1238384Sjkim/* crypto/ec/ecp_nistputil.c */ 2238384Sjkim/* 3238384Sjkim * Written by Bodo Moeller for the OpenSSL project. 4238384Sjkim */ 5238384Sjkim/* Copyright 2011 Google Inc. 6238384Sjkim * 7238384Sjkim * Licensed under the Apache License, Version 2.0 (the "License"); 8238384Sjkim * 9238384Sjkim * you may not use this file except in compliance with the License. 10238384Sjkim * You may obtain a copy of the License at 11238384Sjkim * 12238384Sjkim * http://www.apache.org/licenses/LICENSE-2.0 13238384Sjkim * 14238384Sjkim * Unless required by applicable law or agreed to in writing, software 15238384Sjkim * distributed under the License is distributed on an "AS IS" BASIS, 16238384Sjkim * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 17238384Sjkim * See the License for the specific language governing permissions and 18238384Sjkim * limitations under the License. 19238384Sjkim */ 20238384Sjkim 21238384Sjkim#include <openssl/opensslconf.h> 22238384Sjkim#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128 23238384Sjkim 24238384Sjkim/* 25238384Sjkim * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c. 26238384Sjkim */ 27238384Sjkim 28280304Sjkim# include <stddef.h> 29280304Sjkim# include "ec_lcl.h" 30238384Sjkim 31280304Sjkim/* 32280304Sjkim * Convert an array of points into affine coordinates. (If the point at 33280304Sjkim * infinity is found (Z = 0), it remains unchanged.) This function is 34280304Sjkim * essentially an equivalent to EC_POINTs_make_affine(), but works with the 35280304Sjkim * internal representation of points as used by ecp_nistp###.c rather than 36280304Sjkim * with (BIGNUM-based) EC_POINT data structures. point_array is the 37280304Sjkim * input/output buffer ('num' points in projective form, i.e. three 38280304Sjkim * coordinates each), based on an internal representation of field elements 39280304Sjkim * of size 'felem_size'. tmp_felems needs to point to a temporary array of 40280304Sjkim * 'num'+1 field elements for storage of intermediate values. 41238384Sjkim */ 42238384Sjkimvoid ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array, 43280304Sjkim size_t felem_size, 44280304Sjkim void *tmp_felems, 45280304Sjkim void (*felem_one) (void *out), 46280304Sjkim int (*felem_is_zero) (const void 47280304Sjkim *in), 48280304Sjkim void (*felem_assign) (void *out, 49280304Sjkim const void 50280304Sjkim *in), 51280304Sjkim void (*felem_square) (void *out, 52280304Sjkim const void 53280304Sjkim *in), 54280304Sjkim void (*felem_mul) (void *out, 55280304Sjkim const void 56280304Sjkim *in1, 57280304Sjkim const void 58280304Sjkim *in2), 59280304Sjkim void (*felem_inv) (void *out, 60280304Sjkim const void 61280304Sjkim *in), 62280304Sjkim void (*felem_contract) (void 63280304Sjkim *out, 64280304Sjkim const 65280304Sjkim void 66280304Sjkim *in)) 67280304Sjkim{ 68280304Sjkim int i = 0; 69238384Sjkim 70280304Sjkim# define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size]) 71280304Sjkim# define X(I) (&((char *)point_array)[3*(I) * felem_size]) 72280304Sjkim# define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size]) 73280304Sjkim# define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size]) 74238384Sjkim 75280304Sjkim if (!felem_is_zero(Z(0))) 76280304Sjkim felem_assign(tmp_felem(0), Z(0)); 77280304Sjkim else 78280304Sjkim felem_one(tmp_felem(0)); 79280304Sjkim for (i = 1; i < (int)num; i++) { 80280304Sjkim if (!felem_is_zero(Z(i))) 81280304Sjkim felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i)); 82280304Sjkim else 83280304Sjkim felem_assign(tmp_felem(i), tmp_felem(i - 1)); 84280304Sjkim } 85280304Sjkim /* 86280304Sjkim * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any 87280304Sjkim * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1 88280304Sjkim */ 89238384Sjkim 90280304Sjkim felem_inv(tmp_felem(num - 1), tmp_felem(num - 1)); 91280304Sjkim for (i = num - 1; i >= 0; i--) { 92280304Sjkim if (i > 0) 93280304Sjkim /* 94280304Sjkim * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i) 95280304Sjkim * is the inverse of the product of Z(0) .. Z(i) 96280304Sjkim */ 97280304Sjkim /* 1/Z(i) */ 98280304Sjkim felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i)); 99280304Sjkim else 100280304Sjkim felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */ 101238384Sjkim 102280304Sjkim if (!felem_is_zero(Z(i))) { 103280304Sjkim if (i > 0) 104280304Sjkim /* 105280304Sjkim * For next iteration, replace tmp_felem(i-1) by its inverse 106280304Sjkim */ 107280304Sjkim felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i)); 108238384Sjkim 109280304Sjkim /* 110280304Sjkim * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1) 111280304Sjkim */ 112280304Sjkim felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */ 113280304Sjkim felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */ 114280304Sjkim felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */ 115280304Sjkim felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */ 116280304Sjkim felem_contract(X(i), X(i)); 117280304Sjkim felem_contract(Y(i), Y(i)); 118280304Sjkim felem_one(Z(i)); 119280304Sjkim } else { 120280304Sjkim if (i > 0) 121280304Sjkim /* 122280304Sjkim * For next iteration, replace tmp_felem(i-1) by its inverse 123280304Sjkim */ 124280304Sjkim felem_assign(tmp_felem(i - 1), tmp_felem(i)); 125280304Sjkim } 126280304Sjkim } 127280304Sjkim} 128238384Sjkim 129280304Sjkim/*- 130238384Sjkim * This function looks at 5+1 scalar bits (5 current, 1 adjacent less 131238384Sjkim * significant bit), and recodes them into a signed digit for use in fast point 132238384Sjkim * multiplication: the use of signed rather than unsigned digits means that 133238384Sjkim * fewer points need to be precomputed, given that point inversion is easy 134238384Sjkim * (a precomputed point dP makes -dP available as well). 135238384Sjkim * 136238384Sjkim * BACKGROUND: 137238384Sjkim * 138238384Sjkim * Signed digits for multiplication were introduced by Booth ("A signed binary 139238384Sjkim * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, 140238384Sjkim * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. 141238384Sjkim * Booth's original encoding did not generally improve the density of nonzero 142238384Sjkim * digits over the binary representation, and was merely meant to simplify the 143238384Sjkim * handling of signed factors given in two's complement; but it has since been 144238384Sjkim * shown to be the basis of various signed-digit representations that do have 145238384Sjkim * further advantages, including the wNAF, using the following general approach: 146238384Sjkim * 147238384Sjkim * (1) Given a binary representation 148238384Sjkim * 149238384Sjkim * b_k ... b_2 b_1 b_0, 150238384Sjkim * 151238384Sjkim * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 152238384Sjkim * by using bit-wise subtraction as follows: 153238384Sjkim * 154238384Sjkim * b_k b_(k-1) ... b_2 b_1 b_0 155238384Sjkim * - b_k ... b_3 b_2 b_1 b_0 156238384Sjkim * ------------------------------------- 157238384Sjkim * s_k b_(k-1) ... s_3 s_2 s_1 s_0 158238384Sjkim * 159238384Sjkim * A left-shift followed by subtraction of the original value yields a new 160238384Sjkim * representation of the same value, using signed bits s_i = b_(i+1) - b_i. 161238384Sjkim * This representation from Booth's paper has since appeared in the 162238384Sjkim * literature under a variety of different names including "reversed binary 163238384Sjkim * form", "alternating greedy expansion", "mutual opposite form", and 164238384Sjkim * "sign-alternating {+-1}-representation". 165238384Sjkim * 166238384Sjkim * An interesting property is that among the nonzero bits, values 1 and -1 167238384Sjkim * strictly alternate. 168238384Sjkim * 169238384Sjkim * (2) Various window schemes can be applied to the Booth representation of 170238384Sjkim * integers: for example, right-to-left sliding windows yield the wNAF 171238384Sjkim * (a signed-digit encoding independently discovered by various researchers 172238384Sjkim * in the 1990s), and left-to-right sliding windows yield a left-to-right 173238384Sjkim * equivalent of the wNAF (independently discovered by various researchers 174238384Sjkim * around 2004). 175238384Sjkim * 176238384Sjkim * To prevent leaking information through side channels in point multiplication, 177238384Sjkim * we need to recode the given integer into a regular pattern: sliding windows 178238384Sjkim * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few 179238384Sjkim * decades older: we'll be using the so-called "modified Booth encoding" due to 180238384Sjkim * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 181238384Sjkim * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five 182238384Sjkim * signed bits into a signed digit: 183238384Sjkim * 184238384Sjkim * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) 185238384Sjkim * 186238384Sjkim * The sign-alternating property implies that the resulting digit values are 187238384Sjkim * integers from -16 to 16. 188238384Sjkim * 189238384Sjkim * Of course, we don't actually need to compute the signed digits s_i as an 190238384Sjkim * intermediate step (that's just a nice way to see how this scheme relates 191238384Sjkim * to the wNAF): a direct computation obtains the recoded digit from the 192238384Sjkim * six bits b_(4j + 4) ... b_(4j - 1). 193238384Sjkim * 194238384Sjkim * This function takes those five bits as an integer (0 .. 63), writing the 195238384Sjkim * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute 196238384Sjkim * value, in the range 0 .. 8). Note that this integer essentially provides the 197238384Sjkim * input bits "shifted to the left" by one position: for example, the input to 198238384Sjkim * compute the least significant recoded digit, given that there's no bit b_-1, 199238384Sjkim * has to be b_4 b_3 b_2 b_1 b_0 0. 200238384Sjkim * 201238384Sjkim */ 202280304Sjkimvoid ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, 203280304Sjkim unsigned char *digit, unsigned char in) 204280304Sjkim{ 205280304Sjkim unsigned char s, d; 206238384Sjkim 207280304Sjkim s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 208280304Sjkim * 6-bit value */ 209280304Sjkim d = (1 << 6) - in - 1; 210280304Sjkim d = (d & s) | (in & ~s); 211280304Sjkim d = (d >> 1) + (d & 1); 212238384Sjkim 213280304Sjkim *sign = s & 1; 214280304Sjkim *digit = d; 215280304Sjkim} 216238384Sjkim#else 217280304Sjkimstatic void *dummy = &dummy; 218238384Sjkim#endif 219