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Diffstat (limited to 'src/crypto/ec/util-64.c')
| -rw-r--r-- | src/crypto/ec/util-64.c | 183 |
1 files changed, 183 insertions, 0 deletions
diff --git a/src/crypto/ec/util-64.c b/src/crypto/ec/util-64.c new file mode 100644 index 0000000..171b063 --- /dev/null +++ b/src/crypto/ec/util-64.c @@ -0,0 +1,183 @@ +/* Copyright (c) 2015, Google Inc. + * + * Permission to use, copy, modify, and/or distribute this software for any + * purpose with or without fee is hereby granted, provided that the above + * copyright notice and this permission notice appear in all copies. + * + * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES + * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF + * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY + * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES + * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION + * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN + * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ + +#include <openssl/base.h> + + +#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) + +#include <openssl/ec.h> + +#include "internal.h" + +/* Convert an array of points into affine coordinates. (If the point at + * infinity is found (Z = 0), it remains unchanged.) This function is + * essentially an equivalent to EC_POINTs_make_affine(), but works with the + * internal representation of points as used by ecp_nistp###.c rather than + * with (BIGNUM-based) EC_POINT data structures. point_array is the + * input/output buffer ('num' points in projective form, i.e. three + * coordinates each), based on an internal representation of field elements + * of size 'felem_size'. tmp_felems needs to point to a temporary array of + * 'num'+1 field elements for storage of intermediate values. */ +void ec_GFp_nistp_points_make_affine_internal( + size_t num, void *point_array, size_t felem_size, void *tmp_felems, + void (*felem_one)(void *out), int (*felem_is_zero)(const void *in), + void (*felem_assign)(void *out, const void *in), + void (*felem_square)(void *out, const void *in), + void (*felem_mul)(void *out, const void *in1, const void *in2), + void (*felem_inv)(void *out, const void *in), + void (*felem_contract)(void *out, const void *in)) { + int i = 0; + +#define tmp_felem(I) (&((char *)tmp_felems)[(I)*felem_size]) +#define X(I) (&((char *)point_array)[3 * (I)*felem_size]) +#define Y(I) (&((char *)point_array)[(3 * (I) + 1) * felem_size]) +#define Z(I) (&((char *)point_array)[(3 * (I) + 2) * felem_size]) + + if (!felem_is_zero(Z(0))) { + felem_assign(tmp_felem(0), Z(0)); + } else { + felem_one(tmp_felem(0)); + } + + for (i = 1; i < (int)num; i++) { + if (!felem_is_zero(Z(i))) { + felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i)); + } else { + felem_assign(tmp_felem(i), tmp_felem(i - 1)); + } + } + /* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any + * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1. */ + + felem_inv(tmp_felem(num - 1), tmp_felem(num - 1)); + for (i = num - 1; i >= 0; i--) { + if (i > 0) { + /* tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i) + * is the inverse of the product of Z(0) .. Z(i). */ + /* 1/Z(i) */ + felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i)); + } else { + felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */ + } + + if (!felem_is_zero(Z(i))) { + if (i > 0) { + /* For next iteration, replace tmp_felem(i-1) by its inverse. */ + felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i)); + } + + /* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1). */ + felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */ + felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */ + felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */ + felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */ + felem_contract(X(i), X(i)); + felem_contract(Y(i), Y(i)); + felem_one(Z(i)); + } else { + if (i > 0) { + /* For next iteration, replace tmp_felem(i-1) by its inverse. */ + felem_assign(tmp_felem(i - 1), tmp_felem(i)); + } + } + } +} + +/* This function looks at 5+1 scalar bits (5 current, 1 adjacent less + * significant bit), and recodes them into a signed digit for use in fast point + * multiplication: the use of signed rather than unsigned digits means that + * fewer points need to be precomputed, given that point inversion is easy (a + * precomputed point dP makes -dP available as well). + * + * BACKGROUND: + * + * Signed digits for multiplication were introduced by Booth ("A signed binary + * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, + * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. + * Booth's original encoding did not generally improve the density of nonzero + * digits over the binary representation, and was merely meant to simplify the + * handling of signed factors given in two's complement; but it has since been + * shown to be the basis of various signed-digit representations that do have + * further advantages, including the wNAF, using the following general + * approach: + * + * (1) Given a binary representation + * + * b_k ... b_2 b_1 b_0, + * + * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 + * by using bit-wise subtraction as follows: + * + * b_k b_(k-1) ... b_2 b_1 b_0 + * - b_k ... b_3 b_2 b_1 b_0 + * ------------------------------------- + * s_k b_(k-1) ... s_3 s_2 s_1 s_0 + * + * A left-shift followed by subtraction of the original value yields a new + * representation of the same value, using signed bits s_i = b_(i+1) - b_i. + * This representation from Booth's paper has since appeared in the + * literature under a variety of different names including "reversed binary + * form", "alternating greedy expansion", "mutual opposite form", and + * "sign-alternating {+-1}-representation". + * + * An interesting property is that among the nonzero bits, values 1 and -1 + * strictly alternate. + * + * (2) Various window schemes can be applied to the Booth representation of + * integers: for example, right-to-left sliding windows yield the wNAF + * (a signed-digit encoding independently discovered by various researchers + * in the 1990s), and left-to-right sliding windows yield a left-to-right + * equivalent of the wNAF (independently discovered by various researchers + * around 2004). + * + * To prevent leaking information through side channels in point multiplication, + * we need to recode the given integer into a regular pattern: sliding windows + * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few + * decades older: we'll be using the so-called "modified Booth encoding" due to + * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 + * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five + * signed bits into a signed digit: + * + * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) + * + * The sign-alternating property implies that the resulting digit values are + * integers from -16 to 16. + * + * Of course, we don't actually need to compute the signed digits s_i as an + * intermediate step (that's just a nice way to see how this scheme relates + * to the wNAF): a direct computation obtains the recoded digit from the + * six bits b_(4j + 4) ... b_(4j - 1). + * + * This function takes those five bits as an integer (0 .. 63), writing the + * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute + * value, in the range 0 .. 8). Note that this integer essentially provides the + * input bits "shifted to the left" by one position: for example, the input to + * compute the least significant recoded digit, given that there's no bit b_-1, + * has to be b_4 b_3 b_2 b_1 b_0 0. */ +void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, + uint8_t in) { + uint8_t s, d; + + s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as + * 6-bit value */ + d = (1 << 6) - in - 1; + d = (d & s) | (in & ~s); + d = (d >> 1) + (d & 1); + + *sign = s & 1; + *digit = d; +} + +#endif /* 64_BIT && !WINDOWS */ |