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735 lines
27 KiB
C
735 lines
27 KiB
C
/* Copyright 2008, Google Inc.
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are
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* met:
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*
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* * Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* * Redistributions in binary form must reproduce the above
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* copyright notice, this list of conditions and the following disclaimer
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* in the documentation and/or other materials provided with the
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* distribution.
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* * Neither the name of Google Inc. nor the names of its
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* contributors may be used to endorse or promote products derived from
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* this software without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* curve25519-donna: Curve25519 elliptic curve, public key function
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*
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* http://code.google.com/p/curve25519-donna/
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*
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* Adam Langley <agl@imperialviolet.org>
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*
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* Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
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*
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* More information about curve25519 can be found here
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* http://cr.yp.to/ecdh.html
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*
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* djb's sample implementation of curve25519 is written in a special assembly
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* language called qhasm and uses the floating point registers.
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*
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* This is, almost, a clean room reimplementation from the curve25519 paper. It
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* uses many of the tricks described therein. Only the crecip function is taken
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* from the sample implementation.
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*/
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#include <string.h>
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#include <stdint.h>
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#ifdef _MSC_VER
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#define inline __inline
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#endif
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typedef uint8_t u8;
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typedef int32_t s32;
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typedef int64_t limb;
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/* Field element representation:
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*
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* Field elements are written as an array of signed, 64-bit limbs, least
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* significant first. The value of the field element is:
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* x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
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*
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* i.e. the limbs are 26, 25, 26, 25, ... bits wide.
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*/
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/* Sum two numbers: output += in */
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static void fsum(limb *output, const limb *in) {
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unsigned i;
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for (i = 0; i < 10; i += 2) {
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output[0+i] = (output[0+i] + in[0+i]);
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output[1+i] = (output[1+i] + in[1+i]);
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}
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}
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/* Find the difference of two numbers: output = in - output
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* (note the order of the arguments!)
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*/
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static void fdifference(limb *output, const limb *in) {
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unsigned i;
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for (i = 0; i < 10; ++i) {
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output[i] = (in[i] - output[i]);
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}
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}
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/* Multiply a number by a scalar: output = in * scalar */
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static void fscalar_product(limb *output, const limb *in, const limb scalar) {
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unsigned i;
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for (i = 0; i < 10; ++i) {
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output[i] = in[i] * scalar;
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}
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}
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/* Multiply two numbers: output = in2 * in
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*
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* output must be distinct to both inputs. The inputs are reduced coefficient
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* form, the output is not.
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*/
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static void fproduct(limb *output, const limb *in2, const limb *in) {
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output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]);
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output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) +
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((limb) ((s32) in2[1])) * ((s32) in[0]);
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output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) +
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((limb) ((s32) in2[0])) * ((s32) in[2]) +
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((limb) ((s32) in2[2])) * ((s32) in[0]);
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output[3] = ((limb) ((s32) in2[1])) * ((s32) in[2]) +
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((limb) ((s32) in2[2])) * ((s32) in[1]) +
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((limb) ((s32) in2[0])) * ((s32) in[3]) +
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((limb) ((s32) in2[3])) * ((s32) in[0]);
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output[4] = ((limb) ((s32) in2[2])) * ((s32) in[2]) +
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2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) +
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((limb) ((s32) in2[3])) * ((s32) in[1])) +
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((limb) ((s32) in2[0])) * ((s32) in[4]) +
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((limb) ((s32) in2[4])) * ((s32) in[0]);
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output[5] = ((limb) ((s32) in2[2])) * ((s32) in[3]) +
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((limb) ((s32) in2[3])) * ((s32) in[2]) +
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((limb) ((s32) in2[1])) * ((s32) in[4]) +
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((limb) ((s32) in2[4])) * ((s32) in[1]) +
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((limb) ((s32) in2[0])) * ((s32) in[5]) +
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((limb) ((s32) in2[5])) * ((s32) in[0]);
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output[6] = 2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) +
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((limb) ((s32) in2[1])) * ((s32) in[5]) +
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((limb) ((s32) in2[5])) * ((s32) in[1])) +
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((limb) ((s32) in2[2])) * ((s32) in[4]) +
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((limb) ((s32) in2[4])) * ((s32) in[2]) +
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((limb) ((s32) in2[0])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[0]);
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output[7] = ((limb) ((s32) in2[3])) * ((s32) in[4]) +
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((limb) ((s32) in2[4])) * ((s32) in[3]) +
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((limb) ((s32) in2[2])) * ((s32) in[5]) +
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((limb) ((s32) in2[5])) * ((s32) in[2]) +
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((limb) ((s32) in2[1])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[1]) +
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((limb) ((s32) in2[0])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[0]);
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output[8] = ((limb) ((s32) in2[4])) * ((s32) in[4]) +
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2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) +
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((limb) ((s32) in2[5])) * ((s32) in[3]) +
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((limb) ((s32) in2[1])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[1])) +
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((limb) ((s32) in2[2])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[2]) +
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((limb) ((s32) in2[0])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[0]);
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output[9] = ((limb) ((s32) in2[4])) * ((s32) in[5]) +
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((limb) ((s32) in2[5])) * ((s32) in[4]) +
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((limb) ((s32) in2[3])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[3]) +
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((limb) ((s32) in2[2])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[2]) +
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((limb) ((s32) in2[1])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[1]) +
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((limb) ((s32) in2[0])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[0]);
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output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) +
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((limb) ((s32) in2[3])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[3]) +
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((limb) ((s32) in2[1])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[1])) +
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((limb) ((s32) in2[4])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[4]) +
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((limb) ((s32) in2[2])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[2]);
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output[11] = ((limb) ((s32) in2[5])) * ((s32) in[6]) +
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((limb) ((s32) in2[6])) * ((s32) in[5]) +
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((limb) ((s32) in2[4])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[4]) +
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((limb) ((s32) in2[3])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[3]) +
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((limb) ((s32) in2[2])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[2]);
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output[12] = ((limb) ((s32) in2[6])) * ((s32) in[6]) +
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2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[5]) +
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((limb) ((s32) in2[3])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[3])) +
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((limb) ((s32) in2[4])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[4]);
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output[13] = ((limb) ((s32) in2[6])) * ((s32) in[7]) +
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((limb) ((s32) in2[7])) * ((s32) in[6]) +
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((limb) ((s32) in2[5])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[5]) +
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((limb) ((s32) in2[4])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[4]);
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output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) +
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((limb) ((s32) in2[5])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[5])) +
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((limb) ((s32) in2[6])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[6]);
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output[15] = ((limb) ((s32) in2[7])) * ((s32) in[8]) +
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((limb) ((s32) in2[8])) * ((s32) in[7]) +
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((limb) ((s32) in2[6])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[6]);
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output[16] = ((limb) ((s32) in2[8])) * ((s32) in[8]) +
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2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[7]));
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output[17] = ((limb) ((s32) in2[8])) * ((s32) in[9]) +
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((limb) ((s32) in2[9])) * ((s32) in[8]);
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output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]);
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}
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/* Reduce a long form to a short form by taking the input mod 2^255 - 19. */
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static void freduce_degree(limb *output) {
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/* Each of these shifts and adds ends up multiplying the value by 19. */
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output[8] += output[18] << 4;
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output[8] += output[18] << 1;
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output[8] += output[18];
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output[7] += output[17] << 4;
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output[7] += output[17] << 1;
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output[7] += output[17];
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output[6] += output[16] << 4;
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output[6] += output[16] << 1;
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output[6] += output[16];
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output[5] += output[15] << 4;
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output[5] += output[15] << 1;
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output[5] += output[15];
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output[4] += output[14] << 4;
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output[4] += output[14] << 1;
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output[4] += output[14];
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output[3] += output[13] << 4;
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output[3] += output[13] << 1;
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output[3] += output[13];
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output[2] += output[12] << 4;
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output[2] += output[12] << 1;
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output[2] += output[12];
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output[1] += output[11] << 4;
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output[1] += output[11] << 1;
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output[1] += output[11];
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output[0] += output[10] << 4;
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output[0] += output[10] << 1;
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output[0] += output[10];
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}
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#if (-1 & 3) != 3
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#error "This code only works on a two's complement system"
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#endif
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/* return v / 2^26, using only shifts and adds. */
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static inline limb
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div_by_2_26(const limb v)
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{
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/* High word of v; no shift needed*/
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const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
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/* Set to all 1s if v was negative; else set to 0s. */
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const int32_t sign = ((int32_t) highword) >> 31;
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/* Set to 0x3ffffff if v was negative; else set to 0. */
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const int32_t roundoff = ((uint32_t) sign) >> 6;
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/* Should return v / (1<<26) */
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return (v + roundoff) >> 26;
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}
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/* return v / (2^25), using only shifts and adds. */
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static inline limb
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div_by_2_25(const limb v)
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{
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/* High word of v; no shift needed*/
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const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
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/* Set to all 1s if v was negative; else set to 0s. */
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const int32_t sign = ((int32_t) highword) >> 31;
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/* Set to 0x1ffffff if v was negative; else set to 0. */
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const int32_t roundoff = ((uint32_t) sign) >> 7;
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/* Should return v / (1<<25) */
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return (v + roundoff) >> 25;
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}
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static inline s32
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div_s32_by_2_25(const s32 v)
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{
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const s32 roundoff = ((uint32_t)(v >> 31)) >> 7;
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return (v + roundoff) >> 25;
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}
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/* Reduce all coefficients of the short form input so that |x| < 2^26.
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*
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* On entry: |output[i]| < 2^62
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*/
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static void freduce_coefficients(limb *output) {
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unsigned i;
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output[10] = 0;
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for (i = 0; i < 10; i += 2) {
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limb over = div_by_2_26(output[i]);
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output[i] -= over << 26;
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output[i+1] += over;
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over = div_by_2_25(output[i+1]);
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output[i+1] -= over << 25;
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output[i+2] += over;
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}
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/* Now |output[10]| < 2 ^ 38 and all other coefficients are reduced. */
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output[0] += output[10] << 4;
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output[0] += output[10] << 1;
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output[0] += output[10];
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output[10] = 0;
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/* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19 * 2^38
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* So |over| will be no more than 77825 */
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{
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limb over = div_by_2_26(output[0]);
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output[0] -= over << 26;
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output[1] += over;
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}
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/* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 77825
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* So |over| will be no more than 1. */
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{
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/* output[1] fits in 32 bits, so we can use div_s32_by_2_25 here. */
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s32 over32 = div_s32_by_2_25((s32) output[1]);
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output[1] -= over32 << 25;
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output[2] += over32;
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}
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/* Finally, output[0,1,3..9] are reduced, and output[2] is "nearly reduced":
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* we have |output[2]| <= 2^26. This is good enough for all of our math,
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* but it will require an extra freduce_coefficients before fcontract. */
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}
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/* A helpful wrapper around fproduct: output = in * in2.
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*
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* output must be distinct to both inputs. The output is reduced degree and
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* reduced coefficient.
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*/
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static void
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fmul(limb *output, const limb *in, const limb *in2) {
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limb t[19];
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fproduct(t, in, in2);
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freduce_degree(t);
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freduce_coefficients(t);
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memcpy(output, t, sizeof(limb) * 10);
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}
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static void fsquare_inner(limb *output, const limb *in) {
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output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]);
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output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]);
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output[2] = 2 * (((limb) ((s32) in[1])) * ((s32) in[1]) +
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((limb) ((s32) in[0])) * ((s32) in[2]));
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output[3] = 2 * (((limb) ((s32) in[1])) * ((s32) in[2]) +
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((limb) ((s32) in[0])) * ((s32) in[3]));
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output[4] = ((limb) ((s32) in[2])) * ((s32) in[2]) +
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4 * ((limb) ((s32) in[1])) * ((s32) in[3]) +
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2 * ((limb) ((s32) in[0])) * ((s32) in[4]);
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output[5] = 2 * (((limb) ((s32) in[2])) * ((s32) in[3]) +
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((limb) ((s32) in[1])) * ((s32) in[4]) +
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((limb) ((s32) in[0])) * ((s32) in[5]));
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output[6] = 2 * (((limb) ((s32) in[3])) * ((s32) in[3]) +
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((limb) ((s32) in[2])) * ((s32) in[4]) +
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((limb) ((s32) in[0])) * ((s32) in[6]) +
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2 * ((limb) ((s32) in[1])) * ((s32) in[5]));
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output[7] = 2 * (((limb) ((s32) in[3])) * ((s32) in[4]) +
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((limb) ((s32) in[2])) * ((s32) in[5]) +
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((limb) ((s32) in[1])) * ((s32) in[6]) +
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((limb) ((s32) in[0])) * ((s32) in[7]));
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output[8] = ((limb) ((s32) in[4])) * ((s32) in[4]) +
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2 * (((limb) ((s32) in[2])) * ((s32) in[6]) +
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((limb) ((s32) in[0])) * ((s32) in[8]) +
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2 * (((limb) ((s32) in[1])) * ((s32) in[7]) +
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((limb) ((s32) in[3])) * ((s32) in[5])));
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output[9] = 2 * (((limb) ((s32) in[4])) * ((s32) in[5]) +
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((limb) ((s32) in[3])) * ((s32) in[6]) +
|
|
((limb) ((s32) in[2])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[1])) * ((s32) in[8]) +
|
|
((limb) ((s32) in[0])) * ((s32) in[9]));
|
|
output[10] = 2 * (((limb) ((s32) in[5])) * ((s32) in[5]) +
|
|
((limb) ((s32) in[4])) * ((s32) in[6]) +
|
|
((limb) ((s32) in[2])) * ((s32) in[8]) +
|
|
2 * (((limb) ((s32) in[3])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[1])) * ((s32) in[9])));
|
|
output[11] = 2 * (((limb) ((s32) in[5])) * ((s32) in[6]) +
|
|
((limb) ((s32) in[4])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[3])) * ((s32) in[8]) +
|
|
((limb) ((s32) in[2])) * ((s32) in[9]));
|
|
output[12] = ((limb) ((s32) in[6])) * ((s32) in[6]) +
|
|
2 * (((limb) ((s32) in[4])) * ((s32) in[8]) +
|
|
2 * (((limb) ((s32) in[5])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[3])) * ((s32) in[9])));
|
|
output[13] = 2 * (((limb) ((s32) in[6])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[5])) * ((s32) in[8]) +
|
|
((limb) ((s32) in[4])) * ((s32) in[9]));
|
|
output[14] = 2 * (((limb) ((s32) in[7])) * ((s32) in[7]) +
|
|
((limb) ((s32) in[6])) * ((s32) in[8]) +
|
|
2 * ((limb) ((s32) in[5])) * ((s32) in[9]));
|
|
output[15] = 2 * (((limb) ((s32) in[7])) * ((s32) in[8]) +
|
|
((limb) ((s32) in[6])) * ((s32) in[9]));
|
|
output[16] = ((limb) ((s32) in[8])) * ((s32) in[8]) +
|
|
4 * ((limb) ((s32) in[7])) * ((s32) in[9]);
|
|
output[17] = 2 * ((limb) ((s32) in[8])) * ((s32) in[9]);
|
|
output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]);
|
|
}
|
|
|
|
static void
|
|
fsquare(limb *output, const limb *in) {
|
|
limb t[19];
|
|
fsquare_inner(t, in);
|
|
freduce_degree(t);
|
|
freduce_coefficients(t);
|
|
memcpy(output, t, sizeof(limb) * 10);
|
|
}
|
|
|
|
/* Take a little-endian, 32-byte number and expand it into polynomial form */
|
|
static void
|
|
fexpand(limb *output, const u8 *input) {
|
|
#define F(n,start,shift,mask) \
|
|
output[n] = ((((limb) input[start + 0]) | \
|
|
((limb) input[start + 1]) << 8 | \
|
|
((limb) input[start + 2]) << 16 | \
|
|
((limb) input[start + 3]) << 24) >> shift) & mask;
|
|
F(0, 0, 0, 0x3ffffff);
|
|
F(1, 3, 2, 0x1ffffff);
|
|
F(2, 6, 3, 0x3ffffff);
|
|
F(3, 9, 5, 0x1ffffff);
|
|
F(4, 12, 6, 0x3ffffff);
|
|
F(5, 16, 0, 0x1ffffff);
|
|
F(6, 19, 1, 0x3ffffff);
|
|
F(7, 22, 3, 0x1ffffff);
|
|
F(8, 25, 4, 0x3ffffff);
|
|
F(9, 28, 6, 0x3ffffff);
|
|
#undef F
|
|
}
|
|
|
|
#if (-32 >> 1) != -16
|
|
#error "This code only works when >> does sign-extension on negative numbers"
|
|
#endif
|
|
|
|
/* Take a fully reduced polynomial form number and contract it into a
|
|
* little-endian, 32-byte array
|
|
*/
|
|
static void
|
|
fcontract(u8 *output, limb *input) {
|
|
int i;
|
|
int j;
|
|
|
|
for (j = 0; j < 2; ++j) {
|
|
for (i = 0; i < 9; ++i) {
|
|
if ((i & 1) == 1) {
|
|
/* This calculation is a time-invariant way to make input[i] positive
|
|
by borrowing from the next-larger limb.
|
|
*/
|
|
const s32 mask = (s32)(input[i]) >> 31;
|
|
const s32 carry = -(((s32)(input[i]) & mask) >> 25);
|
|
input[i] = (s32)(input[i]) + (carry << 25);
|
|
input[i+1] = (s32)(input[i+1]) - carry;
|
|
} else {
|
|
const s32 mask = (s32)(input[i]) >> 31;
|
|
const s32 carry = -(((s32)(input[i]) & mask) >> 26);
|
|
input[i] = (s32)(input[i]) + (carry << 26);
|
|
input[i+1] = (s32)(input[i+1]) - carry;
|
|
}
|
|
}
|
|
{
|
|
const s32 mask = (s32)(input[9]) >> 31;
|
|
const s32 carry = -(((s32)(input[9]) & mask) >> 25);
|
|
input[9] = (s32)(input[9]) + (carry << 25);
|
|
input[0] = (s32)(input[0]) - (carry * 19);
|
|
}
|
|
}
|
|
|
|
/* The first borrow-propagation pass above ended with every limb
|
|
except (possibly) input[0] non-negative.
|
|
|
|
Since each input limb except input[0] is decreased by at most 1
|
|
by a borrow-propagation pass, the second borrow-propagation pass
|
|
could only have wrapped around to decrease input[0] again if the
|
|
first pass left input[0] negative *and* input[1] through input[9]
|
|
were all zero. In that case, input[1] is now 2^25 - 1, and this
|
|
last borrow-propagation step will leave input[1] non-negative.
|
|
*/
|
|
{
|
|
const s32 mask = (s32)(input[0]) >> 31;
|
|
const s32 carry = -(((s32)(input[0]) & mask) >> 26);
|
|
input[0] = (s32)(input[0]) + (carry << 26);
|
|
input[1] = (s32)(input[1]) - carry;
|
|
}
|
|
|
|
/* Both passes through the above loop, plus the last 0-to-1 step, are
|
|
necessary: if input[9] is -1 and input[0] through input[8] are 0,
|
|
negative values will remain in the array until the end.
|
|
*/
|
|
|
|
input[1] <<= 2;
|
|
input[2] <<= 3;
|
|
input[3] <<= 5;
|
|
input[4] <<= 6;
|
|
input[6] <<= 1;
|
|
input[7] <<= 3;
|
|
input[8] <<= 4;
|
|
input[9] <<= 6;
|
|
#define F(i, s) \
|
|
output[s+0] |= input[i] & 0xff; \
|
|
output[s+1] = (input[i] >> 8) & 0xff; \
|
|
output[s+2] = (input[i] >> 16) & 0xff; \
|
|
output[s+3] = (input[i] >> 24) & 0xff;
|
|
output[0] = 0;
|
|
output[16] = 0;
|
|
F(0,0);
|
|
F(1,3);
|
|
F(2,6);
|
|
F(3,9);
|
|
F(4,12);
|
|
F(5,16);
|
|
F(6,19);
|
|
F(7,22);
|
|
F(8,25);
|
|
F(9,28);
|
|
#undef F
|
|
}
|
|
|
|
/* Input: Q, Q', Q-Q'
|
|
* Output: 2Q, Q+Q'
|
|
*
|
|
* x2 z3: long form
|
|
* x3 z3: long form
|
|
* x z: short form, destroyed
|
|
* xprime zprime: short form, destroyed
|
|
* qmqp: short form, preserved
|
|
*/
|
|
static void fmonty(limb *x2, limb *z2, /* output 2Q */
|
|
limb *x3, limb *z3, /* output Q + Q' */
|
|
limb *x, limb *z, /* input Q */
|
|
limb *xprime, limb *zprime, /* input Q' */
|
|
const limb *qmqp /* input Q - Q' */) {
|
|
limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
|
|
zzprime[19], zzzprime[19], xxxprime[19];
|
|
|
|
memcpy(origx, x, 10 * sizeof(limb));
|
|
fsum(x, z);
|
|
fdifference(z, origx); // does x - z
|
|
|
|
memcpy(origxprime, xprime, sizeof(limb) * 10);
|
|
fsum(xprime, zprime);
|
|
fdifference(zprime, origxprime);
|
|
fproduct(xxprime, xprime, z);
|
|
fproduct(zzprime, x, zprime);
|
|
freduce_degree(xxprime);
|
|
freduce_coefficients(xxprime);
|
|
freduce_degree(zzprime);
|
|
freduce_coefficients(zzprime);
|
|
memcpy(origxprime, xxprime, sizeof(limb) * 10);
|
|
fsum(xxprime, zzprime);
|
|
fdifference(zzprime, origxprime);
|
|
fsquare(xxxprime, xxprime);
|
|
fsquare(zzzprime, zzprime);
|
|
fproduct(zzprime, zzzprime, qmqp);
|
|
freduce_degree(zzprime);
|
|
freduce_coefficients(zzprime);
|
|
memcpy(x3, xxxprime, sizeof(limb) * 10);
|
|
memcpy(z3, zzprime, sizeof(limb) * 10);
|
|
|
|
fsquare(xx, x);
|
|
fsquare(zz, z);
|
|
fproduct(x2, xx, zz);
|
|
freduce_degree(x2);
|
|
freduce_coefficients(x2);
|
|
fdifference(zz, xx); // does zz = xx - zz
|
|
memset(zzz + 10, 0, sizeof(limb) * 9);
|
|
fscalar_product(zzz, zz, 121665);
|
|
/* No need to call freduce_degree here:
|
|
fscalar_product doesn't increase the degree of its input. */
|
|
freduce_coefficients(zzz);
|
|
fsum(zzz, xx);
|
|
fproduct(z2, zz, zzz);
|
|
freduce_degree(z2);
|
|
freduce_coefficients(z2);
|
|
}
|
|
|
|
/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
|
|
* them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid
|
|
* side-channel attacks.
|
|
*
|
|
* NOTE that this function requires that 'iswap' be 1 or 0; other values give
|
|
* wrong results. Also, the two limb arrays must be in reduced-coefficient,
|
|
* reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
|
|
* and all all values in a[0..9],b[0..9] must have magnitude less than
|
|
* INT32_MAX.
|
|
*/
|
|
static void
|
|
swap_conditional(limb a[19], limb b[19], limb iswap) {
|
|
unsigned i;
|
|
const s32 swap = (s32) -iswap;
|
|
|
|
for (i = 0; i < 10; ++i) {
|
|
const s32 x = swap & ( ((s32)a[i]) ^ ((s32)b[i]) );
|
|
a[i] = ((s32)a[i]) ^ x;
|
|
b[i] = ((s32)b[i]) ^ x;
|
|
}
|
|
}
|
|
|
|
/* Calculates nQ where Q is the x-coordinate of a point on the curve
|
|
*
|
|
* resultx/resultz: the x coordinate of the resulting curve point (short form)
|
|
* n: a little endian, 32-byte number
|
|
* q: a point of the curve (short form)
|
|
*/
|
|
static void
|
|
cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
|
|
limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
|
|
limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
|
|
limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
|
|
limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
|
|
|
|
unsigned i, j;
|
|
|
|
memcpy(nqpqx, q, sizeof(limb) * 10);
|
|
|
|
for (i = 0; i < 32; ++i) {
|
|
u8 byte = n[31 - i];
|
|
for (j = 0; j < 8; ++j) {
|
|
const limb bit = byte >> 7;
|
|
|
|
swap_conditional(nqx, nqpqx, bit);
|
|
swap_conditional(nqz, nqpqz, bit);
|
|
fmonty(nqx2, nqz2,
|
|
nqpqx2, nqpqz2,
|
|
nqx, nqz,
|
|
nqpqx, nqpqz,
|
|
q);
|
|
swap_conditional(nqx2, nqpqx2, bit);
|
|
swap_conditional(nqz2, nqpqz2, bit);
|
|
|
|
t = nqx;
|
|
nqx = nqx2;
|
|
nqx2 = t;
|
|
t = nqz;
|
|
nqz = nqz2;
|
|
nqz2 = t;
|
|
t = nqpqx;
|
|
nqpqx = nqpqx2;
|
|
nqpqx2 = t;
|
|
t = nqpqz;
|
|
nqpqz = nqpqz2;
|
|
nqpqz2 = t;
|
|
|
|
byte <<= 1;
|
|
}
|
|
}
|
|
|
|
memcpy(resultx, nqx, sizeof(limb) * 10);
|
|
memcpy(resultz, nqz, sizeof(limb) * 10);
|
|
}
|
|
|
|
// -----------------------------------------------------------------------------
|
|
// Shamelessly copied from djb's code
|
|
// -----------------------------------------------------------------------------
|
|
static void
|
|
crecip(limb *out, const limb *z) {
|
|
limb z2[10];
|
|
limb z9[10];
|
|
limb z11[10];
|
|
limb z2_5_0[10];
|
|
limb z2_10_0[10];
|
|
limb z2_20_0[10];
|
|
limb z2_50_0[10];
|
|
limb z2_100_0[10];
|
|
limb t0[10];
|
|
limb t1[10];
|
|
int i;
|
|
|
|
/* 2 */ fsquare(z2,z);
|
|
/* 4 */ fsquare(t1,z2);
|
|
/* 8 */ fsquare(t0,t1);
|
|
/* 9 */ fmul(z9,t0,z);
|
|
/* 11 */ fmul(z11,z9,z2);
|
|
/* 22 */ fsquare(t0,z11);
|
|
/* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9);
|
|
|
|
/* 2^6 - 2^1 */ fsquare(t0,z2_5_0);
|
|
/* 2^7 - 2^2 */ fsquare(t1,t0);
|
|
/* 2^8 - 2^3 */ fsquare(t0,t1);
|
|
/* 2^9 - 2^4 */ fsquare(t1,t0);
|
|
/* 2^10 - 2^5 */ fsquare(t0,t1);
|
|
/* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0);
|
|
|
|
/* 2^11 - 2^1 */ fsquare(t0,z2_10_0);
|
|
/* 2^12 - 2^2 */ fsquare(t1,t0);
|
|
/* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
|
|
/* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0);
|
|
|
|
/* 2^21 - 2^1 */ fsquare(t0,z2_20_0);
|
|
/* 2^22 - 2^2 */ fsquare(t1,t0);
|
|
/* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
|
|
/* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0);
|
|
|
|
/* 2^41 - 2^1 */ fsquare(t1,t0);
|
|
/* 2^42 - 2^2 */ fsquare(t0,t1);
|
|
/* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
|
|
/* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0);
|
|
|
|
/* 2^51 - 2^1 */ fsquare(t0,z2_50_0);
|
|
/* 2^52 - 2^2 */ fsquare(t1,t0);
|
|
/* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
|
|
/* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0);
|
|
|
|
/* 2^101 - 2^1 */ fsquare(t1,z2_100_0);
|
|
/* 2^102 - 2^2 */ fsquare(t0,t1);
|
|
/* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
|
|
/* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0);
|
|
|
|
/* 2^201 - 2^1 */ fsquare(t0,t1);
|
|
/* 2^202 - 2^2 */ fsquare(t1,t0);
|
|
/* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
|
|
/* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0);
|
|
|
|
/* 2^251 - 2^1 */ fsquare(t1,t0);
|
|
/* 2^252 - 2^2 */ fsquare(t0,t1);
|
|
/* 2^253 - 2^3 */ fsquare(t1,t0);
|
|
/* 2^254 - 2^4 */ fsquare(t0,t1);
|
|
/* 2^255 - 2^5 */ fsquare(t1,t0);
|
|
/* 2^255 - 21 */ fmul(out,t1,z11);
|
|
}
|
|
|
|
int curve25519_donna(u8 *, const u8 *, const u8 *);
|
|
|
|
int
|
|
curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
|
|
limb bp[10], x[10], z[11], zmone[10];
|
|
uint8_t e[32];
|
|
int i;
|
|
|
|
for (i = 0; i < 32; ++i) e[i] = secret[i];
|
|
e[0] &= 248;
|
|
e[31] &= 127;
|
|
e[31] |= 64;
|
|
|
|
fexpand(bp, basepoint);
|
|
cmult(x, z, e, bp);
|
|
crecip(zmone, z);
|
|
fmul(z, x, zmone);
|
|
freduce_coefficients(z);
|
|
fcontract(mypublic, z);
|
|
return 0;
|
|
}
|