/**************************************************************************** * crypto/curve25519.c * $OpenBSD: curve25519.c,v 1.2 2020/07/22 13:54:30 tobhe Exp $ * * Copyright (C) 2018-2020 Jason A. Donenfeld . All Rights * Reserved. * Copyright (C) 2015-2016 The fiat-crypto Authors. * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. * * This contains two implementation: a machine-generated formally verified * implementation of Curve25519 ECDH from: * . Though originally machine * generated, it has been tweaked to be suitable for use in the kernel. * It is optimized for 32-bit machines and machines that cannot work * efficiently with 128-bit integer types. ****************************************************************************/ /**************************************************************************** * Included Files ****************************************************************************/ #include #include #include #include /**************************************************************************** * Private Type Definitions ****************************************************************************/ /* 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; /* 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; /**************************************************************************** * Private Data ****************************************************************************/ static const uint8_t g_null_point[CURVE25519_KEY_SIZE]; static const uint8_t g_base_point[CURVE25519_KEY_SIZE] = { 9 }; /**************************************************************************** * Private Functions ****************************************************************************/ static uint32_t get_unaligned_le32(FAR const uint8_t *a) { uint32_t l; memcpy(&l, a, sizeof(l)); return letoh32(l); } static void fe_frombytes_impl(uint32_t h[10], FAR const uint8_t *s) { /* Ignores top bit of s. */ uint32_t a0 = get_unaligned_le32(s); uint32_t a1 = get_unaligned_le32(s + 4); uint32_t a2 = get_unaligned_le32(s + 8); uint32_t a3 = get_unaligned_le32(s + 12); uint32_t a4 = get_unaligned_le32(s + 16); uint32_t a5 = get_unaligned_le32(s + 20); uint32_t a6 = get_unaligned_le32(s + 24); uint32_t a7 = get_unaligned_le32(s + 28); /* 26 used, 32 - 26 left. 26 */ h[0] = a0 & ((1 << 26) - 1); /* (32 - 26) + 19 = 6 + 19 = 25 */ h[1] = (a0 >> 26) | ((a1 & ((1 << 19) - 1)) << 6); /* (32 - 19) + 13 = 13 + 13 = 26 */ h[2] = (a1 >> 19) | ((a2 & ((1 << 13) - 1)) << 13); /* (32 - 13) + 6 = 19 + 6 = 25 */ h[3] = (a2 >> 13) | ((a3 & ((1 << 6) - 1)) << 19); /* (32 - 6) = 26 */ h[4] = (a3 >> 6); /* 25 */ h[5] = a4 & ((1 << 25) - 1); /* (32 - 25) + 19 = 7 + 19 = 26 */ h[6] = (a4 >> 25) | ((a5 & ((1 << 19) - 1)) << 7); /* (32 - 19) + 12 = 13 + 12 = 25 */ h[7] = (a5 >> 19) | ((a6 & ((1 << 12) - 1)) << 13); /* (32 - 12) + 6 = 20 + 6 = 26 */ h[8] = (a6 >> 12) | ((a7 & ((1 << 6) - 1)) << 20); /* 25 */ h[9] = (a7 >> 6) & ((1 << 25) - 1); } static void fe_frombytes(FAR fe *h, FAR const uint8_t *s) { fe_frombytes_impl(h->v, s); } /* uint8_t c - bool */ static uint8_t /* bool */ addcarryx_u25(uint8_t c, uint32_t a, uint32_t b, FAR uint32_t *low) { /* 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; } /* uint8_t c - bool */ static uint8_t /* bool */ addcarryx_u26(uint8_t c, uint32_t a, uint32_t b, FAR uint32_t *low) { /* This function extracts 26 bits of result and 1 bit of carry * (27 total), so a 32-bit intermediate is sufficient. */ uint32_t x = a + b + c; *low = x & ((1 << 26) - 1); return (x >> 26) & 1; } /* uint8_t c - bool */ static uint8_t /* bool */ subborrow_u25(uint8_t c, uint32_t a, uint32_t b, FAR uint32_t *low) { /* 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; } /* uint8_t c - bool */ static uint8_t /* bool */ subborrow_u26(uint8_t c, uint32_t a, uint32_t b, FAR uint32_t *low) { /* 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; } static uint32_t cmovznz32(uint32_t t, uint32_t z, uint32_t nz) { t = - !!t; /* all set if nonzero, 0 if 0 */ return (t & nz) | ((~t) & z); } static 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; /* bool */ uint8_t x21 = subborrow_u26(0x0, x2, 0x3ffffed, &x20); uint32_t x23; /* bool */ uint8_t x24 = subborrow_u25(x21, x4, 0x1ffffff, &x23); uint32_t x26; /* bool */ uint8_t x27 = subborrow_u26(x24, x6, 0x3ffffff, &x26); uint32_t x29; /* bool */ uint8_t x30 = subborrow_u25(x27, x8, 0x1ffffff, &x29); uint32_t x32; /* bool */ uint8_t x33 = subborrow_u26(x30, x10, 0x3ffffff, &x32); uint32_t x35; /* bool */ uint8_t x36 = subborrow_u25(x33, x12, 0x1ffffff, &x35); uint32_t x38; /* bool */ uint8_t x39 = subborrow_u26(x36, x14, 0x3ffffff, &x38); uint32_t x41; /* bool */ uint8_t x42 = subborrow_u25(x39, x16, 0x1ffffff, &x41); uint32_t x44; /* bool */ uint8_t x45 = subborrow_u26(x42, x18, 0x3ffffff, &x44); uint32_t x47; /* bool */ uint8_t x48 = subborrow_u25(x45, x17, 0x1ffffff, &x47); uint32_t x49 = cmovznz32(x48, 0x0, 0xffffffff); uint32_t x50 = (x49 & 0x3ffffed); uint32_t x52; /* bool */ uint8_t x53 = addcarryx_u26(0x0, x20, x50, &x52); uint32_t x54 = (x49 & 0x1ffffff); uint32_t x56; /* bool */ uint8_t x57 = addcarryx_u25(x53, x23, x54, &x56); uint32_t x58 = (x49 & 0x3ffffff); uint32_t x60; /* bool */ uint8_t x61 = addcarryx_u26(x57, x26, x58, &x60); uint32_t x62 = (x49 & 0x1ffffff); uint32_t x64; /* bool */ uint8_t x65 = addcarryx_u25(x61, x29, x62, &x64); uint32_t x66 = (x49 & 0x3ffffff); uint32_t x68; /* bool */ uint8_t x69 = addcarryx_u26(x65, x32, x66, &x68); uint32_t x70 = (x49 & 0x1ffffff); uint32_t x72; /* bool */ uint8_t x73 = addcarryx_u25(x69, x35, x70, &x72); uint32_t x74 = (x49 & 0x3ffffff); uint32_t x76; /* bool */ uint8_t x77 = addcarryx_u26(x73, x38, x74, &x76); uint32_t x78 = (x49 & 0x1ffffff); uint32_t x80; /* bool */ uint8_t x81 = addcarryx_u25(x77, x41, x78, &x80); uint32_t x82 = (x49 & 0x3ffffff); uint32_t x84; /* bool */ uint8_t 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; } static void fe_tobytes(uint8_t s[32], FAR 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; } /* h = f */ static void fe_copy(FAR fe *h, FAR const fe *f) { memmove(h, f, sizeof(uint32_t) * 10); } static void fe_copy_lt(FAR fe_loose *h, FAR const fe *f) { memmove(h, f, sizeof(uint32_t) * 10); } /* h = 0 */ static void fe_0(FAR fe *h) { memset(h, 0, sizeof(uint32_t) * 10); } /* h = 1 */ static void fe_1(FAR fe *h) { memset(h, 0, sizeof(uint32_t) * 10); h->v[0] = 1; } static 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); } /* h = f + g * Can overlap h with f or g. */ static void fe_add(FAR fe_loose *h, FAR const fe *f, FAR const fe *g) { fe_add_impl(h->v, f->v, g->v); } static void fe_sub_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] = ((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); } /* h = f - g * Can overlap h with f or g. */ static void fe_sub(FAR fe_loose *h, FAR const fe *f, FAR const fe *g) { fe_sub_impl(h->v, f->v, g->v); } static void fe_mul_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]; 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 fe_mul_ttt(FAR fe *h, FAR const fe *f, FAR const fe *g) { fe_mul_impl(h->v, f->v, g->v); } static void fe_mul_tlt(FAR fe *h, FAR const fe_loose *f, FAR const fe *g) { fe_mul_impl(h->v, f->v, g->v); } static void fe_mul_tll(FAR fe *h, FAR const fe_loose *f, FAR const fe_loose *g) { fe_mul_impl(h->v, f->v, g->v); } static 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; } static void fe_sq_tl(FAR fe *h, FAR const fe_loose *f) { fe_sqr_impl(h->v, f->v); } static void fe_sq_tt(FAR fe *h, FAR const fe *f) { fe_sqr_impl(h->v, f->v); } static void fe_loose_invert(FAR fe *out, FAR const fe_loose *z) { fe t0; fe t1; fe t2; fe t3; int 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); } static void fe_invert(FAR fe *out, FAR const fe *z) { fe_loose l; fe_copy_lt(&l, z); fe_loose_invert(out, &l); } /* Replace (f,g) with (g,f) if b == 1; * replace (f,g) with (f,g) if b == 0. * * Preconditions: b in {0,1} */ static void fe_cswap(FAR fe *f, FAR 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; } } /* NOTE: based on fiat-crypto fe_mul, edited for in2=121666, 0, 0. */ static 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 fe_mul121666(FAR fe *h, FAR const fe_loose *f) { fe_mul_121666_impl(h->v, f->v); } /**************************************************************************** * Public Functions ****************************************************************************/ int curve25519(uint8_t out[CURVE25519_KEY_SIZE], const uint8_t scalar[CURVE25519_KEY_SIZE], const uint8_t point[CURVE25519_KEY_SIZE]) { fe x1, x2, z2, x3, z3; fe_loose x2l, z2l, x3l; unsigned swap = 0; int pos; uint8_t e[32]; memcpy(e, scalar, 32); curve25519_clamp_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) { fe tmp0, tmp1; fe_loose tmp0l, tmp1l; /* 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); fe_invert(&z2, &z2); fe_mul_ttt(&x2, &x2, &z2); fe_tobytes(out, &x2); explicit_bzero(&x1, sizeof(x1)); explicit_bzero(&x2, sizeof(x2)); explicit_bzero(&z2, sizeof(z2)); explicit_bzero(&x3, sizeof(x3)); explicit_bzero(&z3, sizeof(z3)); explicit_bzero(&x2l, sizeof(x2l)); explicit_bzero(&z2l, sizeof(z2l)); explicit_bzero(&x3l, sizeof(x3l)); explicit_bzero(&e, sizeof(e)); return timingsafe_bcmp(out, g_null_point, CURVE25519_KEY_SIZE); } int curve25519_generate_public(uint8_t pub[CURVE25519_KEY_SIZE], const uint8_t secret[CURVE25519_KEY_SIZE]) { if (timingsafe_bcmp(secret, g_null_point, CURVE25519_KEY_SIZE) == 0) { return 0; } return curve25519(pub, secret, g_base_point); }