z3: long form * x z: short form, destroyed * xprime zprime: short form, destroyed * qmqp: short form, preserved */ static void fmonty(limb *x2, limb *z2, /* output 2Q */ limb *x3, limb *z3, /* output Q + Q' */ limb *x, limb *z, /* input Q */ limb *xprime, limb *zprime, /* input Q' */ const limb *qmqp /* input Q - Q' */) { limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5], zzprime[5], zzzprime[5]; memcpy(origx, x, 5 * sizeof(limb)); fsum(x, z); fdifference_backwards(z, origx); // does x - z memcpy(origxprime, xprime, sizeof(limb) * 5); fsum(xprime, zprime); fdifference_backwards(zprime, origxprime); fmul(xxprime, xprime, z); fmul(zzprime, x, zprime); memcpy(origxprime, xxprime, sizeof(limb) * 5); fsum(xxprime, zzprime); fdifference_backwards(zzprime, origxprime); fsquare_times(x3, xxprime, 1); fsquare_times(zzzprime, zzprime, 1); fmul(z3, zzzprime, qmqp); fsquare_times(xx, x, 1); fsquare_times(zz, z, 1); fmul(x2, xx, zz); fdifference_backwards(zz, xx); // does zz = xx - zz fscalar_product(zzz, zz, 121665); fsum(zzz, xx); fmul(z2, zz, zzz); } /* Maybe swap the contents of two limb arrays (@a and @b), each @len elements * long. Perform the swap iff @swap is non-zero. * * This function performs the swap without leaking any side-channel * information. */ static void swap_conditional(limb a[static 5], limb b[static 5], limb iswap) { unsigned int i; const limb swap = -iswap; for (i = 0; i < 5; ++i) { const limb x = swap & (a[i] ^ b[i]); a[i] ^= x; b[i] ^= x; } } /* Calculates nQ where Q is the x-coordinate of a point on the curve * * resultx/resultz: the x coordinate of the resulting curve point (short form) * n: a little endian, 32-byte number * q: a point of the curve (short form) */ static void cmult(limb *resultx, limb *resultz, const uint8_t *n, const limb *q) { limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0}; limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1}; limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; unsigned int i, j; memcpy(nqpqx, q, sizeof(limb) * 5); for (i = 0; i < 32; ++i) { uint8_t byte = n[31 - i]; for (j = 0; j < 8; ++j) { const limb bit = byte >> 7; swap_conditional(nqx, nqpqx, bit); swap_conditional(nqz, nqpqz, bit); fmonty(nqx2, nqz2, nqpqx2, nqpqz2, nqx, nqz, nqpqx, nqpqz, q); swap_conditional(nqx2, nqpqx2, bit); swap_conditional(nqz2, nqpqz2, bit); t = nqx; nqx = nqx2; nqx2 = t; t = nqz; nqz = nqz2; nqz2 = t; t = nqpqx; nqpqx = nqpqx2; nqpqx2 = t; t = nqpqz; nqpqz = nqpqz2; nqpqz2 = t; byte <<= 1; } } memcpy(resultx, nqx, sizeof(limb) * 5); memcpy(resultz, nqz, sizeof(limb) * 5); } static void crecip(felem out, const felem z) { felem a, t0, b, c; /* 2 */ fsquare_times(a, z, 1); // a = 2 /* 8 */ fsquare_times(t0, a, 2); /* 9 */ fmul(b, t0, z); // b = 9 /* 11 */ fmul(a, b, a); // a = 11 /* 22 */ fsquare_times(t0, a, 1); /* 2^5 - 2^0 = 31 */ fmul(b, t0, b); /* 2^10 - 2^5 */ fsquare_times(t0, b, 5); /* 2^10 - 2^0 */ fmul(b, t0, b); /* 2^20 - 2^10 */ fsquare_times(t0, b, 10); /* 2^20 - 2^0 */ fmul(c, t0, b); /* 2^40 - 2^20 */ fsquare_times(t0, c, 20); /* 2^40 - 2^0 */ fmul(t0, t0, c); /* 2^50 - 2^10 */ fsquare_times(t0, t0, 10); /* 2^50 - 2^0 */ fmul(b, t0, b); /* 2^100 - 2^50 */ fsquare_times(t0, b, 50); /* 2^100 - 2^0 */ fmul(c, t0, b); /* 2^200 - 2^100 */ fsquare_times(t0, c, 100); /* 2^200 - 2^0 */ fmul(t0, t0, c); /* 2^250 - 2^50 */ fsquare_times(t0, t0, 50); /* 2^250 - 2^0 */ fmul(t0, t0, b); /* 2^255 - 2^5 */ fsquare_times(t0, t0, 5); /* 2^255 - 21 */ fmul(out, t0, a); } void curve25519(uint8_t mypublic[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE], const uint8_t basepoint[static CURVE25519_POINT_SIZE]) { limb bp[5], x[5], z[5], zmone[5]; uint8_t e[32]; memcpy(e, secret, 32); curve25519_normalize_secret(e); fexpand(bp, basepoint); cmult(x, z, e, bp); crecip(zmone, z); fmul(z, x, zmone); fcontract(mypublic, z); } #else typedef int64_t limb; /* Field element representation: * * Field elements are written as an array of signed, 64-bit limbs, least * significant first. The value of the field element is: * x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ... * * i.e. the limbs are 26, 25, 26, 25, ... bits wide. */ /* Sum two numbers: output += in */ static void fsum(limb *output, const limb *in) { unsigned int i; for (i = 0; i < 10; i += 2) { output[0 + i] = output[0 + i] + in[0 + i]; output[1 + i] = output[1 + i] + in[1 + i]; } } /* Find the difference of two numbers: output = in - output * (note the order of the arguments!). */ static void fdifference(limb *output, const limb *in) { unsigned int i; for (i = 0; i < 10; ++i) { output[i] = in[i] - output[i]; } } /* Multiply a number by a scalar: output = in * scalar */ static void fscalar_product(limb *output, const limb *in, const limb scalar) { unsigned int i; for (i = 0; i < 10; ++i) { output[i] = in[i] * scalar; } } /* Multiply two numbers: output = in2 * in * * output must be distinct to both inputs. The inputs are reduced coefficient * form, the output is not. * * output[x] <= 14 * the largest product of the input limbs. */ static void fproduct(limb *output, const limb *in2, const limb *in) { output[0] = ((limb) ((int32_t) in2[0])) * ((int32_t) in[0]); output[1] = ((limb) ((int32_t) in2[0])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[0]); output[2] = 2 * ((limb) ((int32_t) in2[1])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[0]); output[3] = ((limb) ((int32_t) in2[1])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[0]); output[4] = ((limb) ((int32_t) in2[2])) * ((int32_t) in[2]) + 2 * (((limb) ((int32_t) in2[1])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[1])) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[0]); output[5] = ((limb) ((int32_t) in2[2])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[0]); output[6] = 2 * (((limb) ((int32_t) in2[3])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[1])) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[0]); output[7] = ((limb) ((int32_t) in2[3])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[0]); output[8] = ((limb) ((int32_t) in2[4])) * ((int32_t) in[4]) + 2 * (((limb) ((int32_t) in2[3])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[1])) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[0]); output[9] = ((limb) ((int32_t) in2[4])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[2]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[1]) + ((limb) ((int32_t) in2[0])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[0]); output[10] = 2 * (((limb) ((int32_t) in2[5])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[1])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[1])) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[2]); output[11] = ((limb) ((int32_t) in2[5])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[4]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[3]) + ((limb) ((int32_t) in2[2])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[2]); output[12] = ((limb) ((int32_t) in2[6])) * ((int32_t) in[6]) + 2 * (((limb) ((int32_t) in2[5])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[3])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[3])) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[4]); output[13] = ((limb) ((int32_t) in2[6])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[7])) * ((int32_t) in[6]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[5]) + ((limb) ((int32_t) in2[4])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[4]); output[14] = 2 * (((limb) ((int32_t) in2[7])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[5])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[5])) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[6]); output[15] = ((limb) ((int32_t) in2[7])) * ((int32_t) in[8]) + ((limb) ((int32_t) in2[8])) * ((int32_t) in[7]) + ((limb) ((int32_t) in2[6])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[6]); output[16] = ((limb) ((int32_t) in2[8])) * ((int32_t) in[8]) + 2 * (((limb) ((int32_t) in2[7])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[7])); output[17] = ((limb) ((int32_t) in2[8])) * ((int32_t) in[9]) + ((limb) ((int32_t) in2[9])) * ((int32_t) in[8]); output[18] = 2 * ((limb) ((int32_t) in2[9])) * ((int32_t) in[9]); } /* Reduce a long form to a short form by taking the input mod 2^255 - 19. * * On entry: |output[i]| < 14*2^54 * On exit: |output[0..8]| < 280*2^54 */ static void freduce_degree(limb *output) { /* Each of these shifts and adds ends up multiplying the value by 19. * * For output[0..8], the absolute entry value is < 14*2^54 and we add, at * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */ output[8] += output[18] << 4; output[8] += output[18] << 1; output[8] += output[18]; output[7] += output[17] << 4; output[7] += output[17] << 1; output[7] += output[17]; output[6] += output[16] << 4; output[6] += output[16] << 1; output[6] += output[16]; output[5] += output[15] << 4; output[5] += output[15] << 1; output[5] += output[15]; output[4] += output[14] << 4; output[4] += output[14] << 1; output[4] += output[14]; output[3] += output[13] << 4; output[3] += output[13] << 1; output[3] += output[13]; output[2] += output[12] << 4; output[2] += output[12] << 1; output[2] += output[12]; output[1] += output[11] << 4; output[1] += output[11] << 1; output[1] += output[11]; output[0] += output[10] << 4; output[0] += output[10] << 1; output[0] += output[10]; } #if (-1 & 3) != 3 #error "This code only works on a two's complement system" #endif /* return v / 2^26, using only shifts and adds. * * On entry: v can take any value. */ static inline limb div_by_2_26(const limb v) { /* High word of v; no shift needed. */ const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); /* Set to all 1s if v was negative; else set to 0s. */ const int32_t sign = ((int32_t) highword) >> 31; /* Set to 0x3ffffff if v was negative; else set to 0. */ const int32_t roundoff = ((uint32_t) sign) >> 6; /* Should return v / (1<<26) */ return (v + roundoff) >> 26; } /* return v / (2^25), using only shifts and adds. * * On entry: v can take any value. */ static inline limb div_by_2_25(const limb v) { /* High word of v; no shift needed*/ const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); /* Set to all 1s if v was negative; else set to 0s. */ const int32_t sign = ((int32_t) highword) >> 31; /* Set to 0x1ffffff if v was negative; else set to 0. */ const int32_t roundoff = ((uint32_t) sign) >> 7; /* Should return v / (1<<25) */ return (v + roundoff) >> 25; } /* Reduce all coefficients of the short form input so that |x| < 2^26. * * On entry: |output[i]| < 280*2^54 */ static void freduce_coefficients(limb *output) { unsigned int i; output[10] = 0; for (i = 0; i < 10; i += 2) { limb over = div_by_2_26(output[i]); /* The entry condition (that |output[i]| < 280*2^54) means that over is, at * most, 280*2^28 in the first iteration of this loop. This is added to the * next limb and we can approximate the resulting bound of that limb by * 281*2^54. */ output[i] -= over << 26; output[i+1] += over; /* For the first iteration, |output[i+1]| < 281*2^54, thus |over| < * 281*2^29. When this is added to the next limb, the resulting bound can * be approximated as 281*2^54. * * For subsequent iterations of the loop, 281*2^54 remains a conservative * bound and no overflow occurs. */ over = div_by_2_25(output[i+1]); output[i+1] -= over << 25; output[i+2] += over; } /* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */ output[0] += output[10] << 4; output[0] += output[10] << 1; output[0] += output[10]; output[10] = 0; /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29 * So |over| will be no more than 2^16. */ { limb over = div_by_2_26(output[0]); output[0] -= over << 26; output[1] += over; } /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The * bound on |output[1]| is sufficient to meet our needs. */ } /* A helpful wrapper around fproduct: output = in * in2. * * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27. * * output must be distinct to both inputs. The output is reduced degree * (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */ static void fmul(limb *output, const limb *in, const limb *in2) { limb t[19]; fproduct(t, in, in2); /* |t[i]| < 14*2^54 */ freduce_degree(t); freduce_coefficients(t); /* |t[i]| < 2^26 */ memcpy(output, t, sizeof(limb) * 10); } /* Square a number: output = in**2 * * output must be distinct from the input. The inputs are reduced coefficient * form, the output is not. * * output[x] <= 14 * the largest product of the input limbs. */ static void fsquare_inner(limb *output, const limb *in) { output[0] = ((limb) ((int32_t) in[0])) * ((int32_t) in[0]); output[1] = 2 * ((limb) ((int32_t) in[0])) * ((int32_t) in[1]); output[2] = 2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[1]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[2])); output[3] = 2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[2]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[3])); output[4] = ((limb) ((int32_t) in[2])) * ((int32_t) in[2]) + 4 * ((limb) ((int32_t) in[1])) * ((int32_t) in[3]) + 2 * ((limb) ((int32_t) in[0])) * ((int32_t) in[4]); output[5] = 2 * (((limb) ((int32_t) in[2])) * ((int32_t) in[3]) + ((limb) ((int32_t) in[1])) * ((int32_t) in[4]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[5])); output[6] = 2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[3]) + ((limb) ((int32_t) in[2])) * ((int32_t) in[4]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[6]) + 2 * ((limb) ((int32_t) in[1])) * ((int32_t) in[5])); output[7] = 2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[4]) + ((limb) ((int32_t) in[2])) * ((int32_t) in[5]) + ((limb) ((int32_t) in[1])) * ((int32_t) in[6]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[7])); output[8] = ((limb) ((int32_t) in[4])) * ((int32_t) in[4]) + 2 * (((limb) ((int32_t) in[2])) * ((int32_t) in[6]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[8]) + 2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[3])) * ((int32_t) in[5]))); output[9] = 2 * (((limb) ((int32_t) in[4])) * ((int32_t) in[5]) + ((limb) ((int32_t) in[3])) * ((int32_t) in[6]) + ((limb) ((int32_t) in[2])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[1])) * ((int32_t) in[8]) + ((limb) ((int32_t) in[0])) * ((int32_t) in[9])); output[10] = 2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[5]) + ((limb) ((int32_t) in[4])) * ((int32_t) in[6]) + ((limb) ((int32_t) in[2])) * ((int32_t) in[8]) + 2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[1])) * ((int32_t) in[9]))); output[11] = 2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[6]) + ((limb) ((int32_t) in[4])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[3])) * ((int32_t) in[8]) + ((limb) ((int32_t) in[2])) * ((int32_t) in[9])); output[12] = ((limb) ((int32_t) in[6])) * ((int32_t) in[6]) + 2 * (((limb) ((int32_t) in[4])) * ((int32_t) in[8]) + 2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[3])) * ((int32_t) in[9]))); output[13] = 2 * (((limb) ((int32_t) in[6])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[5])) * ((int32_t) in[8]) + ((limb) ((int32_t) in[4])) * ((int32_t) in[9])); output[14] = 2 * (((limb) ((int32_t) in[7])) * ((int32_t) in[7]) + ((limb) ((int32_t) in[6])) * ((int32_t) in[8]) + 2 * ((limb) ((int32_t) in[5])) * ((int32_t) in[9])); output[15] = 2 * (((limb) ((int32_t) in[7])) * ((int32_t) in[8]) + ((limb) ((int32_t) in[6])) * ((int32_t) in[9])); output[16] = ((limb) ((int32_t) in[8])) * ((int32_t) in[8]) + 4 * ((limb) ((int32_t) in[7])) * ((int32_t) in[9]); output[17] = 2 * ((limb) ((int32_t) in[8])) * ((int32_t) in[9]); output[18] = 2 * ((limb) ((int32_t) in[9])) * ((int32_t) in[9]); } /* fsquare sets output = in^2. * * On entry: The |in| argument is in reduced coefficients form and |in[i]| < * 2^27. * * On exit: The |output| argument is in reduced coefficients form (indeed, one * need only provide storage for 10 limbs) and |out[i]| < 2^26. */ static void fsquare(limb *output, const limb *in) { limb t[19]; fsquare_inner(t, in); /* |t[i]| < 14*2^54 because the largest product of two limbs will be < * 2^(27+27) and fsquare_inner adds together, at most, 14 of those * products. */ freduce_degree(t); freduce_coefficients(t); /* |t[i]| < 2^26 */ memcpy(output, t, sizeof(limb) * 10); } /* Take a little-endian, 32-byte number and expand it into polynomial form */ static void fexpand(limb *output, const uint8_t *input) { #define F(n, start, shift, mask) \ output[n] = ((((limb) input[start + 0]) | \ ((limb) input[start + 1]) << 8 | \ ((limb) input[start + 2]) << 16 | \ ((limb) input[start + 3]) << 24) >> shift) & mask; F(0, 0, 0, 0x3ffffff); F(1, 3, 2, 0x1ffffff); F(2, 6, 3, 0x3ffffff); F(3, 9, 5, 0x1ffffff); F(4, 12, 6, 0x3ffffff); F(5, 16, 0, 0x1ffffff); F(6, 19, 1, 0x3ffffff); F(7, 22, 3, 0x1ffffff); F(8, 25, 4, 0x3ffffff); F(9, 28, 6, 0x1ffffff); #undef F } #if (-32 >> 1) != -16 #error "This code only works when >> does sign-extension on negative numbers" #endif /* int32_t_eq returns 0xffffffff iff a == b and zero otherwise. */ static int32_t int32_t_eq(int32_t a, int32_t b) { a = ~(a ^ b); a &= a << 16; a &= a << 8; a &= a << 4; a &= a << 2; a &= a << 1; return a >> 31; } /* int32_t_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are * both non-negative. */ static int32_t int32_t_gte(int32_t a, int32_t b) { a -= b; /* a >= 0 iff a >= b. */ return ~(a >> 31); } /* Take a fully reduced polynomial form number and contract it into a * little-endian, 32-byte array. * * On entry: |input_limbs[i]| < 2^26 */ static void fcontract(uint8_t *output, limb *input_limbs) { int i; int j; int32_t input[10]; int32_t mask; /* |input_limbs[i]| < 2^26, so it's valid to convert to an int32_t. */ for (i = 0; i < 10; i++) { input[i] = input_limbs[i]; } for (j = 0; j < 2; ++j) { for (i = 0; i < 9; ++i) { if ((i & 1) == 1) { /* This calculation is a time-invariant way to make input[i] * non-negative by borrowing from the next-larger limb. */ const int32_t mask = input[i] >> 31; const int32_t carry = -((input[i] & mask) >> 25); input[i] = input[i] + (carry << 25); input[i+1] = input[i+1] - carry; } else { const int32_t mask = input[i] >> 31; const int32_t carry = -((input[i] & mask) >> 26); input[i] = input[i] + (carry << 26); input[i+1] = input[i+1] - carry; } } /* There's no greater limb for input[9] to borrow from, but we can multiply * by 19 and borrow from input[0], which is valid mod 2^255-19. */ { const int32_t mask = input[9] >> 31; const int32_t carry = -((input[9] & mask) >> 25); input[9] = input[9] + (carry << 25); input[0] = input[0] - (carry * 19); } /* After the first iteration, input[1..9] are non-negative and fit within * 25 or 26 bits, depending on position. However, input[0] may be * negative. */ } /* The first borrow-propagation pass above ended with every limb except (possibly) input[0] non-negative. If input[0] was negative after the first pass, then it was because of a carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most, one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19. In the second pass, each limb is decreased by at most one. Thus the second borrow-propagation pass could only have wrapped around to decrease input[0] again if the first pass left input[0] negative *and* input[1] through input[9] were all zero. In that case, input[1] is now 2^25 - 1, and this last borrow-propagation step will leave input[1] non-negative. */ { const int32_t mask = input[0] >> 31; const int32_t carry = -((input[0] & mask) >> 26); input[0] = input[0] + (carry << 26); input[1] = input[1] - carry; } /* All input[i] are now non-negative. However, there might be values between * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */ for (j = 0; j < 2; j++) { for (i = 0; i < 9; i++) { if ((i & 1) == 1) { const int32_t carry = input[i] >> 25; input[i] &= 0x1ffffff; input[i+1] += carry; } else { const int32_t carry = input[i] >> 26; input[i] &= 0x3ffffff; input[i+1] += carry; } } { const int32_t carry = input[9] >> 25; input[9] &= 0x1ffffff; input[0] += 19*carry; } } /* If the first carry-chain pass, just above, ended up with a carry from * input[9], and that caused input[0] to be out-of-bounds, then input[0] was * < 2^26 + 2*19, because the carry was, at most, two. * * If the second pass carried from input[9] again then input[0] is < 2*19 and * the input[9] -> input[0] carry didn't push input[0] out of bounds. */ /* It still remains the case that input might be between 2^255-19 and 2^255. * In this case, input[1..9] must take their maximum value and input[0] must * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */ mask = int32_t_gte(input[0], 0x3ffffed); for (i = 1; i < 10; i++) { if ((i & 1) == 1) { mask &= int32_t_eq(input[i], 0x1ffffff); } else { mask &= int32_t_eq(input[i], 0x3ffffff); } } /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus * this conditionally subtracts 2^255-19. */ input[0] -= mask & 0x3ffffed; for (i = 1; i < 10; i++) { if ((i & 1) == 1) { input[i] -= mask & 0x1ffffff; } else { input[i] -= mask & 0x3ffffff; } } input[1] <<= 2; input[2] <<= 3; input[3] <<= 5; input[4] <<= 6; input[6] <<= 1; input[7] <<= 3; input[8] <<= 4; input[9] <<= 6; #define F(i, s) \ output[s+0] |= input[i] & 0xff; \ output[s+1] = (input[i] >> 8) & 0xff; \ output[s+2] = (input[i] >> 16) & 0xff; \ output[s+3] = (input[i] >> 24) & 0xff; output[0] = 0; output[16] = 0; F(0, 0); F(1, 3); F(2, 6); F(3, 9); F(4, 12); F(5, 16); F(6, 19); F(7, 22); F(8, 25); F(9, 28); #undef F } /* Input: Q, Q', Q-Q' * Output: 2Q, Q+Q' * * x2 z3: long form * x3 z3: long form * x z: short form, destroyed * xprime zprime: short form, destroyed * qmqp: short form, preserved * * On entry and exit, the absolute value of the limbs of all inputs and outputs * are < 2^26. */ static void fmonty(limb *x2, limb *z2, /* output 2Q */ limb *x3, limb *z3, /* output Q + Q' */ limb *x, limb *z, /* input Q */ limb *xprime, limb *zprime, /* input Q' */ const limb *qmqp /* input Q - Q' */) { limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19], zzprime[19], zzzprime[19], xxxprime[19]; memcpy(origx, x, 10 * sizeof(limb)); fsum(x, z); /* |x[i]| < 2^27 */ fdifference(z, origx); /* does x - z */ /* |z[i]| < 2^27 */ memcpy(origxprime, xprime, sizeof(limb) * 10); fsum(xprime, zprime); /* |xprime[i]| < 2^27 */ fdifference(zprime, origxprime); /* |zprime[i]| < 2^27 */ fproduct(xxprime, xprime, z); /* |xxprime[i]| < 14*2^54: the largest product of two limbs will be < * 2^(27+27) and fproduct adds together, at most, 14 of those products. * (Approximating that to 2^58 doesn't work out.) */ fproduct(zzprime, x, zprime); /* |zzprime[i]| < 14*2^54 */ freduce_degree(xxprime); freduce_coefficients(xxprime); /* |xxprime[i]| < 2^26 */ freduce_degree(zzprime); freduce_coefficients(zzprime); /* |zzprime[i]| < 2^26 */ memcpy(origxprime, xxprime, sizeof(limb) * 10); fsum(xxprime, zzprime); /* |xxprime[i]| < 2^27 */ fdifference(zzprime, origxprime); /* |zzprime[i]| < 2^27 */ fsquare(xxxprime, xxprime); /* |xxxprime[i]| < 2^26 */ fsquare(zzzprime, zzprime); /* |zzzprime[i]| < 2^26 */ fproduct(zzprime, zzzprime, qmqp); /* |zzprime[i]| < 14*2^52 */ freduce_degree(zzprime); freduce_coefficients(zzprime); /* |zzprime[i]| < 2^26 */ memcpy(x3, xxxprime, sizeof(limb) * 10); memcpy(z3, zzprime, sizeof(limb) * 10); fsquare(xx, x); /* |xx[i]| < 2^26 */ fsquare(zz, z); /* |zz[i]| < 2^26 */ fproduct(x2, xx, zz); /* |x2[i]| < 14*2^52 */ freduce_degree(x2); freduce_coefficients(x2); /* |x2[i]| < 2^26 */ fdifference(zz, xx); // does zz = xx - zz /* |zz[i]| < 2^27 */ memset(zzz + 10, 0, sizeof(limb) * 9); fscalar_product(zzz, zz, 121665); /* |zzz[i]| < 2^(27+17) */ /* No need to call freduce_degree here: fscalar_product doesn't increase the degree of its input. */ freduce_coefficients(zzz); /* |zzz[i]| < 2^26 */ fsum(zzz, xx); /* |zzz[i]| < 2^27 */ fproduct(z2, zz, zzz); /* |z2[i]| < 14*2^(26+27) */ freduce_degree(z2); freduce_coefficients(z2); /* |z2|i| < 2^26 */ } /* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave * them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid * side-channel attacks. * * NOTE that this function requires that 'iswap' be 1 or 0; other values give * wrong results. Also, the two limb arrays must be in reduced-coefficient, * reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped, * and all all values in a[0..9],b[0..9] must have magnitude less than * INT32_MAX. */ static void swap_conditional(limb a[static 19], limb b[static 19], limb iswap) { unsigned int i; const int32_t swap = (int32_t) -iswap; for (i = 0; i < 10; ++i) { const int32_t x = swap & (((int32_t)a[i]) ^ ((int32_t)b[i])); a[i] = ((int32_t)a[i]) ^ x; b[i] = ((int32_t)b[i]) ^ x; } } /* Calculates nQ where Q is the x-coordinate of a point on the curve * * resultx/resultz: the x coordinate of the resulting curve point (short form) * n: a little endian, 32-byte number * q: a point of the curve (short form) */ static void cmult(limb *resultx, limb *resultz, const uint8_t *n, const limb *q) { limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0}; limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1}; limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; unsigned int i, j; memcpy(nqpqx, q, sizeof(limb) * 10); for (i = 0; i < 32; ++i) { uint8_t byte = n[31 - i]; for (j = 0; j < 8; ++j) { const limb bit = byte >> 7; swap_conditional(nqx, nqpqx, bit); swap_conditional(nqz, nqpqz, bit); fmonty(nqx2, nqz2, nqpqx2, nqpqz2, nqx, nqz, nqpqx, nqpqz, q); swap_conditional(nqx2, nqpqx2, bit); swap_conditional(nqz2, nqpqz2, bit); t = nqx; nqx = nqx2; nqx2 = t; t = nqz; nqz = nqz2; nqz2 = t; t = nqpqx; nqpqx = nqpqx2; nqpqx2 = t; t = nqpqz; nqpqz = nqpqz2; nqpqz2 = t; byte <<= 1; } } memcpy(resultx, nqx, sizeof(limb) * 10); memcpy(resultz, nqz, sizeof(limb) * 10); } static void crecip(limb *out, const limb *z) { limb z2[10]; limb z9[10]; limb z11[10]; limb z2_5_0[10]; limb z2_10_0[10]; limb z2_20_0[10]; limb z2_50_0[10]; limb z2_100_0[10]; limb t0[10]; limb t1[10]; int i; /* 2 */ fsquare(z2, z); /* 4 */ fsquare(t1, z2); /* 8 */ fsquare(t0, t1); /* 9 */ fmul(z9, t0, z); /* 11 */ fmul(z11, z9, z2); /* 22 */ fsquare(t0, z11); /* 2^5 - 2^0 = 31 */ fmul(z2_5_0, t0, z9); /* 2^6 - 2^1 */ fsquare(t0, z2_5_0); /* 2^7 - 2^2 */ fsquare(t1, t0); /* 2^8 - 2^3 */ fsquare(t0, t1); /* 2^9 - 2^4 */ fsquare(t1, t0); /* 2^10 - 2^5 */ fsquare(t0, t1); /* 2^10 - 2^0 */ fmul(z2_10_0, t0, z2_5_0); /* 2^11 - 2^1 */ fsquare(t0, z2_10_0); /* 2^12 - 2^2 */ fsquare(t1, t0); /* 2^20 - 2^10 */ for (i = 2; i < 10; i += 2) { fsquare(t0, t1); fsquare(t1, t0); } /* 2^20 - 2^0 */ fmul(z2_20_0, t1, z2_10_0); /* 2^21 - 2^1 */ fsquare(t0, z2_20_0); /* 2^22 - 2^2 */ fsquare(t1, t0); /* 2^40 - 2^20 */ for (i = 2; i < 20; i += 2) { fsquare(t0, t1); fsquare(t1, t0); } /* 2^40 - 2^0 */ fmul(t0, t1, z2_20_0); /* 2^41 - 2^1 */ fsquare(t1, t0); /* 2^42 - 2^2 */ fsquare(t0, t1); /* 2^50 - 2^10 */ for (i = 2; i < 10; i += 2) { fsquare(t1, t0); fsquare(t0, t1); } /* 2^50 - 2^0 */ fmul(z2_50_0, t0, z2_10_0); /* 2^51 - 2^1 */ fsquare(t0, z2_50_0); /* 2^52 - 2^2 */ fsquare(t1, t0); /* 2^100 - 2^50 */ for (i = 2; i < 50; i += 2) { fsquare(t0, t1); fsquare(t1, t0); } /* 2^100 - 2^0 */ fmul(z2_100_0, t1, z2_50_0); /* 2^101 - 2^1 */ fsquare(t1, z2_100_0); /* 2^102 - 2^2 */ fsquare(t0, t1); /* 2^200 - 2^100 */ for (i = 2; i < 100; i += 2) { fsquare(t1, t0); fsquare(t0, t1); } /* 2^200 - 2^0 */ fmul(t1, t0, z2_100_0); /* 2^201 - 2^1 */ fsquare(t0, t1); /* 2^202 - 2^2 */ fsquare(t1, t0); /* 2^250 - 2^50 */ for (i = 2; i < 50; i += 2) { fsquare(t0, t1); fsquare(t1, t0); } /* 2^250 - 2^0 */ fmul(t0, t1, z2_50_0); /* 2^251 - 2^1 */ fsquare(t1, t0); /* 2^252 - 2^2 */ fsquare(t0, t1); /* 2^253 - 2^3 */ fsquare(t1, t0); /* 2^254 - 2^4 */ fsquare(t0, t1); /* 2^255 - 2^5 */ fsquare(t1, t0); /* 2^255 - 21 */ fmul(out, t1, z11); } void curve25519(uint8_t mypublic[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE], const uint8_t basepoint[static CURVE25519_POINT_SIZE]) { limb bp[10], x[10], z[11], zmone[10]; uint8_t e[32]; memcpy(e, secret, 32); curve25519_normalize_secret(e); fexpand(bp, basepoint); cmult(x, z, e, bp); crecip(zmone, z); fmul(z, x, zmone); fcontract(mypublic, z); } #endif void curve25519_generate_public(uint8_t pub[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE]) { static const uint8_t basepoint[CURVE25519_POINT_SIZE] = { 9 }; curve25519(pub, secret, basepoint); }