From 7bc05796e5c3ecaf18a51d73d05e2a02c876b4bc Mon Sep 17 00:00:00 2001 From: "Jason A. Donenfeld" Date: Thu, 18 Jan 2018 11:50:49 +0100 Subject: contrib: keygen-html: update curve25519 implementation --- .../examples/keygen-html/src/curve25519_generate.c | 1554 ++++++++++---------- 1 file changed, 769 insertions(+), 785 deletions(-) (limited to 'contrib') diff --git a/contrib/examples/keygen-html/src/curve25519_generate.c b/contrib/examples/keygen-html/src/curve25519_generate.c index 20d3f91..1633275 100644 --- a/contrib/examples/keygen-html/src/curve25519_generate.c +++ b/contrib/examples/keygen-html/src/curve25519_generate.c @@ -1,869 +1,853 @@ /* SPDX-License-Identifier: GPL-2.0 * - * Copyright (C) 2008 Google Inc. All Rights Reserved. - * Copyright (C) 2015-2018 Jason A. Donenfeld . All Rights Reserved. + * Copyright (C) 2015-2016 The fiat-crypto Authors. + * Copyright (C) 2018 Jason A. Donenfeld . All Rights Reserved. + * + * This is a machine-generated formally verified implementation of curve25519 DH from: + * https://github.com/mit-plv/fiat-crypto */ #include +#ifndef __always_inline +#define __always_inline __inline __attribute__((__always_inline__)) +#endif + +#ifndef __aligned +#define __aligned(x) __attribute__((aligned(x))) +#endif + +#if __BYTE_ORDER == __LITTLE_ENDIAN +#define le32toh(x) (x) +#else +#define htole32(x) __builtin_bswap32(x) +#endif + + typedef unsigned long long uint64_t; -typedef long long int64_t; -typedef int int32_t; typedef unsigned int uint32_t; typedef unsigned char uint8_t; -typedef int64_t limb; +/* fe means field element. Here the field is \Z/(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]. + * fe limbs are bounded by 1.125*2^26,1.125*2^25,1.125*2^26,1.125*2^25,etc. + * Multiplication and carrying produce fe from fe_loose. + */ +typedef struct fe { uint32_t v[10]; } fe; -/* 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. +/* fe_loose limbs are bounded by 3.375*2^26,3.375*2^25,3.375*2^26,3.375*2^25,etc. + * Addition and subtraction produce fe_loose from (fe, fe). */ +typedef struct fe_loose { uint32_t v[10]; } fe_loose; -/* Sum two numbers: output += in */ -static void fsum(limb *output, const limb *in) +static __always_inline void fe_frombytes_impl(uint32_t h[10], const uint8_t *s) { - 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]; - } + /* Ignores top bit of s. */ + uint32_t a0 = le32toh(*(uint32_t *)(s)); + uint32_t a1 = le32toh(*(uint32_t *)(s+4)); + uint32_t a2 = le32toh(*(uint32_t *)(s+8)); + uint32_t a3 = le32toh(*(uint32_t *)(s+12)); + uint32_t a4 = le32toh(*(uint32_t *)(s+16)); + uint32_t a5 = le32toh(*(uint32_t *)(s+20)); + uint32_t a6 = le32toh(*(uint32_t *)(s+24)); + uint32_t a7 = le32toh(*(uint32_t *)(s+28)); + h[0] = a0&((1<<26)-1); /* 26 used, 32-26 left. 26 */ + h[1] = (a0>>26) | ((a1&((1<<19)-1))<< 6); /* (32-26) + 19 = 6+19 = 25 */ + h[2] = (a1>>19) | ((a2&((1<<13)-1))<<13); /* (32-19) + 13 = 13+13 = 26 */ + h[3] = (a2>>13) | ((a3&((1<< 6)-1))<<19); /* (32-13) + 6 = 19+ 6 = 25 */ + h[4] = (a3>> 6); /* (32- 6) = 26 */ + h[5] = a4&((1<<25)-1); /* 25 */ + h[6] = (a4>>25) | ((a5&((1<<19)-1))<< 7); /* (32-25) + 19 = 7+19 = 26 */ + h[7] = (a5>>19) | ((a6&((1<<12)-1))<<13); /* (32-19) + 12 = 13+12 = 25 */ + h[8] = (a6>>12) | ((a7&((1<< 6)-1))<<20); /* (32-12) + 6 = 20+ 6 = 26 */ + h[9] = (a7>> 6)&((1<<25)-1); /* 25 */ } -/* Find the difference of two numbers: output = in - output - * (note the order of the arguments!). - */ -static void fdifference(limb *output, const limb *in) +static __always_inline void fe_frombytes(fe *h, const uint8_t *s) { - unsigned int i; - - for (i = 0; i < 10; ++i) { - output[i] = in[i] - output[i]; - } + fe_frombytes_impl(h->v, s); } -/* Multiply a number by a scalar: output = in * scalar */ -static void fscalar_product(limb *output, const limb *in, const limb scalar) +static __always_inline uint8_t /*bool*/ addcarryx_u25(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low) { - unsigned int i; - - for (i = 0; i < 10; ++i) { - output[i] = in[i] * scalar; - } + /* This function extracts 25 bits of result and 1 bit of carry (26 total), so + * a 32-bit intermediate is sufficient. + */ + uint32_t x = a + b + c; + *low = x & ((1 << 25) - 1); + return (x >> 25) & 1; } -/* 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) +static __always_inline uint8_t /*bool*/ addcarryx_u26(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low) { - /* 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. + /* This function extracts 26 bits of result and 1 bit of carry (27 total), so + * a 32-bit intermediate is sufficient. */ - 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 + uint32_t x = a + b + c; + *low = x & ((1 << 26) - 1); + return (x >> 26) & 1; +} -/* 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) +static __always_inline uint8_t /*bool*/ subborrow_u25(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low) { - /* 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; + /* This function extracts 25 bits of result and 1 bit of borrow (26 total), so + * a 32-bit intermediate is sufficient. + */ + uint32_t x = a - b - c; + *low = x & ((1 << 25) - 1); + return x >> 31; } -/* 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) +static __always_inline uint8_t /*bool*/ subborrow_u26(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low) { - /* 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; + /* This function extracts 26 bits of result and 1 bit of borrow (27 total), so + * a 32-bit intermediate is sufficient. + */ + uint32_t x = a - b - c; + *low = x & ((1 << 26) - 1); + return x >> 31; } -/* 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) +static __always_inline uint32_t cmovznz32(uint32_t t, uint32_t z, uint32_t nz) { - unsigned int i; - - output[10] = 0; + t = -!!t; /* all set if nonzero, 0 if 0 */ + return (t&nz) | ((~t)&z); +} - 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]; +static __always_inline void fe_freeze(uint32_t out[10], const uint32_t in1[10]) +{ + { const uint32_t x17 = in1[9]; + { const uint32_t x18 = in1[8]; + { const uint32_t x16 = in1[7]; + { const uint32_t x14 = in1[6]; + { const uint32_t x12 = in1[5]; + { const uint32_t x10 = in1[4]; + { const uint32_t x8 = in1[3]; + { const uint32_t x6 = in1[2]; + { const uint32_t x4 = in1[1]; + { const uint32_t x2 = in1[0]; + { uint32_t x20; uint8_t/*bool*/ x21 = subborrow_u26(0x0, x2, 0x3ffffed, &x20); + { uint32_t x23; uint8_t/*bool*/ x24 = subborrow_u25(x21, x4, 0x1ffffff, &x23); + { uint32_t x26; uint8_t/*bool*/ x27 = subborrow_u26(x24, x6, 0x3ffffff, &x26); + { uint32_t x29; uint8_t/*bool*/ x30 = subborrow_u25(x27, x8, 0x1ffffff, &x29); + { uint32_t x32; uint8_t/*bool*/ x33 = subborrow_u26(x30, x10, 0x3ffffff, &x32); + { uint32_t x35; uint8_t/*bool*/ x36 = subborrow_u25(x33, x12, 0x1ffffff, &x35); + { uint32_t x38; uint8_t/*bool*/ x39 = subborrow_u26(x36, x14, 0x3ffffff, &x38); + { uint32_t x41; uint8_t/*bool*/ x42 = subborrow_u25(x39, x16, 0x1ffffff, &x41); + { uint32_t x44; uint8_t/*bool*/ x45 = subborrow_u26(x42, x18, 0x3ffffff, &x44); + { uint32_t x47; uint8_t/*bool*/ x48 = subborrow_u25(x45, x17, 0x1ffffff, &x47); + { uint32_t x49 = cmovznz32(x48, 0x0, 0xffffffff); + { uint32_t x50 = (x49 & 0x3ffffed); + { uint32_t x52; uint8_t/*bool*/ x53 = addcarryx_u26(0x0, x20, x50, &x52); + { uint32_t x54 = (x49 & 0x1ffffff); + { uint32_t x56; uint8_t/*bool*/ x57 = addcarryx_u25(x53, x23, x54, &x56); + { uint32_t x58 = (x49 & 0x3ffffff); + { uint32_t x60; uint8_t/*bool*/ x61 = addcarryx_u26(x57, x26, x58, &x60); + { uint32_t x62 = (x49 & 0x1ffffff); + { uint32_t x64; uint8_t/*bool*/ x65 = addcarryx_u25(x61, x29, x62, &x64); + { uint32_t x66 = (x49 & 0x3ffffff); + { uint32_t x68; uint8_t/*bool*/ x69 = addcarryx_u26(x65, x32, x66, &x68); + { uint32_t x70 = (x49 & 0x1ffffff); + { uint32_t x72; uint8_t/*bool*/ x73 = addcarryx_u25(x69, x35, x70, &x72); + { uint32_t x74 = (x49 & 0x3ffffff); + { uint32_t x76; uint8_t/*bool*/ x77 = addcarryx_u26(x73, x38, x74, &x76); + { uint32_t x78 = (x49 & 0x1ffffff); + { uint32_t x80; uint8_t/*bool*/ x81 = addcarryx_u25(x77, x41, x78, &x80); + { uint32_t x82 = (x49 & 0x3ffffff); + { uint32_t x84; uint8_t/*bool*/ x85 = addcarryx_u26(x81, x44, x82, &x84); + { uint32_t x86 = (x49 & 0x1ffffff); + { uint32_t x88; addcarryx_u25(x85, x47, x86, &x88); + out[0] = x52; + out[1] = x56; + out[2] = x60; + out[3] = x64; + out[4] = x68; + out[5] = x72; + out[6] = x76; + out[7] = x80; + out[8] = x84; + out[9] = x88; + }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} +} - output[10] = 0; +static __always_inline void fe_tobytes(uint8_t s[32], const fe *f) +{ + uint32_t h[10]; + fe_freeze(h, f->v); + s[0] = h[0] >> 0; + s[1] = h[0] >> 8; + s[2] = h[0] >> 16; + s[3] = (h[0] >> 24) | (h[1] << 2); + s[4] = h[1] >> 6; + s[5] = h[1] >> 14; + s[6] = (h[1] >> 22) | (h[2] << 3); + s[7] = h[2] >> 5; + s[8] = h[2] >> 13; + s[9] = (h[2] >> 21) | (h[3] << 5); + s[10] = h[3] >> 3; + s[11] = h[3] >> 11; + s[12] = (h[3] >> 19) | (h[4] << 6); + s[13] = h[4] >> 2; + s[14] = h[4] >> 10; + s[15] = h[4] >> 18; + s[16] = h[5] >> 0; + s[17] = h[5] >> 8; + s[18] = h[5] >> 16; + s[19] = (h[5] >> 24) | (h[6] << 1); + s[20] = h[6] >> 7; + s[21] = h[6] >> 15; + s[22] = (h[6] >> 23) | (h[7] << 3); + s[23] = h[7] >> 5; + s[24] = h[7] >> 13; + s[25] = (h[7] >> 21) | (h[8] << 4); + s[26] = h[8] >> 4; + s[27] = h[8] >> 12; + s[28] = (h[8] >> 20) | (h[9] << 6); + s[29] = h[9] >> 2; + s[30] = h[9] >> 10; + s[31] = h[9] >> 18; +} - /* 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]); +/* h = f */ +static __always_inline void fe_copy(fe *h, const fe *f) +{ + __builtin_memmove(h, f, sizeof(uint32_t) * 10); +} - output[0] -= over << 26; - output[1] += over; - } +static __always_inline void fe_copy_lt(fe_loose *h, const fe *f) +{ + __builtin_memmove(h, f, sizeof(uint32_t) * 10); +} - /* 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. - */ +/* h = 0 */ +static __always_inline void fe_0(fe *h) +{ + __builtin_memset(h, 0, sizeof(uint32_t) * 10); } -/* 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) +/* h = 1 */ +static __always_inline void fe_1(fe *h) { - limb t[19]; + __builtin_memset(h, 0, sizeof(uint32_t) * 10); + h->v[0] = 1; +} - fproduct(t, in, in2); - /* |t[i]| < 14*2^54 */ - freduce_degree(t); - freduce_coefficients(t); - /* |t[i]| < 2^26 */ - __builtin_memcpy(output, t, sizeof(limb) * 10); +static __always_inline void fe_add_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10]) +{ + { const uint32_t x20 = in1[9]; + { const uint32_t x21 = in1[8]; + { const uint32_t x19 = in1[7]; + { const uint32_t x17 = in1[6]; + { const uint32_t x15 = in1[5]; + { const uint32_t x13 = in1[4]; + { const uint32_t x11 = in1[3]; + { const uint32_t x9 = in1[2]; + { const uint32_t x7 = in1[1]; + { const uint32_t x5 = in1[0]; + { const uint32_t x38 = in2[9]; + { const uint32_t x39 = in2[8]; + { const uint32_t x37 = in2[7]; + { const uint32_t x35 = in2[6]; + { const uint32_t x33 = in2[5]; + { const uint32_t x31 = in2[4]; + { const uint32_t x29 = in2[3]; + { const uint32_t x27 = in2[2]; + { const uint32_t x25 = in2[1]; + { const uint32_t x23 = in2[0]; + out[0] = (x5 + x23); + out[1] = (x7 + x25); + out[2] = (x9 + x27); + out[3] = (x11 + x29); + out[4] = (x13 + x31); + out[5] = (x15 + x33); + out[6] = (x17 + x35); + out[7] = (x19 + x37); + out[8] = (x21 + x39); + out[9] = (x20 + x38); + }}}}}}}}}}}}}}}}}}}} } -/* 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. +/* h = f + g + * Can overlap h with f or g. */ -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) +static __always_inline void fe_add(fe_loose *h, const fe *f, const fe *g) { - 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 */ - __builtin_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 + fe_add_impl(h->v, f->v, g->v); +} -/* int32_t_eq returns 0xffffffff iff a == b and zero otherwise. */ -static int32_t int32_t_eq(int32_t a, int32_t b) +static __always_inline void fe_sub_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10]) { - a = ~(a ^ b); - a &= a << 16; - a &= a << 8; - a &= a << 4; - a &= a << 2; - a &= a << 1; - return a >> 31; + { const uint32_t x20 = in1[9]; + { const uint32_t x21 = in1[8]; + { const uint32_t x19 = in1[7]; + { const uint32_t x17 = in1[6]; + { const uint32_t x15 = in1[5]; + { const uint32_t x13 = in1[4]; + { const uint32_t x11 = in1[3]; + { const uint32_t x9 = in1[2]; + { const uint32_t x7 = in1[1]; + { const uint32_t x5 = in1[0]; + { const uint32_t x38 = in2[9]; + { const uint32_t x39 = in2[8]; + { const uint32_t x37 = in2[7]; + { const uint32_t x35 = in2[6]; + { const uint32_t x33 = in2[5]; + { const uint32_t x31 = in2[4]; + { const uint32_t x29 = in2[3]; + { const uint32_t x27 = in2[2]; + { const uint32_t x25 = in2[1]; + { const uint32_t x23 = in2[0]; + out[0] = ((0x7ffffda + x5) - x23); + out[1] = ((0x3fffffe + x7) - x25); + out[2] = ((0x7fffffe + x9) - x27); + out[3] = ((0x3fffffe + x11) - x29); + out[4] = ((0x7fffffe + x13) - x31); + out[5] = ((0x3fffffe + x15) - x33); + out[6] = ((0x7fffffe + x17) - x35); + out[7] = ((0x3fffffe + x19) - x37); + out[8] = ((0x7fffffe + x21) - x39); + out[9] = ((0x3fffffe + x20) - x38); + }}}}}}}}}}}}}}}}}}}} } -/* int32_t_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are - * both non-negative. +/* h = f - g + * Can overlap h with f or g. */ -static int32_t int32_t_gte(int32_t a, int32_t b) +static __always_inline void fe_sub(fe_loose *h, const fe *f, const fe *g) { - a -= b; - /* a >= 0 iff a >= b. */ - return ~(a >> 31); + fe_sub_impl(h->v, f->v, g->v); } -/* 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) +static __always_inline void fe_mul_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10]) { - 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); - } + { const uint32_t x20 = in1[9]; + { const uint32_t x21 = in1[8]; + { const uint32_t x19 = in1[7]; + { const uint32_t x17 = in1[6]; + { const uint32_t x15 = in1[5]; + { const uint32_t x13 = in1[4]; + { const uint32_t x11 = in1[3]; + { const uint32_t x9 = in1[2]; + { const uint32_t x7 = in1[1]; + { const uint32_t x5 = in1[0]; + { const uint32_t x38 = in2[9]; + { const uint32_t x39 = in2[8]; + { const uint32_t x37 = in2[7]; + { const uint32_t x35 = in2[6]; + { const uint32_t x33 = in2[5]; + { const uint32_t x31 = in2[4]; + { const uint32_t x29 = in2[3]; + { const uint32_t x27 = in2[2]; + { const uint32_t x25 = in2[1]; + { const uint32_t x23 = in2[0]; + { uint64_t x40 = ((uint64_t)x23 * x5); + { uint64_t x41 = (((uint64_t)x23 * x7) + ((uint64_t)x25 * x5)); + { uint64_t x42 = ((((uint64_t)(0x2 * x25) * x7) + ((uint64_t)x23 * x9)) + ((uint64_t)x27 * x5)); + { uint64_t x43 = (((((uint64_t)x25 * x9) + ((uint64_t)x27 * x7)) + ((uint64_t)x23 * x11)) + ((uint64_t)x29 * x5)); + { uint64_t x44 = (((((uint64_t)x27 * x9) + (0x2 * (((uint64_t)x25 * x11) + ((uint64_t)x29 * x7)))) + ((uint64_t)x23 * x13)) + ((uint64_t)x31 * x5)); + { uint64_t x45 = (((((((uint64_t)x27 * x11) + ((uint64_t)x29 * x9)) + ((uint64_t)x25 * x13)) + ((uint64_t)x31 * x7)) + ((uint64_t)x23 * x15)) + ((uint64_t)x33 * x5)); + { uint64_t x46 = (((((0x2 * ((((uint64_t)x29 * x11) + ((uint64_t)x25 * x15)) + ((uint64_t)x33 * x7))) + ((uint64_t)x27 * x13)) + ((uint64_t)x31 * x9)) + ((uint64_t)x23 * x17)) + ((uint64_t)x35 * x5)); + { uint64_t x47 = (((((((((uint64_t)x29 * x13) + ((uint64_t)x31 * x11)) + ((uint64_t)x27 * x15)) + ((uint64_t)x33 * x9)) + ((uint64_t)x25 * x17)) + ((uint64_t)x35 * x7)) + ((uint64_t)x23 * x19)) + ((uint64_t)x37 * x5)); + { uint64_t x48 = (((((((uint64_t)x31 * x13) + (0x2 * (((((uint64_t)x29 * x15) + ((uint64_t)x33 * x11)) + ((uint64_t)x25 * x19)) + ((uint64_t)x37 * x7)))) + ((uint64_t)x27 * x17)) + ((uint64_t)x35 * x9)) + ((uint64_t)x23 * x21)) + ((uint64_t)x39 * x5)); + { uint64_t x49 = (((((((((((uint64_t)x31 * x15) + ((uint64_t)x33 * x13)) + ((uint64_t)x29 * x17)) + ((uint64_t)x35 * x11)) + ((uint64_t)x27 * x19)) + ((uint64_t)x37 * x9)) + ((uint64_t)x25 * x21)) + ((uint64_t)x39 * x7)) + ((uint64_t)x23 * x20)) + ((uint64_t)x38 * x5)); + { uint64_t x50 = (((((0x2 * ((((((uint64_t)x33 * x15) + ((uint64_t)x29 * x19)) + ((uint64_t)x37 * x11)) + ((uint64_t)x25 * x20)) + ((uint64_t)x38 * x7))) + ((uint64_t)x31 * x17)) + ((uint64_t)x35 * x13)) + ((uint64_t)x27 * x21)) + ((uint64_t)x39 * x9)); + { uint64_t x51 = (((((((((uint64_t)x33 * x17) + ((uint64_t)x35 * x15)) + ((uint64_t)x31 * x19)) + ((uint64_t)x37 * x13)) + ((uint64_t)x29 * x21)) + ((uint64_t)x39 * x11)) + ((uint64_t)x27 * x20)) + ((uint64_t)x38 * x9)); + { uint64_t x52 = (((((uint64_t)x35 * x17) + (0x2 * (((((uint64_t)x33 * x19) + ((uint64_t)x37 * x15)) + ((uint64_t)x29 * x20)) + ((uint64_t)x38 * x11)))) + ((uint64_t)x31 * x21)) + ((uint64_t)x39 * x13)); + { uint64_t x53 = (((((((uint64_t)x35 * x19) + ((uint64_t)x37 * x17)) + ((uint64_t)x33 * x21)) + ((uint64_t)x39 * x15)) + ((uint64_t)x31 * x20)) + ((uint64_t)x38 * x13)); + { uint64_t x54 = (((0x2 * ((((uint64_t)x37 * x19) + ((uint64_t)x33 * x20)) + ((uint64_t)x38 * x15))) + ((uint64_t)x35 * x21)) + ((uint64_t)x39 * x17)); + { uint64_t x55 = (((((uint64_t)x37 * x21) + ((uint64_t)x39 * x19)) + ((uint64_t)x35 * x20)) + ((uint64_t)x38 * x17)); + { uint64_t x56 = (((uint64_t)x39 * x21) + (0x2 * (((uint64_t)x37 * x20) + ((uint64_t)x38 * x19)))); + { uint64_t x57 = (((uint64_t)x39 * x20) + ((uint64_t)x38 * x21)); + { uint64_t x58 = ((uint64_t)(0x2 * x38) * x20); + { uint64_t x59 = (x48 + (x58 << 0x4)); + { uint64_t x60 = (x59 + (x58 << 0x1)); + { uint64_t x61 = (x60 + x58); + { uint64_t x62 = (x47 + (x57 << 0x4)); + { uint64_t x63 = (x62 + (x57 << 0x1)); + { uint64_t x64 = (x63 + x57); + { uint64_t x65 = (x46 + (x56 << 0x4)); + { uint64_t x66 = (x65 + (x56 << 0x1)); + { uint64_t x67 = (x66 + x56); + { uint64_t x68 = (x45 + (x55 << 0x4)); + { uint64_t x69 = (x68 + (x55 << 0x1)); + { uint64_t x70 = (x69 + x55); + { uint64_t x71 = (x44 + (x54 << 0x4)); + { uint64_t x72 = (x71 + (x54 << 0x1)); + { uint64_t x73 = (x72 + x54); + { uint64_t x74 = (x43 + (x53 << 0x4)); + { uint64_t x75 = (x74 + (x53 << 0x1)); + { uint64_t x76 = (x75 + x53); + { uint64_t x77 = (x42 + (x52 << 0x4)); + { uint64_t x78 = (x77 + (x52 << 0x1)); + { uint64_t x79 = (x78 + x52); + { uint64_t x80 = (x41 + (x51 << 0x4)); + { uint64_t x81 = (x80 + (x51 << 0x1)); + { uint64_t x82 = (x81 + x51); + { uint64_t x83 = (x40 + (x50 << 0x4)); + { uint64_t x84 = (x83 + (x50 << 0x1)); + { uint64_t x85 = (x84 + x50); + { uint64_t x86 = (x85 >> 0x1a); + { uint32_t x87 = ((uint32_t)x85 & 0x3ffffff); + { uint64_t x88 = (x86 + x82); + { uint64_t x89 = (x88 >> 0x19); + { uint32_t x90 = ((uint32_t)x88 & 0x1ffffff); + { uint64_t x91 = (x89 + x79); + { uint64_t x92 = (x91 >> 0x1a); + { uint32_t x93 = ((uint32_t)x91 & 0x3ffffff); + { uint64_t x94 = (x92 + x76); + { uint64_t x95 = (x94 >> 0x19); + { uint32_t x96 = ((uint32_t)x94 & 0x1ffffff); + { uint64_t x97 = (x95 + x73); + { uint64_t x98 = (x97 >> 0x1a); + { uint32_t x99 = ((uint32_t)x97 & 0x3ffffff); + { uint64_t x100 = (x98 + x70); + { uint64_t x101 = (x100 >> 0x19); + { uint32_t x102 = ((uint32_t)x100 & 0x1ffffff); + { uint64_t x103 = (x101 + x67); + { uint64_t x104 = (x103 >> 0x1a); + { uint32_t x105 = ((uint32_t)x103 & 0x3ffffff); + { uint64_t x106 = (x104 + x64); + { uint64_t x107 = (x106 >> 0x19); + { uint32_t x108 = ((uint32_t)x106 & 0x1ffffff); + { uint64_t x109 = (x107 + x61); + { uint64_t x110 = (x109 >> 0x1a); + { uint32_t x111 = ((uint32_t)x109 & 0x3ffffff); + { uint64_t x112 = (x110 + x49); + { uint64_t x113 = (x112 >> 0x19); + { uint32_t x114 = ((uint32_t)x112 & 0x1ffffff); + { uint64_t x115 = (x87 + (0x13 * x113)); + { uint32_t x116 = (uint32_t) (x115 >> 0x1a); + { uint32_t x117 = ((uint32_t)x115 & 0x3ffffff); + { uint32_t x118 = (x116 + x90); + { uint32_t x119 = (x118 >> 0x19); + { uint32_t x120 = (x118 & 0x1ffffff); + out[0] = x117; + out[1] = x120; + out[2] = (x119 + x93); + out[3] = x96; + out[4] = x99; + out[5] = x102; + out[6] = x105; + out[7] = x108; + out[8] = x111; + out[9] = x114; + }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} +} - /* 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. - */ - } +static __always_inline void fe_mul_ttt(fe *h, const fe *f, const fe *g) +{ + fe_mul_impl(h->v, f->v, g->v); +} - /* 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; - } +static __always_inline void fe_mul_tlt(fe *h, const fe_loose *f, const fe *g) +{ + fe_mul_impl(h->v, f->v, g->v); +} - /* 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; - } - } +static __always_inline void fe_mul_tll(fe *h, const fe_loose *f, const fe_loose *g) +{ + fe_mul_impl(h->v, f->v, g->v); +} - /* 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. - */ +static __always_inline void fe_sqr_impl(uint32_t out[10], const uint32_t in1[10]) +{ + { const uint32_t x17 = in1[9]; + { const uint32_t x18 = in1[8]; + { const uint32_t x16 = in1[7]; + { const uint32_t x14 = in1[6]; + { const uint32_t x12 = in1[5]; + { const uint32_t x10 = in1[4]; + { const uint32_t x8 = in1[3]; + { const uint32_t x6 = in1[2]; + { const uint32_t x4 = in1[1]; + { const uint32_t x2 = in1[0]; + { uint64_t x19 = ((uint64_t)x2 * x2); + { uint64_t x20 = ((uint64_t)(0x2 * x2) * x4); + { uint64_t x21 = (0x2 * (((uint64_t)x4 * x4) + ((uint64_t)x2 * x6))); + { uint64_t x22 = (0x2 * (((uint64_t)x4 * x6) + ((uint64_t)x2 * x8))); + { uint64_t x23 = ((((uint64_t)x6 * x6) + ((uint64_t)(0x4 * x4) * x8)) + ((uint64_t)(0x2 * x2) * x10)); + { uint64_t x24 = (0x2 * ((((uint64_t)x6 * x8) + ((uint64_t)x4 * x10)) + ((uint64_t)x2 * x12))); + { uint64_t x25 = (0x2 * (((((uint64_t)x8 * x8) + ((uint64_t)x6 * x10)) + ((uint64_t)x2 * x14)) + ((uint64_t)(0x2 * x4) * x12))); + { uint64_t x26 = (0x2 * (((((uint64_t)x8 * x10) + ((uint64_t)x6 * x12)) + ((uint64_t)x4 * x14)) + ((uint64_t)x2 * x16))); + { uint64_t x27 = (((uint64_t)x10 * x10) + (0x2 * ((((uint64_t)x6 * x14) + ((uint64_t)x2 * x18)) + (0x2 * (((uint64_t)x4 * x16) + ((uint64_t)x8 * x12)))))); + { uint64_t x28 = (0x2 * ((((((uint64_t)x10 * x12) + ((uint64_t)x8 * x14)) + ((uint64_t)x6 * x16)) + ((uint64_t)x4 * x18)) + ((uint64_t)x2 * x17))); + { uint64_t x29 = (0x2 * (((((uint64_t)x12 * x12) + ((uint64_t)x10 * x14)) + ((uint64_t)x6 * x18)) + (0x2 * (((uint64_t)x8 * x16) + ((uint64_t)x4 * x17))))); + { uint64_t x30 = (0x2 * (((((uint64_t)x12 * x14) + ((uint64_t)x10 * x16)) + ((uint64_t)x8 * x18)) + ((uint64_t)x6 * x17))); + { uint64_t x31 = (((uint64_t)x14 * x14) + (0x2 * (((uint64_t)x10 * x18) + (0x2 * (((uint64_t)x12 * x16) + ((uint64_t)x8 * x17)))))); + { uint64_t x32 = (0x2 * ((((uint64_t)x14 * x16) + ((uint64_t)x12 * x18)) + ((uint64_t)x10 * x17))); + { uint64_t x33 = (0x2 * ((((uint64_t)x16 * x16) + ((uint64_t)x14 * x18)) + ((uint64_t)(0x2 * x12) * x17))); + { uint64_t x34 = (0x2 * (((uint64_t)x16 * x18) + ((uint64_t)x14 * x17))); + { uint64_t x35 = (((uint64_t)x18 * x18) + ((uint64_t)(0x4 * x16) * x17)); + { uint64_t x36 = ((uint64_t)(0x2 * x18) * x17); + { uint64_t x37 = ((uint64_t)(0x2 * x17) * x17); + { uint64_t x38 = (x27 + (x37 << 0x4)); + { uint64_t x39 = (x38 + (x37 << 0x1)); + { uint64_t x40 = (x39 + x37); + { uint64_t x41 = (x26 + (x36 << 0x4)); + { uint64_t x42 = (x41 + (x36 << 0x1)); + { uint64_t x43 = (x42 + x36); + { uint64_t x44 = (x25 + (x35 << 0x4)); + { uint64_t x45 = (x44 + (x35 << 0x1)); + { uint64_t x46 = (x45 + x35); + { uint64_t x47 = (x24 + (x34 << 0x4)); + { uint64_t x48 = (x47 + (x34 << 0x1)); + { uint64_t x49 = (x48 + x34); + { uint64_t x50 = (x23 + (x33 << 0x4)); + { uint64_t x51 = (x50 + (x33 << 0x1)); + { uint64_t x52 = (x51 + x33); + { uint64_t x53 = (x22 + (x32 << 0x4)); + { uint64_t x54 = (x53 + (x32 << 0x1)); + { uint64_t x55 = (x54 + x32); + { uint64_t x56 = (x21 + (x31 << 0x4)); + { uint64_t x57 = (x56 + (x31 << 0x1)); + { uint64_t x58 = (x57 + x31); + { uint64_t x59 = (x20 + (x30 << 0x4)); + { uint64_t x60 = (x59 + (x30 << 0x1)); + { uint64_t x61 = (x60 + x30); + { uint64_t x62 = (x19 + (x29 << 0x4)); + { uint64_t x63 = (x62 + (x29 << 0x1)); + { uint64_t x64 = (x63 + x29); + { uint64_t x65 = (x64 >> 0x1a); + { uint32_t x66 = ((uint32_t)x64 & 0x3ffffff); + { uint64_t x67 = (x65 + x61); + { uint64_t x68 = (x67 >> 0x19); + { uint32_t x69 = ((uint32_t)x67 & 0x1ffffff); + { uint64_t x70 = (x68 + x58); + { uint64_t x71 = (x70 >> 0x1a); + { uint32_t x72 = ((uint32_t)x70 & 0x3ffffff); + { uint64_t x73 = (x71 + x55); + { uint64_t x74 = (x73 >> 0x19); + { uint32_t x75 = ((uint32_t)x73 & 0x1ffffff); + { uint64_t x76 = (x74 + x52); + { uint64_t x77 = (x76 >> 0x1a); + { uint32_t x78 = ((uint32_t)x76 & 0x3ffffff); + { uint64_t x79 = (x77 + x49); + { uint64_t x80 = (x79 >> 0x19); + { uint32_t x81 = ((uint32_t)x79 & 0x1ffffff); + { uint64_t x82 = (x80 + x46); + { uint64_t x83 = (x82 >> 0x1a); + { uint32_t x84 = ((uint32_t)x82 & 0x3ffffff); + { uint64_t x85 = (x83 + x43); + { uint64_t x86 = (x85 >> 0x19); + { uint32_t x87 = ((uint32_t)x85 & 0x1ffffff); + { uint64_t x88 = (x86 + x40); + { uint64_t x89 = (x88 >> 0x1a); + { uint32_t x90 = ((uint32_t)x88 & 0x3ffffff); + { uint64_t x91 = (x89 + x28); + { uint64_t x92 = (x91 >> 0x19); + { uint32_t x93 = ((uint32_t)x91 & 0x1ffffff); + { uint64_t x94 = (x66 + (0x13 * x92)); + { uint32_t x95 = (uint32_t) (x94 >> 0x1a); + { uint32_t x96 = ((uint32_t)x94 & 0x3ffffff); + { uint32_t x97 = (x95 + x69); + { uint32_t x98 = (x97 >> 0x19); + { uint32_t x99 = (x97 & 0x1ffffff); + out[0] = x96; + out[1] = x99; + out[2] = (x98 + x72); + out[3] = x75; + out[4] = x78; + out[5] = x81; + out[6] = x84; + out[7] = x87; + out[8] = x90; + out[9] = x93; + }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} +} - /* 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); - } - } +static __always_inline void fe_sq_tl(fe *h, const fe_loose *f) +{ + fe_sqr_impl(h->v, f->v); +} - /* 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; - } - } +static __always_inline void fe_sq_tt(fe *h, const fe *f) +{ + fe_sqr_impl(h->v, f->v); +} - 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]; - - __builtin_memcpy(origx, x, 10 * sizeof(limb)); - fsum(x, z); - /* |x[i]| < 2^27 */ - fdifference(z, origx); /* does x - z */ - /* |z[i]| < 2^27 */ - - __builtin_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 */ - __builtin_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 */ - __builtin_memcpy(x3, xxxprime, sizeof(limb) * 10); - __builtin_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 */ - __builtin_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) +static __always_inline void fe_loose_invert(fe *out, const fe_loose *z) { - unsigned int i; - const int32_t swap = (int32_t) -iswap; + fe t0; + fe t1; + fe t2; + fe t3; + int i; - for (i = 0; i < 10; ++i) { - const int32_t x = swap & (((int32_t)a[i]) ^ ((int32_t)b[i])); + fe_sq_tl(&t0, z); + fe_sq_tt(&t1, &t0); + for (i = 1; i < 2; ++i) + fe_sq_tt(&t1, &t1); + fe_mul_tlt(&t1, z, &t1); + fe_mul_ttt(&t0, &t0, &t1); + fe_sq_tt(&t2, &t0); + fe_mul_ttt(&t1, &t1, &t2); + fe_sq_tt(&t2, &t1); + for (i = 1; i < 5; ++i) + fe_sq_tt(&t2, &t2); + fe_mul_ttt(&t1, &t2, &t1); + fe_sq_tt(&t2, &t1); + for (i = 1; i < 10; ++i) + fe_sq_tt(&t2, &t2); + fe_mul_ttt(&t2, &t2, &t1); + fe_sq_tt(&t3, &t2); + for (i = 1; i < 20; ++i) + fe_sq_tt(&t3, &t3); + fe_mul_ttt(&t2, &t3, &t2); + fe_sq_tt(&t2, &t2); + for (i = 1; i < 10; ++i) + fe_sq_tt(&t2, &t2); + fe_mul_ttt(&t1, &t2, &t1); + fe_sq_tt(&t2, &t1); + for (i = 1; i < 50; ++i) + fe_sq_tt(&t2, &t2); + fe_mul_ttt(&t2, &t2, &t1); + fe_sq_tt(&t3, &t2); + for (i = 1; i < 100; ++i) + fe_sq_tt(&t3, &t3); + fe_mul_ttt(&t2, &t3, &t2); + fe_sq_tt(&t2, &t2); + for (i = 1; i < 50; ++i) + fe_sq_tt(&t2, &t2); + fe_mul_ttt(&t1, &t2, &t1); + fe_sq_tt(&t1, &t1); + for (i = 1; i < 5; ++i) + fe_sq_tt(&t1, &t1); + fe_mul_ttt(out, &t1, &t0); +} - a[i] = ((int32_t)a[i]) ^ x; - b[i] = ((int32_t)b[i]) ^ x; - } +static __always_inline void fe_invert(fe *out, const fe *z) +{ + fe_loose l; + fe_copy_lt(&l, z); + fe_loose_invert(out, &l); } -/* Calculates nQ where Q is the x-coordinate of a point on the curve +/* Replace (f,g) with (g,f) if b == 1; + * replace (f,g) with (f,g) if b == 0. * - * 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) + * Preconditions: b in {0,1} */ -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; - - __builtin_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; - } +static __always_inline void fe_cswap(fe *f, fe *g, unsigned int b) +{ + unsigned i; + b = 0-b; + for (i = 0; i < 10; i++) { + uint32_t x = f->v[i] ^ g->v[i]; + x &= b; + f->v[i] ^= x; + g->v[i] ^= x; } +} - __builtin_memcpy(resultx, nqx, sizeof(limb) * 10); - __builtin_memcpy(resultz, nqz, sizeof(limb) * 10); +/* NOTE: based on fiat-crypto fe_mul, edited for in2=121666, 0, 0.*/ +static __always_inline void fe_mul_121666_impl(uint32_t out[10], const uint32_t in1[10]) +{ + { const uint32_t x20 = in1[9]; + { const uint32_t x21 = in1[8]; + { const uint32_t x19 = in1[7]; + { const uint32_t x17 = in1[6]; + { const uint32_t x15 = in1[5]; + { const uint32_t x13 = in1[4]; + { const uint32_t x11 = in1[3]; + { const uint32_t x9 = in1[2]; + { const uint32_t x7 = in1[1]; + { const uint32_t x5 = in1[0]; + { const uint32_t x38 = 0; + { const uint32_t x39 = 0; + { const uint32_t x37 = 0; + { const uint32_t x35 = 0; + { const uint32_t x33 = 0; + { const uint32_t x31 = 0; + { const uint32_t x29 = 0; + { const uint32_t x27 = 0; + { const uint32_t x25 = 0; + { const uint32_t x23 = 121666; + { uint64_t x40 = ((uint64_t)x23 * x5); + { uint64_t x41 = (((uint64_t)x23 * x7) + ((uint64_t)x25 * x5)); + { uint64_t x42 = ((((uint64_t)(0x2 * x25) * x7) + ((uint64_t)x23 * x9)) + ((uint64_t)x27 * x5)); + { uint64_t x43 = (((((uint64_t)x25 * x9) + ((uint64_t)x27 * x7)) + ((uint64_t)x23 * x11)) + ((uint64_t)x29 * x5)); + { uint64_t x44 = (((((uint64_t)x27 * x9) + (0x2 * (((uint64_t)x25 * x11) + ((uint64_t)x29 * x7)))) + ((uint64_t)x23 * x13)) + ((uint64_t)x31 * x5)); + { uint64_t x45 = (((((((uint64_t)x27 * x11) + ((uint64_t)x29 * x9)) + ((uint64_t)x25 * x13)) + ((uint64_t)x31 * x7)) + ((uint64_t)x23 * x15)) + ((uint64_t)x33 * x5)); + { uint64_t x46 = (((((0x2 * ((((uint64_t)x29 * x11) + ((uint64_t)x25 * x15)) + ((uint64_t)x33 * x7))) + ((uint64_t)x27 * x13)) + ((uint64_t)x31 * x9)) + ((uint64_t)x23 * x17)) + ((uint64_t)x35 * x5)); + { uint64_t x47 = (((((((((uint64_t)x29 * x13) + ((uint64_t)x31 * x11)) + ((uint64_t)x27 * x15)) + ((uint64_t)x33 * x9)) + ((uint64_t)x25 * x17)) + ((uint64_t)x35 * x7)) + ((uint64_t)x23 * x19)) + ((uint64_t)x37 * x5)); + { uint64_t x48 = (((((((uint64_t)x31 * x13) + (0x2 * (((((uint64_t)x29 * x15) + ((uint64_t)x33 * x11)) + ((uint64_t)x25 * x19)) + ((uint64_t)x37 * x7)))) + ((uint64_t)x27 * x17)) + ((uint64_t)x35 * x9)) + ((uint64_t)x23 * x21)) + ((uint64_t)x39 * x5)); + { uint64_t x49 = (((((((((((uint64_t)x31 * x15) + ((uint64_t)x33 * x13)) + ((uint64_t)x29 * x17)) + ((uint64_t)x35 * x11)) + ((uint64_t)x27 * x19)) + ((uint64_t)x37 * x9)) + ((uint64_t)x25 * x21)) + ((uint64_t)x39 * x7)) + ((uint64_t)x23 * x20)) + ((uint64_t)x38 * x5)); + { uint64_t x50 = (((((0x2 * ((((((uint64_t)x33 * x15) + ((uint64_t)x29 * x19)) + ((uint64_t)x37 * x11)) + ((uint64_t)x25 * x20)) + ((uint64_t)x38 * x7))) + ((uint64_t)x31 * x17)) + ((uint64_t)x35 * x13)) + ((uint64_t)x27 * x21)) + ((uint64_t)x39 * x9)); + { uint64_t x51 = (((((((((uint64_t)x33 * x17) + ((uint64_t)x35 * x15)) + ((uint64_t)x31 * x19)) + ((uint64_t)x37 * x13)) + ((uint64_t)x29 * x21)) + ((uint64_t)x39 * x11)) + ((uint64_t)x27 * x20)) + ((uint64_t)x38 * x9)); + { uint64_t x52 = (((((uint64_t)x35 * x17) + (0x2 * (((((uint64_t)x33 * x19) + ((uint64_t)x37 * x15)) + ((uint64_t)x29 * x20)) + ((uint64_t)x38 * x11)))) + ((uint64_t)x31 * x21)) + ((uint64_t)x39 * x13)); + { uint64_t x53 = (((((((uint64_t)x35 * x19) + ((uint64_t)x37 * x17)) + ((uint64_t)x33 * x21)) + ((uint64_t)x39 * x15)) + ((uint64_t)x31 * x20)) + ((uint64_t)x38 * x13)); + { uint64_t x54 = (((0x2 * ((((uint64_t)x37 * x19) + ((uint64_t)x33 * x20)) + ((uint64_t)x38 * x15))) + ((uint64_t)x35 * x21)) + ((uint64_t)x39 * x17)); + { uint64_t x55 = (((((uint64_t)x37 * x21) + ((uint64_t)x39 * x19)) + ((uint64_t)x35 * x20)) + ((uint64_t)x38 * x17)); + { uint64_t x56 = (((uint64_t)x39 * x21) + (0x2 * (((uint64_t)x37 * x20) + ((uint64_t)x38 * x19)))); + { uint64_t x57 = (((uint64_t)x39 * x20) + ((uint64_t)x38 * x21)); + { uint64_t x58 = ((uint64_t)(0x2 * x38) * x20); + { uint64_t x59 = (x48 + (x58 << 0x4)); + { uint64_t x60 = (x59 + (x58 << 0x1)); + { uint64_t x61 = (x60 + x58); + { uint64_t x62 = (x47 + (x57 << 0x4)); + { uint64_t x63 = (x62 + (x57 << 0x1)); + { uint64_t x64 = (x63 + x57); + { uint64_t x65 = (x46 + (x56 << 0x4)); + { uint64_t x66 = (x65 + (x56 << 0x1)); + { uint64_t x67 = (x66 + x56); + { uint64_t x68 = (x45 + (x55 << 0x4)); + { uint64_t x69 = (x68 + (x55 << 0x1)); + { uint64_t x70 = (x69 + x55); + { uint64_t x71 = (x44 + (x54 << 0x4)); + { uint64_t x72 = (x71 + (x54 << 0x1)); + { uint64_t x73 = (x72 + x54); + { uint64_t x74 = (x43 + (x53 << 0x4)); + { uint64_t x75 = (x74 + (x53 << 0x1)); + { uint64_t x76 = (x75 + x53); + { uint64_t x77 = (x42 + (x52 << 0x4)); + { uint64_t x78 = (x77 + (x52 << 0x1)); + { uint64_t x79 = (x78 + x52); + { uint64_t x80 = (x41 + (x51 << 0x4)); + { uint64_t x81 = (x80 + (x51 << 0x1)); + { uint64_t x82 = (x81 + x51); + { uint64_t x83 = (x40 + (x50 << 0x4)); + { uint64_t x84 = (x83 + (x50 << 0x1)); + { uint64_t x85 = (x84 + x50); + { uint64_t x86 = (x85 >> 0x1a); + { uint32_t x87 = ((uint32_t)x85 & 0x3ffffff); + { uint64_t x88 = (x86 + x82); + { uint64_t x89 = (x88 >> 0x19); + { uint32_t x90 = ((uint32_t)x88 & 0x1ffffff); + { uint64_t x91 = (x89 + x79); + { uint64_t x92 = (x91 >> 0x1a); + { uint32_t x93 = ((uint32_t)x91 & 0x3ffffff); + { uint64_t x94 = (x92 + x76); + { uint64_t x95 = (x94 >> 0x19); + { uint32_t x96 = ((uint32_t)x94 & 0x1ffffff); + { uint64_t x97 = (x95 + x73); + { uint64_t x98 = (x97 >> 0x1a); + { uint32_t x99 = ((uint32_t)x97 & 0x3ffffff); + { uint64_t x100 = (x98 + x70); + { uint64_t x101 = (x100 >> 0x19); + { uint32_t x102 = ((uint32_t)x100 & 0x1ffffff); + { uint64_t x103 = (x101 + x67); + { uint64_t x104 = (x103 >> 0x1a); + { uint32_t x105 = ((uint32_t)x103 & 0x3ffffff); + { uint64_t x106 = (x104 + x64); + { uint64_t x107 = (x106 >> 0x19); + { uint32_t x108 = ((uint32_t)x106 & 0x1ffffff); + { uint64_t x109 = (x107 + x61); + { uint64_t x110 = (x109 >> 0x1a); + { uint32_t x111 = ((uint32_t)x109 & 0x3ffffff); + { uint64_t x112 = (x110 + x49); + { uint64_t x113 = (x112 >> 0x19); + { uint32_t x114 = ((uint32_t)x112 & 0x1ffffff); + { uint64_t x115 = (x87 + (0x13 * x113)); + { uint32_t x116 = (uint32_t) (x115 >> 0x1a); + { uint32_t x117 = ((uint32_t)x115 & 0x3ffffff); + { uint32_t x118 = (x116 + x90); + { uint32_t x119 = (x118 >> 0x19); + { uint32_t x120 = (x118 & 0x1ffffff); + out[0] = x117; + out[1] = x120; + out[2] = (x119 + x93); + out[3] = x96; + out[4] = x99; + out[5] = x102; + out[6] = x105; + out[7] = x108; + out[8] = x111; + out[9] = x114; + }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} } -static void crecip(limb *out, const limb *z) +static __always_inline void fe_mul121666(fe *h, const fe_loose *f) { - 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; + fe_mul_121666_impl(h->v, f->v); +} - /* 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); -} - -static inline void curve25519_normalize_secret(uint8_t secret[static 32]) +static __always_inline void normalize_secret(uint8_t secret[static 32]) { secret[0] &= 248; secret[31] &= 127; secret[31] |= 64; } -static inline void curve25519(uint8_t mypublic[static 32], const uint8_t secret[static 32], const uint8_t basepoint[static 32]) + +static void curve25519(uint8_t out[static 32], const uint8_t scalar[static 32], const uint8_t point[static 32]) { - limb bp[10], x[10], z[11], zmone[10]; + fe x1, x2, z2, x3, z3, tmp0, tmp1; + fe_loose x2l, z2l, x3l, tmp0l, tmp1l; + unsigned swap = 0; + int pos; uint8_t e[32]; - __builtin_memcpy(e, secret, 32); - curve25519_normalize_secret(e); + __builtin_memcpy(e, scalar, 32); + normalize_secret(e); + + /* The following implementation was transcribed to Coq and proven to + * correspond to unary scalar multiplication in affine coordinates given that + * x1 != 0 is the x coordinate of some point on the curve. It was also checked + * in Coq that doing a ladderstep with x1 = x3 = 0 gives z2' = z3' = 0, and z2 + * = z3 = 0 gives z2' = z3' = 0. The statement was quantified over the + * underlying field, so it applies to Curve25519 itself and the quadratic + * twist of Curve25519. It was not proven in Coq that prime-field arithmetic + * correctly simulates extension-field arithmetic on prime-field values. + * The decoding of the byte array representation of e was not considered. + * Specification of Montgomery curves in affine coordinates: + * + * Proof that these form a group that is isomorphic to a Weierstrass curve: + * + * Coq transcription and correctness proof of the loop (where scalarbits=255): + * + * + * preconditions: 0 <= e < 2^255 (not necessarily e < order), fe_invert(0) = 0 + */ + fe_frombytes(&x1, point); + fe_1(&x2); + fe_0(&z2); + fe_copy(&x3, &x1); + fe_1(&z3); + + for (pos = 254; pos >= 0; --pos) { + /* loop invariant as of right before the test, for the case where x1 != 0: + * pos >= -1; if z2 = 0 then x2 is nonzero; if z3 = 0 then x3 is nonzero + * let r := e >> (pos+1) in the following equalities of projective points: + * to_xz (r*P) === if swap then (x3, z3) else (x2, z2) + * to_xz ((r+1)*P) === if swap then (x2, z2) else (x3, z3) + * x1 is the nonzero x coordinate of the nonzero point (r*P-(r+1)*P) + */ + unsigned b = 1 & (e[pos / 8] >> (pos & 7)); + swap ^= b; + fe_cswap(&x2, &x3, swap); + fe_cswap(&z2, &z3, swap); + swap = b; + /* Coq transcription of ladderstep formula (called from transcribed loop): + * + * + * x1 != 0 + * x1 = 0 + */ + fe_sub(&tmp0l, &x3, &z3); + fe_sub(&tmp1l, &x2, &z2); + fe_add(&x2l, &x2, &z2); + fe_add(&z2l, &x3, &z3); + fe_mul_tll(&z3, &tmp0l, &x2l); + fe_mul_tll(&z2, &z2l, &tmp1l); + fe_sq_tl(&tmp0, &tmp1l); + fe_sq_tl(&tmp1, &x2l); + fe_add(&x3l, &z3, &z2); + fe_sub(&z2l, &z3, &z2); + fe_mul_ttt(&x2, &tmp1, &tmp0); + fe_sub(&tmp1l, &tmp1, &tmp0); + fe_sq_tl(&z2, &z2l); + fe_mul121666(&z3, &tmp1l); + fe_sq_tl(&x3, &x3l); + fe_add(&tmp0l, &tmp0, &z3); + fe_mul_ttt(&z3, &x1, &z2); + fe_mul_tll(&z2, &tmp1l, &tmp0l); + } + /* here pos=-1, so r=e, so to_xz (e*P) === if swap then (x3, z3) else (x2, z2) */ + fe_cswap(&x2, &x3, swap); + fe_cswap(&z2, &z3, swap); - fexpand(bp, basepoint); - cmult(x, z, e, bp); - crecip(zmone, z); - fmul(z, x, zmone); - fcontract(mypublic, z); + fe_invert(&z2, &z2); + fe_mul_ttt(&x2, &x2, &z2); + fe_tobytes(out, &x2); } EMSCRIPTEN_KEEPALIVE void curve25519_generate_public(uint8_t public[static 32], const uint8_t private[static 32]) @@ -889,7 +873,7 @@ EMSCRIPTEN_KEEPALIVE void curve25519_generate_private(uint8_t private[static 32] for (i = 0; i < 32; ++i) private[i] = EM_ASM_INT_V({ return Module.getRandomValue(); }); - curve25519_normalize_secret(private); + normalize_secret(private); } static inline void encode_base64(char dest[4], const uint8_t src[3]) -- cgit v1.2.3-59-g8ed1b