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Diffstat (limited to 'crypto/ec/ecp_nistp224.c')
| -rw-r--r-- | crypto/ec/ecp_nistp224.c | 1471 |
1 files changed, 1471 insertions, 0 deletions
diff --git a/crypto/ec/ecp_nistp224.c b/crypto/ec/ecp_nistp224.c new file mode 100644 index 0000000000..72ce299725 --- /dev/null +++ b/crypto/ec/ecp_nistp224.c @@ -0,0 +1,1471 @@ +/* crypto/ec/ecp_nistp224.c */ +/* + * Written by Emilia Kasper (Google) for the OpenSSL project. + */ +/* ==================================================================== + * Copyright (c) 2000-2010 The OpenSSL Project. All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * + * 1. Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in + * the documentation and/or other materials provided with the + * distribution. + * + * 3. All advertising materials mentioning features or use of this + * software must display the following acknowledgment: + * "This product includes software developed by the OpenSSL Project + * for use in the OpenSSL Toolkit. (http://www.OpenSSL.org/)" + * + * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to + * endorse or promote products derived from this software without + * prior written permission. For written permission, please contact + * licensing@OpenSSL.org. + * + * 5. Products derived from this software may not be called "OpenSSL" + * nor may "OpenSSL" appear in their names without prior written + * permission of the OpenSSL Project. + * + * 6. Redistributions of any form whatsoever must retain the following + * acknowledgment: + * "This product includes software developed by the OpenSSL Project + * for use in the OpenSSL Toolkit (http://www.OpenSSL.org/)" + * + * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY + * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE + * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR + * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR + * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, + * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; + * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) + * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, + * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) + * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED + * OF THE POSSIBILITY OF SUCH DAMAGE. + * ==================================================================== + * + * This product includes cryptographic software written by Eric Young + * (eay@cryptsoft.com). This product includes software written by Tim + * Hudson (tjh@cryptsoft.com). + * + */ + +/* + * 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 + */ +#ifdef EC_NISTP224_64_GCC_128 +#include <stdint.h> +#include <string.h> +#include <openssl/err.h> +#include "ec_lcl.h" + +typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */ + +typedef uint8_t u8; + +static const u8 nistp224_curve_params[5*28] = { + 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 +}; + +/******************************************************************************/ +/* INTERNAL REPRESENTATION OF FIELD ELEMENTS + * + * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 + * where each slice a_i is a 64-bit word, i.e., a field element is an fslice + * array a with 4 elements, where a[i] = a_i. + * Outputs from multiplications are represented as unreduced polynomials + * 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 + * where each b_i is a 128-bit word. We ensure that inputs to each field + * 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 a_i < 2^57. We only reduce to the unique minimal + * representation at the end of the computation. + * + */ + +typedef uint64_t fslice; + +/* Field element size (and group order size), in bytes: 28*8 = 224 */ +static const unsigned fElemSize = 28; + +/* Precomputed multiples of the standard generator + * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for + * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity, + * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc. + * Points are given in Jacobian projective coordinates: words 0-3 represent the + * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the + * Y-coordinate and words 8-11 represent the Z-coordinate. */ +static const fslice gmul[16][3][4] = { + {{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, + {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, + {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, + {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, + {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, + {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, + {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, + {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, + {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, + {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, + {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, + {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, + {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, + {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, + {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, + {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, + {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, + {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, + {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, + {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}} +}; + +/* Precomputation for the group generator. */ +typedef struct { + fslice g_pre_comp[16][3][4]; + int references; +} NISTP224_PRE_COMP; + +const EC_METHOD *EC_GFp_nistp224_method(void) + { + static const EC_METHOD ret = { + NID_X9_62_prime_field, + ec_GFp_nistp224_group_init, + ec_GFp_simple_group_finish, + ec_GFp_simple_group_clear_finish, + ec_GFp_nist_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, + ec_GFp_simple_set_compressed_coordinates, + ec_GFp_simple_point2oct, + ec_GFp_simple_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, + ec_GFp_nistp224_precompute_mult, + ec_GFp_nistp224_have_precompute_mult, + ec_GFp_nist_field_mul, + ec_GFp_nist_field_sqr, + 0 /* field_div */, + 0 /* field_encode */, + 0 /* field_decode */, + 0 /* field_set_to_one */ }; + + return &ret; + } + +/* Helper functions to convert field elements to/from internal representation */ +static void bin28_to_felem(fslice out[4], 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+21))) & 0x00ffffffffffffff; + } + +static void felem_to_bin28(u8 out[28], const fslice in[4]) + { + 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(fslice out[4], const BIGNUM *bn) + { + u8 b_in[fElemSize]; + u8 b_out[fElemSize]; + /* BN_bn2bin eats leading zeroes */ + memset(b_out, 0, fElemSize); + unsigned num_bytes = BN_num_bytes(bn); + if (num_bytes > fElemSize) + { + ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); + return 0; + } + if (BN_is_negative(bn)) + { + ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); + return 0; + } + 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 fslice in[4]) + { + u8 b_in[fElemSize], b_out[fElemSize]; + felem_to_bin28(b_in, in); + flip_endian(b_out, b_in, fElemSize); + return BN_bin2bn(b_out, fElemSize, out); + } + +/******************************************************************************/ +/* FIELD OPERATIONS + * + * 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. + * + */ + +/* Sum two field elements: out += in */ +static void felem_sum64(fslice out[4], const fslice in[4]) + { + out[0] += in[0]; + out[1] += in[1]; + out[2] += in[2]; + out[3] += in[3]; + } + +/* Subtract field elements: out -= in */ +/* Assumes in[i] < 2^57 */ +static void felem_diff64(fslice out[4], const fslice in[4]) + { + static const uint64_t two58p2 = (1l << 58) + (1l << 2); + static const uint64_t two58m2 = (1l << 58) - (1l << 2); + static const uint64_t two58m42m2 = (1l << 58) - (1l << 42) - (1l << 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: out128 -= in128 */ +/* Assumes in[i] < 2^119 */ +static void felem_diff128(uint128_t out[7], const uint128_t in[4]) + { + static const uint128_t two120 = ((uint128_t) 1) << 120; + static const uint128_t two120m64 = (((uint128_t) 1) << 120) - + (((uint128_t) 1) << 64); + static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) - + (((uint128_t) 1) << 104) - (((uint128_t) 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(uint128_t out[7], const fslice in[4]) + { + static const uint128_t two64p8 = (((uint128_t) 1) << 64) + + (((uint128_t) 1) << 8); + static const uint128_t two64m8 = (((uint128_t) 1) << 64) - + (((uint128_t) 1) << 8); + static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) - + (((uint128_t) 1) << 48) - (((uint128_t) 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: out64 = out64 * scalar + * The scalars we actually use are small, so results fit without overflow */ +static void felem_scalar64(fslice out[4], const fslice scalar) + { + out[0] *= scalar; + out[1] *= scalar; + out[2] *= scalar; + out[3] *= scalar; + } + +/* Multiply an unreduced field element by a scalar: out128 = out128 * scalar + * The scalars we actually use are small, so results fit without overflow */ +static void felem_scalar128(uint128_t out[7], const uint128_t 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(uint128_t out[7], const fslice in[4]) + { + out[0] = ((uint128_t) in[0]) * in[0]; + out[1] = ((uint128_t) in[0]) * in[1] * 2; + out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1]; + out[3] = ((uint128_t) in[0]) * in[3] * 2 + + ((uint128_t) in[1]) * in[2] * 2; + out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2]; + out[5] = ((uint128_t) in[2]) * in[3] * 2; + out[6] = ((uint128_t) in[3]) * in[3]; + } + +/* Multiply two field elements: out = in1 * in2 */ +static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4]) + { + out[0] = ((uint128_t) in1[0]) * in2[0]; + out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0]; + out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] + + ((uint128_t) in1[2]) * in2[0]; + out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] + + ((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0]; + out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] + + ((uint128_t) in1[3]) * in2[1]; + out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2]; + out[6] = ((uint128_t) in1[3]) * in2[3]; + } + +/* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126, + * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */ +static void felem_reduce(fslice out[4], const uint128_t in[7]) + { + static const uint128_t two127p15 = (((uint128_t) 1) << 127) + + (((uint128_t) 1) << 15); + static const uint128_t two127m71 = (((uint128_t) 1) << 127) - + (((uint128_t) 1) << 71); + static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) - + (((uint128_t) 1) << 71) - (((uint128_t) 1) << 55); + uint128_t 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]; + output[4] = 0; + + /* 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 */ + + /* Eliminate output[4] */ + output[2] += output[4] >> 16; + 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; + out[1] = output[1] & 0x00ffffffffffffff; + output[3] += output[2] >> 56; + out[2] = output[2] & 0x00ffffffffffffff; + + /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, + * out[3] < 2^57 (due to final carry) */ + out[3] = output[3]; + } + +/* Reduce to unique minimal representation */ +static void felem_contract(fslice out[4], const fslice in[4]) + { + static const int64_t two56 = (1l << 56); + /* 0 <= in < 2^225 */ + /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ + int64_t tmp[4], a; + tmp[0] = (int64_t) in[0] - (in[3] >> 56); + tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000); + tmp[2] = (int64_t) in[2]; + tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff; + + /* eliminate negative coefficients */ + a = tmp[0] >> 63; + tmp[0] += two56 & a; + tmp[1] -= 1 & a; + + a = tmp[1] >> 63; + tmp[1] += two56 & a; + tmp[2] -= 1 & a; + + a = tmp[2] >> 63; + tmp[2] += two56 & a; + tmp[3] -= 1 & a; + + a = tmp[3] >> 63; + tmp[3] += two56 & a; + tmp[0] += 1 & a; + tmp[1] -= (1 & a) << 40; + + /* carry 1 -> 2 -> 3 */ + tmp[2] += tmp[1] >> 56; + tmp[1] &= 0x00ffffffffffffff; + + tmp[3] += tmp[2] >> 56; + tmp[2] &= 0x00ffffffffffffff; + + /* 0 <= in < 2^224 + 2^96 - 1 */ + /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ + tmp[0] -= (tmp[3] >> 56); + tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000); + tmp[3] &= 0x00ffffffffffffff; + + /* eliminate negative coefficients */ + a = tmp[0] >> 63; + tmp[0] += two56 & a; + tmp[1] -= 1 & a; + + a = tmp[1] >> 63; + tmp[1] += two56 & a; + tmp[2] -= 1 & a; + + a = tmp[2] >> 63; + tmp[2] += two56 & a; + tmp[3] -= 1 & a; + + a = tmp[3] >> 63; + tmp[3] += two56 & a; + tmp[0] += 1 & a; + tmp[1] -= (1 & a) << 40; + + /* carry 1 -> 2 -> 3 */ + tmp[2] += tmp[1] >> 56; + tmp[1] &= 0x00ffffffffffffff; + + tmp[3] += tmp[2] >> 56; + tmp[2] &= 0x00ffffffffffffff; + + /* Now 0 <= in < 2^224 */ + + /* if in > 2^224 - 2^96, reduce */ + /* a = 0 iff in > 2^224 - 2^96, i.e., + * the high 128 bits are all 1 and the lower part is non-zero */ + a = (tmp[3] + 1) | (tmp[2] + 1) | + ((tmp[1] | 0x000000ffffffffff) + 1) | + ((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63)); + /* turn a into an all-one mask (if a = 0) or an all-zero mask */ + a = ((a & 0x00ffffffffffffff) - 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; + + 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 fslice felem_is_zero(const fslice in[4]) + { + fslice zero = (in[0] | in[1] | in[2] | in[3]); + zero = (((int64_t)(zero) - 1) >> 63) & 1; + fslice two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000) + | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff); + two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1; + fslice two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000) + | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff); + two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1; + return (zero | two224m96p1 | two225m97p2); + } + +/* Invert a field element */ +/* Computation chain copied from djb's code */ +static void felem_inv(fslice out[4], const fslice in[4]) + { + fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4]; + uint128_t tmp[7]; + 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(fslice *out, const fslice *in, unsigned len, fslice icopy) + { + unsigned i; + /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ + const fslice copy = -icopy; + for (i = 0; i < len; ++i) + { + const fslice tmp = copy & (in[i] ^ out[i]); + out[i] ^= tmp; + } + } + +/* Copy in constant time: + * if isel == 1, copy in2 to out, + * if isel == 0, copy in1 to out. */ +static void select_conditional(fslice *out, const fslice *in1, const fslice *in2, + unsigned len, fslice isel) + { + unsigned i; + /* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */ + const fslice sel = -isel; + for (i = 0; i < len; ++i) + { + const fslice tmp = sel & (in1[i] ^ in2[i]); + out[i] = in1[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(fslice x_out[4], fslice y_out[4], fslice z_out[4], + const fslice x_in[4], const fslice y_in[4], const fslice z_in[4]) + { + uint128_t tmp[7], tmp2[7]; + fslice delta[4]; + fslice gamma[4]; + fslice beta[4]; + fslice alpha[4]; + fslice ftmp[4], ftmp2[4]; + memcpy(ftmp, x_in, 4 * sizeof(fslice)); + memcpy(ftmp2, x_in, 4 * sizeof(fslice)); + + /* 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_diff64(ftmp, delta); + /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ + felem_sum64(ftmp2, delta); + /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ + felem_scalar64(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 */ + memcpy(ftmp, beta, 4 * sizeof(fslice)); + felem_scalar64(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_sum64(delta, gamma); + /* delta[i] < 2^57 + 2^57 = 2^58 */ + memcpy(ftmp, y_in, 4 * sizeof(fslice)); + felem_sum64(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_scalar64(beta, 4); + /* beta[i] < 4 * 2^57 = 2^59 */ + felem_diff64(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 */ + felem_scalar128(tmp2, 8); + /* tmp2[i] < 8 * 2^116 = 2^119 */ + felem_diff128(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 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(fslice x3[4], fslice y3[4], fslice z3[4], + const fslice x1[4], const fslice y1[4], const fslice z1[4], + const fslice x2[4], const fslice y2[4], const fslice z2[4]) + { + fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4]; + uint128_t tmp[7], tmp2[7]; + fslice z1_is_zero, z2_is_zero, x_equal, y_equal; + + /* ftmp = z1^2 */ + felem_square(tmp, z1); + felem_reduce(ftmp, tmp); + + /* ftmp2 = z2^2 */ + felem_square(tmp, z2); + felem_reduce(ftmp2, tmp); + + /* ftmp3 = z1^3 */ + felem_mul(tmp, ftmp, z1); + felem_reduce(ftmp3, tmp); + + /* ftmp4 = z2^3 */ + felem_mul(tmp, ftmp2, z2); + felem_reduce(ftmp4, tmp); + + /* ftmp3 = z1^3*y2 */ + felem_mul(tmp, ftmp3, y2); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + + /* ftmp4 = z2^3*y1 */ + felem_mul(tmp2, ftmp4, y1); + felem_reduce(ftmp4, tmp2); + + /* 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); + + /* ftmp = z1^2*x2 */ + felem_mul(tmp, ftmp, x2); + /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ + + /* ftmp2 =z2^2*x1 */ + felem_mul(tmp2, ftmp2, x1); + felem_reduce(ftmp2, tmp2); + + /* ftmp = z1^2*x2 - z2^2*x1 */ + felem_diff128(tmp, tmp2); + /* 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 */ + felem_mul(tmp, z1, z2); + felem_reduce(ftmp5, tmp); + + /* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */ + felem_mul(tmp, ftmp, ftmp5); + felem_reduce(z3, tmp); + + /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ + memcpy(ftmp5, ftmp, 4 * sizeof(fslice)); + 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); + + /* ftmp4 = 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 */ + memcpy(ftmp5, ftmp2, 4 * sizeof(fslice)); + felem_scalar64(ftmp5, 2); + /* ftmp5[i] < 2 * 2^57 = 2^58 */ + + /* x3 = (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(x3, tmp2); + + /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */ + felem_diff64(ftmp2, x3); + /* 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 - x3) */ + felem_mul(tmp2, ftmp3, ftmp2); + /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ + + /* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) - + z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ + felem_diff128(tmp2, tmp); + /* tmp2[i] < 2^118 + 2^120 < 2^121 */ + felem_reduce(y3, tmp2); + + /* the result (x3, y3, z3) 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(x3, x2, 4, z1_is_zero); + copy_conditional(x3, x1, 4, z2_is_zero); + copy_conditional(y3, y2, 4, z1_is_zero); + copy_conditional(y3, y1, 4, z2_is_zero); + copy_conditional(z3, z2, 4, z1_is_zero); + copy_conditional(z3, z1, 4, z2_is_zero); + } + +/* Select a point from an array of 16 precomputed point multiples, + * in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point + * pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */ +static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4], + fslice out[12]) + { + fslice tmp[5][12]; + select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]); + select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]); + select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); + select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]); + select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]); + select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); + select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]); + select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]); + select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]); + select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); + select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]); + select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]); + select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); + select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]); + select_conditional(out, tmp[1], tmp[4], 12, bits[0]); + } + +/* Interleaved point multiplication using precomputed point multiples: + * The small point multiples 0*P, 1*P, ..., 15*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(fslice x_out[4], fslice y_out[4], fslice z_out[4], + const u8 scalars[][fElemSize], const unsigned num_points, const u8 *g_scalar, + const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4]) + { + unsigned i, j, num; + unsigned gen_mul = (g_scalar != NULL); + fslice nq[12], nqt[12], tmp[12]; + /* set nq to the point at infinity */ + memset(nq, 0, 12 * sizeof(fslice)); + fslice bits[4]; + u8 byte; + + /* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble, + * double 4 times, then add the precomputed point multiples. + * If we are also adding multiples of the generator, then interleave + * these additions with the last 56 doublings. */ + for (i = (num_points ? 28 : 7); i > 0; --i) + { + for (j = 0; j < 8; ++j) + { + /* double once */ |