From f0258fe956843e4efc7c502e44d09b21668b096e Mon Sep 17 00:00:00 2001 From: Adam Langley Date: Tue, 6 Oct 2015 17:49:40 -0700 Subject: [PATCH] Add optimised version of P-224. MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit This imports the Google-authored P-224 implementation by Emilia Käsper and Bodo Möller that is also in upstream OpenSSL. Change-Id: I16005c74a2a3e374fb136d36f3f6569dab9d8919 Reviewed-on: https://boringssl-review.googlesource.com/6145 Reviewed-by: Adam Langley --- crypto/ec/CMakeLists.txt | 1 + crypto/ec/ec.c | 22 +- crypto/ec/internal.h | 1 + crypto/ec/p224-64.c | 1365 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 1383 insertions(+), 6 deletions(-) create mode 100644 crypto/ec/p224-64.c diff --git a/crypto/ec/CMakeLists.txt b/crypto/ec/CMakeLists.txt index 38a91f89..2b767590 100644 --- a/crypto/ec/CMakeLists.txt +++ b/crypto/ec/CMakeLists.txt @@ -10,6 +10,7 @@ add_library( ec_key.c ec_montgomery.c oct.c + p224-64.c p256-64.c util-64.c simple.c diff --git a/crypto/ec/ec.c b/crypto/ec/ec.c index 3117f16e..a73044e7 100644 --- a/crypto/ec/ec.c +++ b/crypto/ec/ec.c @@ -218,15 +218,25 @@ static const struct curve_data P521 = { 0xA5, 0xD0, 0x3B, 0xB5, 0xC9, 0xB8, 0x89, 0x9C, 0x47, 0xAE, 0xBB, 0x6F, 0xB7, 0x1E, 0x91, 0x38, 0x64, 0x09}}; +/* MSan appears to have a bug that causes code to be miscompiled in opt mode. + * While that is being looked at, don't run the uint128_t code under MSan. */ +#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ + !defined(MEMORY_SANITIZER) +#define BORINGSSL_USE_INT128_CODE +#endif + const struct built_in_curve OPENSSL_built_in_curves[] = { - {NID_secp224r1, &P224, 0}, + { + NID_secp224r1, &P224, +#if defined(BORINGSSL_USE_INT128_CODE) + EC_GFp_nistp224_method, +#else + 0, +#endif + }, { NID_X9_62_prime256v1, &P256, - /* MSAN appears to have a bug that causes this P-256 code to be miscompiled - * in opt mode. While that is being looked at, don't run the uint128_t - * P-256 code under MSAN for now. */ -#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ - !defined(MEMORY_SANITIZER) +#if defined(BORINGSSL_USE_INT128_CODE) EC_GFp_nistp256_method, #else 0, diff --git a/crypto/ec/internal.h b/crypto/ec/internal.h index 71062c16..4b78c1b3 100644 --- a/crypto/ec/internal.h +++ b/crypto/ec/internal.h @@ -319,6 +319,7 @@ void ec_GFp_nistp_points_make_affine_internal( void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, uint8_t in); +const EC_METHOD *EC_GFp_nistp224_method(void); const EC_METHOD *EC_GFp_nistp256_method(void); struct ec_key_st { diff --git a/crypto/ec/p224-64.c b/crypto/ec/p224-64.c new file mode 100644 index 00000000..f8527bca --- /dev/null +++ b/crypto/ec/p224-64.c @@ -0,0 +1,1365 @@ +/* Copyright (c) 2015, Google Inc. + * + * Permission to use, copy, modify, and/or 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. */ + +/* A 64-bit implementation of the NIST P-224 elliptic curve point multiplication + * + * Inspired by Daniel J. Bernstein's public domain nistp224 implementation + * and Adam Langley's public domain 64-bit C implementation of curve25519. */ + +#include + +#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) + +#include +#include +#include +#include +#include + +#include + +#include "internal.h" + + +typedef uint8_t u8; +typedef uint64_t u64; +typedef int64_t s64; + +/* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 + * using 64-bit coefficients called 'limbs', and sometimes (for multiplication + * results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + + * 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-limb + * representation is an 'felem'; a 7-widelimb representation is a 'widefelem'. + * Even within felems, bits of adjacent limbs overlap, and we don't always + * reduce the representations: we ensure that inputs to each felem + * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, and + * fit into a 128-bit word without overflow. The coefficients are then again + * partially reduced to obtain an felem satisfying a_i < 2^57. We only reduce + * to the unique minimal representation at the end of the computation. */ + +typedef uint64_t limb; +typedef __uint128_t widelimb; + +typedef limb felem[4]; +typedef widelimb widefelem[7]; + +/* Field element represented as a byte arrary. 28*8 = 224 bits is also the + * group order size for the elliptic curve, and we also use this type for + * scalars for point multiplication. */ +typedef u8 felem_bytearray[28]; + +static const felem_bytearray nistp224_curve_params[5] = { + {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x01}, + {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE}, + {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */ + 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA, 0x27, 0x0B, + 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4}, + {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */ + 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22, 0x34, 0x32, + 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21}, + {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */ + 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64, 0x44, 0xd5, + 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}}; + +/* Precomputed multiples of the standard generator + * Points are given in coordinates (X, Y, Z) where Z normally is 1 + * (0 for the point at infinity). + * For each field element, slice a_0 is word 0, etc. + * + * The table has 2 * 16 elements, starting with the following: + * index | bits | point + * ------+---------+------------------------------ + * 0 | 0 0 0 0 | 0G + * 1 | 0 0 0 1 | 1G + * 2 | 0 0 1 0 | 2^56G + * 3 | 0 0 1 1 | (2^56 + 1)G + * 4 | 0 1 0 0 | 2^112G + * 5 | 0 1 0 1 | (2^112 + 1)G + * 6 | 0 1 1 0 | (2^112 + 2^56)G + * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G + * 8 | 1 0 0 0 | 2^168G + * 9 | 1 0 0 1 | (2^168 + 1)G + * 10 | 1 0 1 0 | (2^168 + 2^56)G + * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G + * 12 | 1 1 0 0 | (2^168 + 2^112)G + * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G + * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G + * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G + * followed by a copy of this with each element multiplied by 2^28. + * + * The reason for this is so that we can clock bits into four different + * locations when doing simple scalar multiplies against the base point, + * and then another four locations using the second 16 elements. */ +static const felem gmul[2][16][3] = { + {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, + {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, + {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, + {1, 0, 0, 0}}, + {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, + {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, + {1, 0, 0, 0}}, + {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, + {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, + {1, 0, 0, 0}}, + {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, + {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, + {1, 0, 0, 0}}, + {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, + {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, + {1, 0, 0, 0}}, + {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, + {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, + {1, 0, 0, 0}}, + {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, + {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, + {1, 0, 0, 0}}, + {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, + {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, + {1, 0, 0, 0}}, + {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, + {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, + {1, 0, 0, 0}}, + {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, + {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, + {1, 0, 0, 0}}, + {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, + {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, + {1, 0, 0, 0}}, + {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, + {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, + {1, 0, 0, 0}}, + {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, + {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, + {1, 0, 0, 0}}, + {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, + {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, + {1, 0, 0, 0}}, + {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, + {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, + {1, 0, 0, 0}}}, + {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, + {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, + {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, + {1, 0, 0, 0}}, + {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, + {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, + {1, 0, 0, 0}}, + {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, + {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, + {1, 0, 0, 0}}, + {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, + {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, + {1, 0, 0, 0}}, + {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, + {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, + {1, 0, 0, 0}}, + {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, + {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, + {1, 0, 0, 0}}, + {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, + {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, + {1, 0, 0, 0}}, + {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, + {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, + {1, 0, 0, 0}}, + {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, + {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, + {1, 0, 0, 0}}, + {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, + {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, + {1, 0, 0, 0}}, + {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, + {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, + {1, 0, 0, 0}}, + {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, + {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, + {1, 0, 0, 0}}, + {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, + {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, + {1, 0, 0, 0}}, + {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, + {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, + {1, 0, 0, 0}}, + {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, + {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, + {1, 0, 0, 0}}}}; + +/* Helper functions to convert field elements to/from internal representation */ +static void bin28_to_felem(felem out, const u8 in[28]) { + out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff; + out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff; + out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff; + out[3] = (*((const uint64_t *)(in + 20))) >> 8; +} + +static void felem_to_bin28(u8 out[28], const felem in) { + unsigned i; + for (i = 0; i < 7; ++i) { + out[i] = in[0] >> (8 * i); + out[i + 7] = in[1] >> (8 * i); + out[i + 14] = in[2] >> (8 * i); + out[i + 21] = in[3] >> (8 * i); + } +} + +/* To preserve endianness when using BN_bn2bin and BN_bin2bn */ +static void flip_endian(u8 *out, const u8 *in, unsigned len) { + unsigned i; + for (i = 0; i < len; ++i) { + out[i] = in[len - 1 - i]; + } +} + +/* From OpenSSL BIGNUM to internal representation */ +static int BN_to_felem(felem out, const BIGNUM *bn) { + /* BN_bn2bin eats leading zeroes */ + felem_bytearray b_out; + memset(b_out, 0, sizeof(b_out)); + unsigned num_bytes = BN_num_bytes(bn); + if (num_bytes > sizeof(b_out) || + BN_is_negative(bn)) { + OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); + return 0; + } + + felem_bytearray b_in; + num_bytes = BN_bn2bin(bn, b_in); + flip_endian(b_out, b_in, num_bytes); + bin28_to_felem(out, b_out); + return 1; +} + +/* From internal representation to OpenSSL BIGNUM */ +static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) { + felem_bytearray b_in, b_out; + felem_to_bin28(b_in, in); + flip_endian(b_out, b_in, sizeof(b_out)); + return BN_bin2bn(b_out, sizeof(b_out), out); +} + +/* Field operations, using the internal representation of field elements. + * NB! These operations are specific to our point multiplication and cannot be + * expected to be correct in general - e.g., multiplication with a large scalar + * will cause an overflow. */ + +static void felem_one(felem out) { + out[0] = 1; + out[1] = 0; + out[2] = 0; + out[3] = 0; +} + +static void felem_assign(felem out, const felem in) { + out[0] = in[0]; + out[1] = in[1]; + out[2] = in[2]; + out[3] = in[3]; +} + +/* Sum two field elements: out += in */ +static void felem_sum(felem out, const felem in) { + out[0] += in[0]; + out[1] += in[1]; + out[2] += in[2]; + out[3] += in[3]; +} + +/* Get negative value: out = -in */ +/* Assumes in[i] < 2^57 */ +static void felem_neg(felem out, const felem in) { + static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); + static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); + static const limb two58m42m2 = + (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); + + /* Set to 0 mod 2^224-2^96+1 to ensure out > in */ + out[0] = two58p2 - in[0]; + out[1] = two58m42m2 - in[1]; + out[2] = two58m2 - in[2]; + out[3] = two58m2 - in[3]; +} + +/* Subtract field elements: out -= in */ +/* Assumes in[i] < 2^57 */ +static void felem_diff(felem out, const felem in) { + static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2); + static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2); + static const limb two58m42m2 = + (((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2); + + /* Add 0 mod 2^224-2^96+1 to ensure out > in */ + out[0] += two58p2; + out[1] += two58m42m2; + out[2] += two58m2; + out[3] += two58m2; + + out[0] -= in[0]; + out[1] -= in[1]; + out[2] -= in[2]; + out[3] -= in[3]; +} + +/* Subtract in unreduced 128-bit mode: out -= in */ +/* Assumes in[i] < 2^119 */ +static void widefelem_diff(widefelem out, const widefelem in) { + static const widelimb two120 = ((widelimb)1) << 120; + static const widelimb two120m64 = + (((widelimb)1) << 120) - (((widelimb)1) << 64); + static const widelimb two120m104m64 = + (((widelimb)1) << 120) - (((widelimb)1) << 104) - (((widelimb)1) << 64); + + /* Add 0 mod 2^224-2^96+1 to ensure out > in */ + out[0] += two120; + out[1] += two120m64; + out[2] += two120m64; + out[3] += two120; + out[4] += two120m104m64; + out[5] += two120m64; + out[6] += two120m64; + + out[0] -= in[0]; + out[1] -= in[1]; + out[2] -= in[2]; + out[3] -= in[3]; + out[4] -= in[4]; + out[5] -= in[5]; + out[6] -= in[6]; +} + +/* Subtract in mixed mode: out128 -= in64 */ +/* in[i] < 2^63 */ +static void felem_diff_128_64(widefelem out, const felem in) { + static const widelimb two64p8 = (((widelimb)1) << 64) + (((widelimb)1) << 8); + static const widelimb two64m8 = (((widelimb)1) << 64) - (((widelimb)1) << 8); + static const widelimb two64m48m8 = + (((widelimb)1) << 64) - (((widelimb)1) << 48) - (((widelimb)1) << 8); + + /* Add 0 mod 2^224-2^96+1 to ensure out > in */ + out[0] += two64p8; + out[1] += two64m48m8; + out[2] += two64m8; + out[3] += two64m8; + + out[0] -= in[0]; + out[1] -= in[1]; + out[2] -= in[2]; + out[3] -= in[3]; +} + +/* Multiply a field element by a scalar: out = out * scalar + * The scalars we actually use are small, so results fit without overflow */ +static void felem_scalar(felem out, const limb scalar) { + out[0] *= scalar; + out[1] *= scalar; + out[2] *= scalar; + out[3] *= scalar; +} + +/* Multiply an unreduced field element by a scalar: out = out * scalar + * The scalars we actually use are small, so results fit without overflow */ +static void widefelem_scalar(widefelem out, const widelimb scalar) { + out[0] *= scalar; + out[1] *= scalar; + out[2] *= scalar; + out[3] *= scalar; + out[4] *= scalar; + out[5] *= scalar; + out[6] *= scalar; +} + +/* Square a field element: out = in^2 */ +static void felem_square(widefelem out, const felem in) { + limb tmp0, tmp1, tmp2; + tmp0 = 2 * in[0]; + tmp1 = 2 * in[1]; + tmp2 = 2 * in[2]; + out[0] = ((widelimb)in[0]) * in[0]; + out[1] = ((widelimb)in[0]) * tmp1; + out[2] = ((widelimb)in[0]) * tmp2 + ((widelimb)in[1]) * in[1]; + out[3] = ((widelimb)in[3]) * tmp0 + ((widelimb)in[1]) * tmp2; + out[4] = ((widelimb)in[3]) * tmp1 + ((widelimb)in[2]) * in[2]; + out[5] = ((widelimb)in[3]) * tmp2; + out[6] = ((widelimb)in[3]) * in[3]; +} + +/* Multiply two field elements: out = in1 * in2 */ +static void felem_mul(widefelem out, const felem in1, const felem in2) { + out[0] = ((widelimb)in1[0]) * in2[0]; + out[1] = ((widelimb)in1[0]) * in2[1] + ((widelimb)in1[1]) * in2[0]; + out[2] = ((widelimb)in1[0]) * in2[2] + ((widelimb)in1[1]) * in2[1] + + ((widelimb)in1[2]) * in2[0]; + out[3] = ((widelimb)in1[0]) * in2[3] + ((widelimb)in1[1]) * in2[2] + + ((widelimb)in1[2]) * in2[1] + ((widelimb)in1[3]) * in2[0]; + out[4] = ((widelimb)in1[1]) * in2[3] + ((widelimb)in1[2]) * in2[2] + + ((widelimb)in1[3]) * in2[1]; + out[5] = ((widelimb)in1[2]) * in2[3] + ((widelimb)in1[3]) * in2[2]; + out[6] = ((widelimb)in1[3]) * in2[3]; +} + +/* Reduce seven 128-bit coefficients to four 64-bit coefficients. + * Requires in[i] < 2^126, + * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */ +static void felem_reduce(felem out, const widefelem in) { + static const widelimb two127p15 = + (((widelimb)1) << 127) + (((widelimb)1) << 15); + static const widelimb two127m71 = + (((widelimb)1) << 127) - (((widelimb)1) << 71); + static const widelimb two127m71m55 = + (((widelimb)1) << 127) - (((widelimb)1) << 71) - (((widelimb)1) << 55); + widelimb output[5]; + + /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */ + output[0] = in[0] + two127p15; + output[1] = in[1] + two127m71m55; + output[2] = in[2] + two127m71; + output[3] = in[3]; + output[4] = in[4]; + + /* Eliminate in[4], in[5], in[6] */ + output[4] += in[6] >> 16; + output[3] += (in[6] & 0xffff) << 40; + output[2] -= in[6]; + + output[3] += in[5] >> 16; + output[2] += (in[5] & 0xffff) << 40; + output[1] -= in[5]; + + output[2] += output[4] >> 16; + output[1] += (output[4] & 0xffff) << 40; + output[0] -= output[4]; + + /* Carry 2 -> 3 -> 4 */ + output[3] += output[2] >> 56; + output[2] &= 0x00ffffffffffffff; + + output[4] = output[3] >> 56; + output[3] &= 0x00ffffffffffffff; + + /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */ + + /* Eliminate output[4] */ + output[2] += output[4] >> 16; + /* output[2] < 2^56 + 2^56 = 2^57 */ + output[1] += (output[4] & 0xffff) << 40; + output[0] -= output[4]; + + /* Carry 0 -> 1 -> 2 -> 3 */ + output[1] += output[0] >> 56; + out[0] = output[0] & 0x00ffffffffffffff; + + output[2] += output[1] >> 56; + /* output[2] < 2^57 + 2^72 */ + out[1] = output[1] & 0x00ffffffffffffff; + output[3] += output[2] >> 56; + /* output[3] <= 2^56 + 2^16 */ + out[2] = output[2] & 0x00ffffffffffffff; + + /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, + * out[3] <= 2^56 + 2^16 (due to final carry), + * so out < 2*p */ + out[3] = output[3]; +} + +static void felem_square_reduce(felem out, const felem in) { + widefelem tmp; + felem_square(tmp, in); + felem_reduce(out, tmp); +} + +static void felem_mul_reduce(felem out, const felem in1, const felem in2) { + widefelem tmp; + felem_mul(tmp, in1, in2); + felem_reduce(out, tmp); +} + +/* Reduce to unique minimal representation. + * Requires 0 <= in < 2*p (always call felem_reduce first) */ +static void felem_contract(felem out, const felem in) { + static const int64_t two56 = ((limb)1) << 56; + /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */ + /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */ + int64_t tmp[4], a; + tmp[0] = in[0]; + tmp[1] = in[1]; + tmp[2] = in[2]; + tmp[3] = in[3]; + /* Case 1: a = 1 iff in >= 2^224 */ + a = (in[3] >> 56); + tmp[0] -= a; + tmp[1] += a << 40; + tmp[3] &= 0x00ffffffffffffff; + /* Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1 and + * the lower part is non-zero */ + a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | + (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); + a &= 0x00ffffffffffffff; + /* turn a into an all-one mask (if a = 0) or an all-zero mask */ + a = (a - 1) >> 63; + /* subtract 2^224 - 2^96 + 1 if a is all-one */ + tmp[3] &= a ^ 0xffffffffffffffff; + tmp[2] &= a ^ 0xffffffffffffffff; + tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; + tmp[0] -= 1 & a; + + /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must + * be non-zero, so we only need one step */ + a = tmp[0] >> 63; + tmp[0] += two56 & a; + tmp[1] -= 1 & a; + + /* carry 1 -> 2 -> 3 */ + tmp[2] += tmp[1] >> 56; + tmp[1] &= 0x00ffffffffffffff; + + tmp[3] += tmp[2] >> 56; + tmp[2] &= 0x00ffffffffffffff; + + /* Now 0 <= out < p */ + out[0] = tmp[0]; + out[1] = tmp[1]; + out[2] = tmp[2]; + out[3] = tmp[3]; +} + +/* Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field + * elements are reduced to in < 2^225, so we only need to check three cases: 0, + * 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2 */ +static limb felem_is_zero(const felem in) { + limb zero = in[0] | in[1] | in[2] | in[3]; + zero = (((int64_t)(zero)-1) >> 63) & 1; + + limb two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) | + (in[2] ^ 0x00ffffffffffffff) | + (in[3] ^ 0x00ffffffffffffff); + two224m96p1 = (((int64_t)(two224m96p1)-1) >> 63) & 1; + limb two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) | + (in[2] ^ 0x00ffffffffffffff) | + (in[3] ^ 0x01ffffffffffffff); + two225m97p2 = (((int64_t)(two225m97p2)-1) >> 63) & 1; + return (zero | two224m96p1 | two225m97p2); +} + +static limb felem_is_zero_int(const felem in) { + return (int)(felem_is_zero(in) & ((limb)1)); +} + +/* Invert a field element */ +/* Computation chain copied from djb's code */ +static void felem_inv(felem out, const felem in) { + felem ftmp, ftmp2, ftmp3, ftmp4; + widefelem tmp; + unsigned i; + + felem_square(tmp, in); + felem_reduce(ftmp, tmp); /* 2 */ + felem_mul(tmp, in, ftmp); + felem_reduce(ftmp, tmp); /* 2^2 - 1 */ + felem_square(tmp, ftmp); + felem_reduce(ftmp, tmp); /* 2^3 - 2 */ + felem_mul(tmp, in, ftmp); + felem_reduce(ftmp, tmp); /* 2^3 - 1 */ + felem_square(tmp, ftmp); + felem_reduce(ftmp2, tmp); /* 2^4 - 2 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp2, tmp); /* 2^5 - 4 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp2, tmp); /* 2^6 - 8 */ + felem_mul(tmp, ftmp2, ftmp); + felem_reduce(ftmp, tmp); /* 2^6 - 1 */ + felem_square(tmp, ftmp); + felem_reduce(ftmp2, tmp); /* 2^7 - 2 */ + for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp2, tmp); + } + felem_mul(tmp, ftmp2, ftmp); + felem_reduce(ftmp2, tmp); /* 2^12 - 1 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp3, tmp); /* 2^13 - 2 */ + for (i = 0; i < 11; ++i) {/* 2^24 - 2^12 */ + felem_square(tmp, ftmp3); + felem_reduce(ftmp3, tmp); + } + felem_mul(tmp, ftmp3, ftmp2); + felem_reduce(ftmp2, tmp); /* 2^24 - 1 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp3, tmp); /* 2^25 - 2 */ + for (i = 0; i < 23; ++i) {/* 2^48 - 2^24 */ + felem_square(tmp, ftmp3); + felem_reduce(ftmp3, tmp); + } + felem_mul(tmp, ftmp3, ftmp2); + felem_reduce(ftmp3, tmp); /* 2^48 - 1 */ + felem_square(tmp, ftmp3); + felem_reduce(ftmp4, tmp); /* 2^49 - 2 */ + for (i = 0; i < 47; ++i) {/* 2^96 - 2^48 */ + felem_square(tmp, ftmp4); + felem_reduce(ftmp4, tmp); + } + felem_mul(tmp, ftmp3, ftmp4); + felem_reduce(ftmp3, tmp); /* 2^96 - 1 */ + felem_square(tmp, ftmp3); + felem_reduce(ftmp4, tmp); /* 2^97 - 2 */ + for (i = 0; i < 23; ++i) {/* 2^120 - 2^24 */ + felem_square(tmp, ftmp4); + felem_reduce(ftmp4, tmp); + } + felem_mul(tmp, ftmp2, ftmp4); + felem_reduce(ftmp2, tmp); /* 2^120 - 1 */ + for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */ + felem_square(tmp, ftmp2); + felem_reduce(ftmp2, tmp); + } + felem_mul(tmp, ftmp2, ftmp); + felem_reduce(ftmp, tmp); /* 2^126 - 1 */ + felem_square(tmp, ftmp); + felem_reduce(ftmp, tmp); /* 2^127 - 2 */ + felem_mul(tmp, ftmp, in); + felem_reduce(ftmp, tmp); /* 2^127 - 1 */ + for (i = 0; i < 97; ++i) {/* 2^224 - 2^97 */ + felem_square(tmp, ftmp); + felem_reduce(ftmp, tmp); + } + felem_mul(tmp, ftmp, ftmp3); + felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */ +} + +/* Copy in constant time: + * if icopy == 1, copy in to out, + * if icopy == 0, copy out to itself. */ +static void copy_conditional(felem out, const felem in, limb icopy) { + unsigned i; + /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ + const limb copy = -icopy; + for (i = 0; i < 4; ++i) { + const limb tmp = copy & (in[i] ^ out[i]); + out[i] ^= tmp; + } +} + +/* ELLIPTIC CURVE POINT OPERATIONS + * + * Points are represented in Jacobian projective coordinates: + * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3), + * or to the point at infinity if Z == 0. */ + +/* Double an elliptic curve point: + * (X', Y', Z') = 2 * (X, Y, Z), where + * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2 + * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2 + * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z + * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, + * while x_out == y_in is not (maybe this works, but it's not tested). */ +static void point_double(felem x_out, felem y_out, felem z_out, + const felem x_in, const felem y_in, const felem z_in) { + widefelem tmp, tmp2; + felem delta, gamma, beta, alpha, ftmp, ftmp2; + + felem_assign(ftmp, x_in); + felem_assign(ftmp2, x_in); + + /* delta = z^2 */ + felem_square(tmp, z_in); + felem_reduce(delta, tmp); + + /* gamma = y^2 */ + felem_square(tmp, y_in); + felem_reduce(gamma, tmp); + + /* beta = x*gamma */ + felem_mul(tmp, x_in, gamma); + felem_reduce(beta, tmp); + + /* alpha = 3*(x-delta)*(x+delta) */ + felem_diff(ftmp, delta); + /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ + felem_sum(ftmp2, delta); + /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ + felem_scalar(ftmp2, 3); + /* ftmp2[i] < 3 * 2^58 < 2^60 */ + felem_mul(tmp, ftmp, ftmp2); + /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */ + felem_reduce(alpha, tmp); + + /* x' = alpha^2 - 8*beta */ + felem_square(tmp, alpha); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + felem_assign(ftmp, beta); + felem_scalar(ftmp, 8); + /* ftmp[i] < 8 * 2^57 = 2^60 */ + felem_diff_128_64(tmp, ftmp); + /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ + felem_reduce(x_out, tmp); + + /* z' = (y + z)^2 - gamma - delta */ + felem_sum(delta, gamma); + /* delta[i] < 2^57 + 2^57 = 2^58 */ + felem_assign(ftmp, y_in); + felem_sum(ftmp, z_in); + /* ftmp[i] < 2^57 + 2^57 = 2^58 */ + felem_square(tmp, ftmp); + /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */ + felem_diff_128_64(tmp, delta); + /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */ + felem_reduce(z_out, tmp); + + /* y' = alpha*(4*beta - x') - 8*gamma^2 */ + felem_scalar(beta, 4); + /* beta[i] < 4 * 2^57 = 2^59 */ + felem_diff(beta, x_out); + /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */ + felem_mul(tmp, alpha, beta); + /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */ + felem_square(tmp2, gamma); + /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */ + widefelem_scalar(tmp2, 8); + /* tmp2[i] < 8 * 2^116 = 2^119 */ + widefelem_diff(tmp, tmp2); + /* tmp[i] < 2^119 + 2^120 < 2^121 */ + felem_reduce(y_out, tmp); +} + +/* Add two elliptic curve points: + * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where + * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - + * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 + * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * + * X_1)^2 - X_3) - + * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 + * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) + * + * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. */ + +/* This function is not entirely constant-time: it includes a branch for + * checking whether the two input points are equal, (while not equal to the + * point at infinity). This case never happens during single point + * multiplication, so there is no timing leak for ECDH or ECDSA signing. */ +static void point_add(felem x3, felem y3, felem z3, const felem x1, + const felem y1, const felem z1, const int mixed, + const felem x2, const felem y2, const felem z2) { + felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; + widefelem tmp, tmp2; + limb z1_is_zero, z2_is_zero, x_equal, y_equal; + + if (!mixed) { + /* ftmp2 = z2^2 */ + felem_square(tmp, z2); + felem_reduce(ftmp2, tmp); + + /* ftmp4 = z2^3 */ + felem_mul(tmp, ftmp2, z2); + felem_reduce(ftmp4, tmp); + + /* ftmp4 = z2^3*y1 */ + felem_mul(tmp2, ftmp4, y1); + felem_reduce(ftmp4, tmp2); + + /* ftmp2 = z2^2*x1 */ + felem_mul(tmp2, ftmp2, x1); + felem_reduce(ftmp2, tmp2); + } else { + /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */ + + /* ftmp4 = z2^3*y1 */ + felem_assign(ftmp4, y1); + + /* ftmp2 = z2^2*x1 */ + felem_assign(ftmp2, x1); + } + + /* ftmp = z1^2 */ + felem_square(tmp, z1); + felem_reduce(ftmp, tmp); + + /* ftmp3 = z1^3 */ + felem_mul(tmp, ftmp, z1); + felem_reduce(ftmp3, tmp); + + /* tmp = z1^3*y2 */ + felem_mul(tmp, ftmp3, y2); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + + /* ftmp3 = z1^3*y2 - z2^3*y1 */ + felem_diff_128_64(tmp, ftmp4); + /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ + felem_reduce(ftmp3, tmp); + + /* tmp = z1^2*x2 */ + felem_mul(tmp, ftmp, x2); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + + /* ftmp = z1^2*x2 - z2^2*x1 */ + felem_diff_128_64(tmp, ftmp2); + /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ + felem_reduce(ftmp, tmp); + + /* the formulae are incorrect if the points are equal + * so we check for this and do doubling if this happens */ + x_equal = felem_is_zero(ftmp); + y_equal = felem_is_zero(ftmp3); + z1_is_zero = felem_is_zero(z1); + z2_is_zero = felem_is_zero(z2); + /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */ + if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { + point_double(x3, y3, z3, x1, y1, z1); + return; + } + + /* ftmp5 = z1*z2 */ + if (!mixed) { + felem_mul(tmp, z1, z2); + felem_reduce(ftmp5, tmp); + } else { + /* special case z2 = 0 is handled later */ + felem_assign(ftmp5, z1); + } + + /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */ + felem_mul(tmp, ftmp, ftmp5); + felem_reduce(z_out, tmp); + + /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ + felem_assign(ftmp5, ftmp); + felem_square(tmp, ftmp); + felem_reduce(ftmp, tmp); + + /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */ + felem_mul(tmp, ftmp, ftmp5); + felem_reduce(ftmp5, tmp); + + /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ + felem_mul(tmp, ftmp2, ftmp); + felem_reduce(ftmp2, tmp); + + /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ + felem_mul(tmp, ftmp4, ftmp5); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + + /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */ + felem_square(tmp2, ftmp3); + /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */ + + /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */ + felem_diff_128_64(tmp2, ftmp5); + /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */ + + /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ + felem_assign(ftmp5, ftmp2); + felem_scalar(ftmp5, 2); + /* ftmp5[i] < 2 * 2^57 = 2^58 */ + + /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - + 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ + felem_diff_128_64(tmp2, ftmp5); + /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */ + felem_reduce(x_out, tmp2); + + /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */ + felem_diff(ftmp2, x_out); + /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */ + + /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */ + felem_mul(tmp2, ftmp3, ftmp2); + /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ + + /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - + z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ + widefelem_diff(tmp2, tmp); + /* tmp2[i] < 2^118 + 2^120 < 2^121 */ + felem_reduce(y_out, tmp2); + + /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is + * the point at infinity, so we need to check for this separately */ + + /* if point 1 is at infinity, copy point 2 to output, and vice versa */ + copy_conditional(x_out, x2, z1_is_zero); + copy_conditional(x_out, x1, z2_is_zero); + copy_conditional(y_out, y2, z1_is_zero); + copy_conditional(y_out, y1, z2_is_zero); + copy_conditional(z_out, z2, z1_is_zero); + copy_conditional(z_out, z1, z2_is_zero); + felem_assign(x3, x_out); + felem_assign(y3, y_out); + felem_assign(z3, z_out); +} + +/* select_point selects the |idx|th point from a precomputation table and + * copies it to out. */ +static void select_point(const u64 idx, unsigned int size, + const felem pre_comp[/*size*/][3], felem out[3]) { + unsigned i, j; + limb *outlimbs = &out[0][0]; + memset(outlimbs, 0, 3 * sizeof(felem)); + + for (i = 0; i < size; i++) { + const limb *inlimbs = &pre_comp[i][0][0]; + u64 mask = i ^ idx; + mask |= mask >> 4; + mask |= mask >> 2; + mask |= mask >> 1; + mask &= 1; + mask--; + for (j = 0; j < 4 * 3; j++) { + outlimbs[j] |= inlimbs[j] & mask; + } + } +} + +/* get_bit returns the |i|th bit in |in| */ +static char get_bit(const felem_bytearray in, unsigned i) { + if (i >= 224) { + return 0; + } + return (in[i >> 3] >> (i & 7)) & 1; +} + +/* Interleaved point multiplication using precomputed point multiples: + * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], + * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple + * of the generator, using certain (large) precomputed multiples in g_pre_comp. + * Output point (X, Y, Z) is stored in x_out, y_out, z_out */ +static void batch_mul(felem x_out, felem y_out, felem z_out, + const felem_bytearray scalars[], + const unsigned num_points, const u8 *g_scalar, + const int mixed, const felem pre_comp[][17][3], + const felem g_pre_comp[2][16][3]) { + int i, skip; + unsigned num; + unsigned gen_mul = (g_scalar != NULL); + felem nq[3], tmp[4]; + u64 bits; + u8 sign, digit; + + /* set nq to the point at infinity */ + memset(nq, 0, 3 * sizeof(felem)); + + /* Loop over all scalars msb-to-lsb, interleaving additions + * of multiples of the generator (two in each of the last 28 rounds) + * and additions of other points multiples (every 5th round). */ + skip = 1; /* save two point operations in the first round */ + for (i = (num_points ? 220 : 27); i >= 0; --i) { + /* double */ + if (!skip) { + point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); + } + + /* add multiples of the generator */ + if (gen_mul && (i <= 27)) { + /* first, look 28 bits upwards */ + bits = get_bit(g_scalar, i + 196) << 3; + bits |= get_bit(g_scalar, i + 140) << 2; + bits |= get_bit(g_scalar, i + 84) << 1; + bits |= get_bit(g_scalar, i + 28); + /* select the point to add, in constant time */ + select_point(bits, 16, g_pre_comp[1], tmp); + + if (!skip) { + point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, + tmp[0], tmp[1], tmp[2]); + } else { + memcpy(nq, tmp, 3 * sizeof(felem)); + skip = 0; + } + + /* second, look at the current position */ + bits = get_bit(g_scalar, i + 168) << 3; + bits |= get_bit(g_scalar, i + 112) << 2; + bits |= get_bit(g_scalar, i + 56) << 1; + bits |= get_bit(g_scalar, i); + /* select the point to add, in constant time */ + select_point(bits, 16, g_pre_comp[0], tmp); + point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], + tmp[1], tmp[2]); + } + + /* do other additions every 5 doublings */ + if (num_points && (i % 5 == 0)) { + /* loop over all scalars */ + for (num = 0; num < num_points; ++num) { + bits = get_bit(scalars[num], i + 4) << 5; + bits |= get_bit(scalars[num], i + 3) << 4; + bits |= get_bit(scalars[num], i + 2) << 3; + bits |= get_bit(scalars[num], i + 1) << 2; + bits |= get_bit(scalars[num], i) << 1; + bits |= get_bit(scalars[num], i - 1); + ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); + + /* select the point to add or subtract */ + select_point(digit, 17, pre_comp[num], tmp); + felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */ + copy_conditional(tmp[1], tmp[3], sign); + + if (!skip) { + point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], mixed, tmp[0], + tmp[1], tmp[2]); + } else { + memcpy(nq, tmp, 3 * sizeof(felem)); + skip = 0; + } + } + } + } + felem_assign(x_out, nq[0]); + felem_assign(y_out, nq[1]); + felem_assign(z_out, nq[2]); +} + +int ec_GFp_nistp224_group_init(EC_GROUP *group) { + int ret; + ret = ec_GFp_simple_group_init(group); + group->a_is_minus3 = 1; + return ret; +} + +int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p, + const BIGNUM *a, const BIGNUM *b, + BN_CTX *ctx) { + int ret = 0; + BN_CTX *new_ctx = NULL; + BIGNUM *curve_p, *curve_a, *curve_b; + + if (ctx == NULL) { + ctx = BN_CTX_new(); + new_ctx = ctx; + if (ctx == NULL) { + return 0; + } + } + BN_CTX_start(ctx); + if (((curve_p = BN_CTX_get(ctx)) == NULL) || + ((curve_a = BN_CTX_get(ctx)) == NULL) || + ((curve_b = BN_CTX_get(ctx)) == NULL)) { + goto err; + } + BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p); + BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a); + BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b); + if (BN_cmp(curve_p, p) || + BN_cmp(curve_a, a) || + BN_cmp(curve_b, b)) { + OPENSSL_PUT_ERROR(EC, EC_R_WRONG_CURVE_PARAMETERS); + goto err; + } + ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); + +err: + BN_CTX_end(ctx); + BN_CTX_free(new_ctx); + return ret; +} + +/* Takes the Jacobian coordinates (X, Y, Z) of a point and returns + * (X', Y') = (X/Z^2, Y/Z^3) */ +int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, + const EC_POINT *point, + BIGNUM *x, BIGNUM *y, + BN_CTX *ctx) { + felem z1, z2, x_in, y_in, x_out, y_out; + widefelem tmp; + + if (EC_POINT_is_at_infinity(group, point)) { + OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); + return 0; + } + + if (!BN_to_felem(x_in, &point->X) || + !BN_to_felem(y_in, &point->Y) || + !BN_to_felem(z1, &point->Z)) { + return 0; + } + + felem_inv(z2, z1); + felem_square(tmp, z2); + felem_reduce(z1, tmp); + felem_mul(tmp, x_in, z1); + felem_reduce(x_in, tmp); + felem_contract(x_out, x_in); + if (x != NULL && !felem_to_BN(x, x_out)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + return 0; + } + + felem_mul(tmp, z1, z2); + felem_reduce(z1, tmp); + felem_mul(tmp, y_in, z1); + felem_reduce(y_in, tmp); + felem_contract(y_out, y_in); + if (y != NULL && !felem_to_BN(y, y_out)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + return 0; + } + + return 1; +} + +static void make_points_affine(size_t num, felem points[/*num*/][3], + felem tmp_felems[/*num+1*/]) { + /* Runs in constant time, unless an input is the point at infinity + * (which normally shouldn't happen). */ + ec_GFp_nistp_points_make_affine_internal( + num, points, sizeof(felem), tmp_felems, (void (*)(void *))felem_one, + (int (*)(const void *))felem_is_zero_int, + (void (*)(void *, const void *))felem_assign, + (void (*)(void *, const void *))felem_square_reduce, + (void (*)(void *, const void *, const void *))felem_mul_reduce, + (void (*)(void *, const void *))felem_inv, + (void (*)(void *, const void *))felem_contract); +} + +/* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values + * Result is stored in r (r can equal one of the inputs). */ +int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, + const BIGNUM *scalar, size_t num, + const EC_POINT *points[], + const BIGNUM *scalars[], BN_CTX *ctx) { + int ret = 0; + int j; + unsigned i; + int mixed = 0; + BN_CTX *new_ctx = NULL; + BIGNUM *x, *y, *z, *tmp_scalar; + felem_bytearray g_secret; + felem_bytearray *secrets = NULL; + felem(*pre_comp)[17][3] = NULL; + felem *tmp_felems = NULL; + felem_bytearray tmp; + unsigned num_bytes; + int have_pre_comp = 0; + size_t num_points = num; + felem x_in, y_in, z_in, x_out, y_out, z_out; + const felem(*g_pre_comp)[16][3] = NULL; + EC_POINT *generator = NULL; + const EC_POINT *p = NULL; + const BIGNUM *p_scalar = NULL; + + if (ctx == NULL) { + ctx = BN_CTX_new(); + new_ctx = ctx; + if (ctx == NULL) { + return 0; + } + } + + BN_CTX_start(ctx); + if ((x = BN_CTX_get(ctx)) == NULL || + (y = BN_CTX_get(ctx)) == NULL || + (z = BN_CTX_get(ctx)) == NULL || + (tmp_scalar = BN_CTX_get(ctx)) == NULL) { + goto err; + } + + if (scalar != NULL) { + /* try to use the standard precomputation */ + g_pre_comp = &gmul[0]; + generator = EC_POINT_new(group); + if (generator == NULL) { + goto err; + } + /* get the generator from precomputation */ + if (!felem_to_BN(x, g_pre_comp[0][1][0]) || + !felem_to_BN(y, g_pre_comp[0][1][1]) || + !felem_to_BN(z, g_pre_comp[0][1][2])) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + goto err; + } + if (!ec_point_set_Jprojective_coordinates_GFp(group, generator, x, y, z, + ctx)) { + goto err; + } + + if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { + /* precomputation matches generator */ + have_pre_comp = 1; + } else { + /* we don't have valid precomputation: + * treat the generator as a random point */ + num_points = num_points + 1; + } + } + + if (num_points > 0) { + if (num_points >= 3) { + /* unless we precompute multiples for just one or two points, + * converting those into affine form is time well spent */ + mixed = 1; + } + secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); + pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem)); + if (mixed) { + tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem)); + } + if (secrets == NULL || + pre_comp == NULL || + (mixed && tmp_felems == NULL)) { + OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); + goto err; + } + + /* we treat NULL scalars as 0, and NULL points as points at infinity, + * i.e., they contribute nothing to the linear combination */ + memset(secrets, 0, num_points * sizeof(felem_bytearray)); + memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem)); + for (i = 0; i < num_points; ++i) { + if (i == num) { + /* the generator */ + p = EC_GROUP_get0_generator(group); + p_scalar = scalar; + } else { + /* the i^th point */ + p = points[i]; + p_scalar = scalars[i]; + } + + if (p_scalar != NULL && p != NULL) { + /* reduce scalar to 0 <= scalar < 2^224 */ + if (BN_num_bits(p_scalar) > 224 || BN_is_negative(p_scalar)) { + /* this is an unusual input, and we don't guarantee + * constant-timeness */ + if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + goto err; + } + num_bytes = BN_bn2bin(tmp_scalar, tmp); + } else { + num_bytes = BN_bn2bin(p_scalar, tmp); + } + + flip_endian(secrets[i], tmp, num_bytes); + /* precompute multiples */ + if (!BN_to_felem(x_out, &p->X) || + !BN_to_felem(y_out, &p->Y) || + !BN_to_felem(z_out, &p->Z)) { + goto err; + } + + felem_assign(pre_comp[i][1][0], x_out); + felem_assign(pre_comp[i][1][1], y_out); + felem_assign(pre_comp[i][1][2], z_out); + + for (j = 2; j <= 16; ++j) { + if (j & 1) { + point_add(pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2], + pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2], + 0, pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], + pre_comp[i][j - 1][2]); + } else { + point_double(pre_comp[i][j][0], pre_comp[i][j][1], + pre_comp[i][j][2], pre_comp[i][j / 2][0], + pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); + } + } + } + } + + if (mixed) { + make_points_affine(num_points * 17, pre_comp[0], tmp_felems); + } + } + + /* the scalar for the generator */ + if (scalar != NULL && have_pre_comp) { + memset(g_secret, 0, sizeof(g_secret)); + /* reduce scalar to 0 <= scalar < 2^224 */ + if (BN_num_bits(scalar) > 224 || BN_is_negative(scalar)) { + /* this is an unusual input, and we don't guarantee constant-timeness */ + if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + goto err; + } + num_bytes = BN_bn2bin(tmp_scalar, tmp); + } else { + num_bytes = BN_bn2bin(scalar, tmp); + } + + flip_endian(g_secret, tmp, num_bytes); + /* do the multiplication with generator precomputation */ + batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, + num_points, g_secret, mixed, (const felem(*)[17][3])pre_comp, + g_pre_comp); + } else { + /* do the multiplication without generator precomputation */ + batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, + num_points, NULL, mixed, (const felem(*)[17][3])pre_comp, NULL); + } + + /* reduce the output to its unique minimal representation */ + felem_contract(x_in, x_out); + felem_contract(y_in, y_out); + felem_contract(z_in, z_out); + if (!felem_to_BN(x, x_in) || + !felem_to_BN(y, y_in) || + !felem_to_BN(z, z_in)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + goto err; + } + ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); + +err: + BN_CTX_end(ctx); + EC_POINT_free(generator); + BN_CTX_free(new_ctx); + OPENSSL_free(secrets); + OPENSSL_free(pre_comp); + OPENSSL_free(tmp_felems); + return ret; +} + +const EC_METHOD *EC_GFp_nistp224_method(void) { + static const EC_METHOD ret = {EC_FLAGS_DEFAULT_OCT, + ec_GFp_nistp224_group_init, + ec_GFp_simple_group_finish, + ec_GFp_simple_group_clear_finish, + ec_GFp_simple_group_copy, + ec_GFp_nistp224_group_set_curve, + ec_GFp_simple_group_get_curve, + ec_GFp_simple_group_get_degree, + ec_GFp_simple_group_check_discriminant, + ec_GFp_simple_point_init, + ec_GFp_simple_point_finish, + ec_GFp_simple_point_clear_finish, + ec_GFp_simple_point_copy, + ec_GFp_simple_point_set_to_infinity, + ec_GFp_simple_set_Jprojective_coordinates_GFp, + ec_GFp_simple_get_Jprojective_coordinates_GFp, + ec_GFp_simple_point_set_affine_coordinates, + ec_GFp_nistp224_point_get_affine_coordinates, + 0 /* point_set_compressed_coordinates */, + 0 /* point2oct */, + 0 /* oct2point */, + ec_GFp_simple_add, + ec_GFp_simple_dbl, + ec_GFp_simple_invert, + ec_GFp_simple_is_at_infinity, + ec_GFp_simple_is_on_curve, + ec_GFp_simple_cmp, + ec_GFp_simple_make_affine, + ec_GFp_simple_points_make_affine, + ec_GFp_nistp224_points_mul, + 0 /* precompute_mult */, + 0 /* have_precompute_mult */, + ec_GFp_simple_field_mul, + ec_GFp_simple_field_sqr, + 0 /* field_div */, + 0 /* field_encode */, + 0 /* field_decode */, + 0 /* field_set_to_one */}; + + return &ret; +} + +#endif