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 /* 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 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) #include #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 */
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