217 lines
9.5 KiB
C
217 lines
9.5 KiB
C
/*
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* Written by Bodo Moeller for the OpenSSL project.
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*/
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/* Copyright 2011 Google Inc.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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*
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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#include <openssl/opensslconf.h>
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#ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
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NON_EMPTY_TRANSLATION_UNIT
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#else
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/*
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* Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
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*/
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# include <stddef.h>
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# include "ec_lcl.h"
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/*
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* Convert an array of points into affine coordinates. (If the point at
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* infinity is found (Z = 0), it remains unchanged.) This function is
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* essentially an equivalent to EC_POINTs_make_affine(), but works with the
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* internal representation of points as used by ecp_nistp###.c rather than
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* with (BIGNUM-based) EC_POINT data structures. point_array is the
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* input/output buffer ('num' points in projective form, i.e. three
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* coordinates each), based on an internal representation of field elements
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* of size 'felem_size'. tmp_felems needs to point to a temporary array of
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* 'num'+1 field elements for storage of intermediate values.
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*/
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void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
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size_t felem_size,
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void *tmp_felems,
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void (*felem_one) (void *out),
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int (*felem_is_zero) (const void
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*in),
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void (*felem_assign) (void *out,
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const void
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*in),
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void (*felem_square) (void *out,
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const void
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*in),
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void (*felem_mul) (void *out,
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const void
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*in1,
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const void
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*in2),
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void (*felem_inv) (void *out,
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const void
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*in),
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void (*felem_contract) (void
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*out,
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const
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void
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*in))
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{
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int i = 0;
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# define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
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# define X(I) (&((char *)point_array)[3*(I) * felem_size])
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# define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
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# define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
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if (!felem_is_zero(Z(0)))
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felem_assign(tmp_felem(0), Z(0));
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else
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felem_one(tmp_felem(0));
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for (i = 1; i < (int)num; i++) {
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if (!felem_is_zero(Z(i)))
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felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
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else
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felem_assign(tmp_felem(i), tmp_felem(i - 1));
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}
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/*
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* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
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* zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
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*/
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felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
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for (i = num - 1; i >= 0; i--) {
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if (i > 0)
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/*
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* tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
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* is the inverse of the product of Z(0) .. Z(i)
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*/
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/* 1/Z(i) */
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felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
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else
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felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
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if (!felem_is_zero(Z(i))) {
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if (i > 0)
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/*
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* For next iteration, replace tmp_felem(i-1) by its inverse
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*/
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felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
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/*
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* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
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*/
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felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
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felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
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felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
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felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
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felem_contract(X(i), X(i));
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felem_contract(Y(i), Y(i));
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felem_one(Z(i));
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} else {
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if (i > 0)
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/*
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* For next iteration, replace tmp_felem(i-1) by its inverse
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*/
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felem_assign(tmp_felem(i - 1), tmp_felem(i));
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}
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}
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}
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/*-
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* This function looks at 5+1 scalar bits (5 current, 1 adjacent less
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* significant bit), and recodes them into a signed digit for use in fast point
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* multiplication: the use of signed rather than unsigned digits means that
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* fewer points need to be precomputed, given that point inversion is easy
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* (a precomputed point dP makes -dP available as well).
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*
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* BACKGROUND:
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*
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* Signed digits for multiplication were introduced by Booth ("A signed binary
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* multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
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* pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
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* Booth's original encoding did not generally improve the density of nonzero
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* digits over the binary representation, and was merely meant to simplify the
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* handling of signed factors given in two's complement; but it has since been
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* shown to be the basis of various signed-digit representations that do have
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* further advantages, including the wNAF, using the following general approach:
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*
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* (1) Given a binary representation
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*
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* b_k ... b_2 b_1 b_0,
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*
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* of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
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* by using bit-wise subtraction as follows:
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*
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* b_k b_(k-1) ... b_2 b_1 b_0
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* - b_k ... b_3 b_2 b_1 b_0
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* -------------------------------------
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* s_k b_(k-1) ... s_3 s_2 s_1 s_0
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*
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* A left-shift followed by subtraction of the original value yields a new
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* representation of the same value, using signed bits s_i = b_(i+1) - b_i.
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* This representation from Booth's paper has since appeared in the
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* literature under a variety of different names including "reversed binary
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* form", "alternating greedy expansion", "mutual opposite form", and
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* "sign-alternating {+-1}-representation".
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*
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* An interesting property is that among the nonzero bits, values 1 and -1
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* strictly alternate.
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*
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* (2) Various window schemes can be applied to the Booth representation of
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* integers: for example, right-to-left sliding windows yield the wNAF
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* (a signed-digit encoding independently discovered by various researchers
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* in the 1990s), and left-to-right sliding windows yield a left-to-right
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* equivalent of the wNAF (independently discovered by various researchers
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* around 2004).
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*
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* To prevent leaking information through side channels in point multiplication,
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* we need to recode the given integer into a regular pattern: sliding windows
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* as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
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* decades older: we'll be using the so-called "modified Booth encoding" due to
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* MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
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* (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
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* signed bits into a signed digit:
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*
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* s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
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*
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* The sign-alternating property implies that the resulting digit values are
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* integers from -16 to 16.
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*
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* Of course, we don't actually need to compute the signed digits s_i as an
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* intermediate step (that's just a nice way to see how this scheme relates
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* to the wNAF): a direct computation obtains the recoded digit from the
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* six bits b_(4j + 4) ... b_(4j - 1).
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*
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* This function takes those five bits as an integer (0 .. 63), writing the
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* recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
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* value, in the range 0 .. 8). Note that this integer essentially provides the
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* input bits "shifted to the left" by one position: for example, the input to
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* compute the least significant recoded digit, given that there's no bit b_-1,
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* has to be b_4 b_3 b_2 b_1 b_0 0.
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*
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*/
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void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
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unsigned char *digit, unsigned char in)
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{
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unsigned char s, d;
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s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
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* 6-bit value */
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d = (1 << 6) - in - 1;
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d = (d & s) | (in & ~s);
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d = (d >> 1) + (d & 1);
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*sign = s & 1;
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*digit = d;
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}
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#endif
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