081e3f34a2
Having |Z_is_one| be out of sync with |Z| could potentially be a very bad thing, and in the past there have been multiple bugs of this sort, including one currently in p256-x86_64.c (type confusion: Montgomery- encoded vs unencoded). Avoid the issue entirely by getting rid of |Z_is_one|. Change-Id: Icb5aa0342df41d6bc443f15f952734295d0ee4ba Reviewed-on: https://boringssl-review.googlesource.com/6576 Reviewed-by: David Benjamin <davidben@google.com>
1244 lines
42 KiB
C
1244 lines
42 KiB
C
/* Copyright (c) 2015, Google Inc.
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*
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* Permission to use, copy, modify, and/or distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
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* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
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* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
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* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
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/* A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
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*
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* Inspired by Daniel J. Bernstein's public domain nistp224 implementation
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* and Adam Langley's public domain 64-bit C implementation of curve25519. */
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#include <openssl/base.h>
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#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \
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!defined(OPENSSL_SMALL)
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#include <openssl/bn.h>
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#include <openssl/ec.h>
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#include <openssl/err.h>
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#include <openssl/mem.h>
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#include <openssl/obj.h>
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#include <string.h>
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#include "internal.h"
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#include "../internal.h"
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typedef uint8_t u8;
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typedef uint64_t u64;
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typedef int64_t s64;
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/* Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
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* using 64-bit coefficients called 'limbs', and sometimes (for multiplication
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* results) as b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 +
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* 2^336*b_6 using 128-bit coefficients called 'widelimbs'. A 4-limb
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* representation is an 'felem'; a 7-widelimb representation is a 'widefelem'.
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* Even within felems, bits of adjacent limbs overlap, and we don't always
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* reduce the representations: we ensure that inputs to each felem
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* multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, and
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* fit into a 128-bit word without overflow. The coefficients are then again
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* partially reduced to obtain an felem satisfying a_i < 2^57. We only reduce
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* to the unique minimal representation at the end of the computation. */
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typedef uint64_t limb;
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typedef uint128_t widelimb;
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typedef limb felem[4];
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typedef widelimb widefelem[7];
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/* Field element represented as a byte arrary. 28*8 = 224 bits is also the
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* group order size for the elliptic curve, and we also use this type for
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* scalars for point multiplication. */
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typedef u8 felem_bytearray[28];
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/* Precomputed multiples of the standard generator
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* Points are given in coordinates (X, Y, Z) where Z normally is 1
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* (0 for the point at infinity).
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* For each field element, slice a_0 is word 0, etc.
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*
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* The table has 2 * 16 elements, starting with the following:
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* index | bits | point
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* ------+---------+------------------------------
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* 0 | 0 0 0 0 | 0G
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* 1 | 0 0 0 1 | 1G
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* 2 | 0 0 1 0 | 2^56G
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* 3 | 0 0 1 1 | (2^56 + 1)G
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* 4 | 0 1 0 0 | 2^112G
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* 5 | 0 1 0 1 | (2^112 + 1)G
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* 6 | 0 1 1 0 | (2^112 + 2^56)G
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* 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
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* 8 | 1 0 0 0 | 2^168G
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* 9 | 1 0 0 1 | (2^168 + 1)G
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* 10 | 1 0 1 0 | (2^168 + 2^56)G
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* 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
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* 12 | 1 1 0 0 | (2^168 + 2^112)G
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* 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
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* 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
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* 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
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* followed by a copy of this with each element multiplied by 2^28.
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*
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* The reason for this is so that we can clock bits into four different
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* locations when doing simple scalar multiplies against the base point,
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* and then another four locations using the second 16 elements. */
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static const felem g_pre_comp[2][16][3] = {
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{{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
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{{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
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{0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
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{1, 0, 0, 0}},
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{{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
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{0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
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{1, 0, 0, 0}},
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{{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
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{0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
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{1, 0, 0, 0}},
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{{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
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{0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
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{1, 0, 0, 0}},
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{{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
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{0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
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{1, 0, 0, 0}},
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{{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
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{0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
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{1, 0, 0, 0}},
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{{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
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{0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
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{1, 0, 0, 0}},
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{{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
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{0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
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{1, 0, 0, 0}},
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{{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
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{0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
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{1, 0, 0, 0}},
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{{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
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{0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
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{1, 0, 0, 0}},
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{{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
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{0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
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{1, 0, 0, 0}},
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{{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
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{0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
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{1, 0, 0, 0}},
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{{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
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{0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
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{1, 0, 0, 0}},
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{{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
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{0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
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{1, 0, 0, 0}},
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{{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
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{0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
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{1, 0, 0, 0}}},
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{{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
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{{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
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{0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
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{1, 0, 0, 0}},
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{{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
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{0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
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{1, 0, 0, 0}},
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{{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
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{0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
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{1, 0, 0, 0}},
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{{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
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{0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
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{1, 0, 0, 0}},
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{{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
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{0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
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{1, 0, 0, 0}},
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{{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
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{0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
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{1, 0, 0, 0}},
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{{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
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{0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
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{1, 0, 0, 0}},
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{{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
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{0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
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{1, 0, 0, 0}},
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{{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
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{0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
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{1, 0, 0, 0}},
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{{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
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{0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
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{1, 0, 0, 0}},
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{{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
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{0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
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{1, 0, 0, 0}},
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{{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
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{0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
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{1, 0, 0, 0}},
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{{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
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{0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
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{1, 0, 0, 0}},
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{{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
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{0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
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{1, 0, 0, 0}},
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{{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
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{0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
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{1, 0, 0, 0}}}};
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/* Helper functions to convert field elements to/from internal representation */
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static void bin28_to_felem(felem out, const u8 in[28]) {
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out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
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out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
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out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
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out[3] = (*((const uint64_t *)(in + 20))) >> 8;
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}
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static void felem_to_bin28(u8 out[28], const felem in) {
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unsigned i;
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for (i = 0; i < 7; ++i) {
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out[i] = in[0] >> (8 * i);
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out[i + 7] = in[1] >> (8 * i);
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out[i + 14] = in[2] >> (8 * i);
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out[i + 21] = in[3] >> (8 * i);
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}
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}
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/* To preserve endianness when using BN_bn2bin and BN_bin2bn */
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static void flip_endian(u8 *out, const u8 *in, unsigned len) {
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unsigned i;
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for (i = 0; i < len; ++i) {
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out[i] = in[len - 1 - i];
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}
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}
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/* From OpenSSL BIGNUM to internal representation */
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static int BN_to_felem(felem out, const BIGNUM *bn) {
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/* BN_bn2bin eats leading zeroes */
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felem_bytearray b_out;
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memset(b_out, 0, sizeof(b_out));
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unsigned num_bytes = BN_num_bytes(bn);
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if (num_bytes > sizeof(b_out) ||
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BN_is_negative(bn)) {
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OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE);
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return 0;
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}
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felem_bytearray b_in;
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num_bytes = BN_bn2bin(bn, b_in);
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flip_endian(b_out, b_in, num_bytes);
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bin28_to_felem(out, b_out);
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return 1;
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}
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/* From internal representation to OpenSSL BIGNUM */
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static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) {
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felem_bytearray b_in, b_out;
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felem_to_bin28(b_in, in);
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flip_endian(b_out, b_in, sizeof(b_out));
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return BN_bin2bn(b_out, sizeof(b_out), out);
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}
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/* Field operations, using the internal representation of field elements.
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* NB! These operations are specific to our point multiplication and cannot be
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* expected to be correct in general - e.g., multiplication with a large scalar
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* will cause an overflow. */
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static void felem_one(felem out) {
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out[0] = 1;
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out[1] = 0;
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out[2] = 0;
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out[3] = 0;
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}
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static void felem_assign(felem out, const felem in) {
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out[0] = in[0];
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out[1] = in[1];
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out[2] = in[2];
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out[3] = in[3];
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}
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/* Sum two field elements: out += in */
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static void felem_sum(felem out, const felem in) {
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out[0] += in[0];
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out[1] += in[1];
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out[2] += in[2];
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out[3] += in[3];
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}
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/* Get negative value: out = -in */
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/* Assumes in[i] < 2^57 */
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static void felem_neg(felem out, const felem in) {
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static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
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static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
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static const limb two58m42m2 =
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(((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
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/* Set to 0 mod 2^224-2^96+1 to ensure out > in */
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out[0] = two58p2 - in[0];
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out[1] = two58m42m2 - in[1];
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out[2] = two58m2 - in[2];
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out[3] = two58m2 - in[3];
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}
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/* Subtract field elements: out -= in */
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/* Assumes in[i] < 2^57 */
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static void felem_diff(felem out, const felem in) {
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static const limb two58p2 = (((limb)1) << 58) + (((limb)1) << 2);
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static const limb two58m2 = (((limb)1) << 58) - (((limb)1) << 2);
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static const limb two58m42m2 =
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(((limb)1) << 58) - (((limb)1) << 42) - (((limb)1) << 2);
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/* Add 0 mod 2^224-2^96+1 to ensure out > in */
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out[0] += two58p2;
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out[1] += two58m42m2;
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out[2] += two58m2;
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out[3] += two58m2;
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out[0] -= in[0];
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out[1] -= in[1];
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out[2] -= in[2];
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out[3] -= in[3];
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}
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/* Subtract in unreduced 128-bit mode: out -= in */
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/* Assumes in[i] < 2^119 */
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static void widefelem_diff(widefelem out, const widefelem in) {
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static const widelimb two120 = ((widelimb)1) << 120;
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static const widelimb two120m64 =
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(((widelimb)1) << 120) - (((widelimb)1) << 64);
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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]) {
|
|
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]);
|
|
}
|
|
|
|
/* 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);
|
|
}
|
|
|
|
int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
|
|
const BIGNUM *g_scalar, const EC_POINT *p_,
|
|
const BIGNUM *p_scalar_, BN_CTX *ctx) {
|
|
/* TODO: This function used to take |points| and |scalars| as arrays of
|
|
* |num| elements. The code below should be simplified to work in terms of
|
|
* |p_| and |p_scalar_|. */
|
|
size_t num = p_ != NULL ? 1 : 0;
|
|
const EC_POINT **points = p_ != NULL ? &p_ : NULL;
|
|
BIGNUM const *const *scalars = p_ != NULL ? &p_scalar_ : NULL;
|
|
|
|
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;
|
|
size_t num_points = num;
|
|
felem x_in, y_in, z_in, x_out, y_out, z_out;
|
|
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 (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 * sizeof(felem[17][3]));
|
|
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 = g_scalar;
|
|
} else {
|
|
/* the i^th point */
|
|
p = points[i];
|
|
p_scalar = scalars[i];
|
|
}
|
|
|
|
if (p_scalar != NULL && p != NULL) {
|
|
/* reduce g_scalar to 0 <= g_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);
|
|
}
|
|
}
|
|
|
|
if (g_scalar != NULL) {
|
|
memset(g_secret, 0, sizeof(g_secret));
|
|
/* reduce g_scalar to 0 <= g_scalar < 2^224 */
|
|
if (BN_num_bits(g_scalar) > 224 || BN_is_negative(g_scalar)) {
|
|
/* this is an unusual input, and we don't guarantee constant-timeness */
|
|
if (!BN_nnmod(tmp_scalar, g_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(g_scalar, tmp);
|
|
}
|
|
|
|
flip_endian(g_secret, tmp, num_bytes);
|
|
}
|
|
batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets,
|
|
num_points, g_scalar != NULL ? g_secret : NULL, mixed,
|
|
(const felem(*)[17][3])pre_comp);
|
|
|
|
/* 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);
|
|
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_GFp_simple_group_init,
|
|
ec_GFp_simple_group_finish,
|
|
ec_GFp_simple_group_copy,
|
|
ec_GFp_simple_group_set_curve,
|
|
ec_GFp_nistp224_point_get_affine_coordinates,
|
|
ec_GFp_nistp224_points_mul,
|
|
0 /* check_pub_key_order */,
|
|
ec_GFp_simple_field_mul,
|
|
ec_GFp_simple_field_sqr,
|
|
0 /* field_encode */,
|
|
0 /* field_decode */};
|
|
|
|
return &ret;
|
|
}
|
|
|
|
#endif /* 64_BIT && !WINDOWS && !SMALL */
|