| /* crypto/ec/ecp_nistputil.c */ |
| /* |
| * Written by Bodo Moeller for the OpenSSL project. |
| */ |
| /* Copyright 2011 Google Inc. |
| * |
| * Licensed under the Apache License, Version 2.0 (the "License"); |
| * |
| * you may not use this file except in compliance with the License. |
| * You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| |
| #include <openssl/opensslconf.h> |
| #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128 |
| |
| /* |
| * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c. |
| */ |
| |
| # include <stddef.h> |
| # include "ec_lcl.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(unsigned char *sign, |
| unsigned char *digit, unsigned char in) |
| { |
| unsigned char 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; |
| } |
| #else |
| static void *dummy = &dummy; |
| #endif |
