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-/* Copyright 2008, Google Inc.
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions are
- * met:
- *
- * * Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * * Redistributions in binary form must reproduce the above
- * copyright notice, this list of conditions and the following disclaimer
- * in the documentation and/or other materials provided with the
- * distribution.
- * * Neither the name of Google Inc. nor the names of its
- * contributors may be used to endorse or promote products derived from
- * this software without specific prior written permission.
- *
- * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
- * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
- * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
- * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
- * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
- * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
- * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
- * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
- * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
- * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
- * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
- *
- * curve25519-donna: Curve25519 elliptic curve, public key function
- *
- * http://code.google.com/p/curve25519-donna/
- *
- * Adam Langley <agl@imperialviolet.org>
- *
- * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
- *
- * More information about curve25519 can be found here
- * http://cr.yp.to/ecdh.html
- *
- * djb's sample implementation of curve25519 is written in a special assembly
- * language called qhasm and uses the floating point registers.
- *
- * This is, almost, a clean room reimplementation from the curve25519 paper. It
- * uses many of the tricks described therein. Only the crecip function is taken
- * from the sample implementation. */
-
-#include <string.h>
-#include <stdint.h>
-
-#ifdef _MSC_VER
-#define inline __inline
-#endif
-
-typedef uint8_t u8;
-typedef int32_t s32;
-typedef int64_t limb;
-
-/* Field element representation:
- *
- * Field elements are written as an array of signed, 64-bit limbs, least
- * significant first. The value of the field element is:
- * x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
- *
- * i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
-
-/* Sum two numbers: output += in */
-static void fsum(limb *output, const limb *in) {
- unsigned i;
- for (i = 0; i < 10; i += 2) {
- output[0+i] = output[0+i] + in[0+i];
- output[1+i] = output[1+i] + in[1+i];
- }
-}
-
-/* Find the difference of two numbers: output = in - output
- * (note the order of the arguments!). */
-static void fdifference(limb *output, const limb *in) {
- unsigned i;
- for (i = 0; i < 10; ++i) {
- output[i] = in[i] - output[i];
- }
-}
-
-/* Multiply a number by a scalar: output = in * scalar */
-static void fscalar_product(limb *output, const limb *in, const limb scalar) {
- unsigned i;
- for (i = 0; i < 10; ++i) {
- output[i] = in[i] * scalar;
- }
-}
-
-/* Multiply two numbers: output = in2 * in
- *
- * output must be distinct to both inputs. The inputs are reduced coefficient
- * form, the output is not.
- *
- * output[x] <= 14 * the largest product of the input limbs. */
-static void fproduct(limb *output, const limb *in2, const limb *in) {
- output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]);
- output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) +
- ((limb) ((s32) in2[1])) * ((s32) in[0]);
- output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) +
- ((limb) ((s32) in2[0])) * ((s32) in[2]) +
- ((limb) ((s32) in2[2])) * ((s32) in[0]);
- output[3] = ((limb) ((s32) in2[1])) * ((s32) in[2]) +
- ((limb) ((s32) in2[2])) * ((s32) in[1]) +
- ((limb) ((s32) in2[0])) * ((s32) in[3]) +
- ((limb) ((s32) in2[3])) * ((s32) in[0]);
- output[4] = ((limb) ((s32) in2[2])) * ((s32) in[2]) +
- 2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) +
- ((limb) ((s32) in2[3])) * ((s32) in[1])) +
- ((limb) ((s32) in2[0])) * ((s32) in[4]) +
- ((limb) ((s32) in2[4])) * ((s32) in[0]);
- output[5] = ((limb) ((s32) in2[2])) * ((s32) in[3]) +
- ((limb) ((s32) in2[3])) * ((s32) in[2]) +
- ((limb) ((s32) in2[1])) * ((s32) in[4]) +
- ((limb) ((s32) in2[4])) * ((s32) in[1]) +
- ((limb) ((s32) in2[0])) * ((s32) in[5]) +
- ((limb) ((s32) in2[5])) * ((s32) in[0]);
- output[6] = 2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) +
- ((limb) ((s32) in2[1])) * ((s32) in[5]) +
- ((limb) ((s32) in2[5])) * ((s32) in[1])) +
- ((limb) ((s32) in2[2])) * ((s32) in[4]) +
- ((limb) ((s32) in2[4])) * ((s32) in[2]) +
- ((limb) ((s32) in2[0])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[0]);
- output[7] = ((limb) ((s32) in2[3])) * ((s32) in[4]) +
- ((limb) ((s32) in2[4])) * ((s32) in[3]) +
- ((limb) ((s32) in2[2])) * ((s32) in[5]) +
- ((limb) ((s32) in2[5])) * ((s32) in[2]) +
- ((limb) ((s32) in2[1])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[1]) +
- ((limb) ((s32) in2[0])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[0]);
- output[8] = ((limb) ((s32) in2[4])) * ((s32) in[4]) +
- 2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) +
- ((limb) ((s32) in2[5])) * ((s32) in[3]) +
- ((limb) ((s32) in2[1])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[1])) +
- ((limb) ((s32) in2[2])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[2]) +
- ((limb) ((s32) in2[0])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[0]);
- output[9] = ((limb) ((s32) in2[4])) * ((s32) in[5]) +
- ((limb) ((s32) in2[5])) * ((s32) in[4]) +
- ((limb) ((s32) in2[3])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[3]) +
- ((limb) ((s32) in2[2])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[2]) +
- ((limb) ((s32) in2[1])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[1]) +
- ((limb) ((s32) in2[0])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[0]);
- output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) +
- ((limb) ((s32) in2[3])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[3]) +
- ((limb) ((s32) in2[1])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[1])) +
- ((limb) ((s32) in2[4])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[4]) +
- ((limb) ((s32) in2[2])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[2]);
- output[11] = ((limb) ((s32) in2[5])) * ((s32) in[6]) +
- ((limb) ((s32) in2[6])) * ((s32) in[5]) +
- ((limb) ((s32) in2[4])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[4]) +
- ((limb) ((s32) in2[3])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[3]) +
- ((limb) ((s32) in2[2])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[2]);
- output[12] = ((limb) ((s32) in2[6])) * ((s32) in[6]) +
- 2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[5]) +
- ((limb) ((s32) in2[3])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[3])) +
- ((limb) ((s32) in2[4])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[4]);
- output[13] = ((limb) ((s32) in2[6])) * ((s32) in[7]) +
- ((limb) ((s32) in2[7])) * ((s32) in[6]) +
- ((limb) ((s32) in2[5])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[5]) +
- ((limb) ((s32) in2[4])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[4]);
- output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) +
- ((limb) ((s32) in2[5])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[5])) +
- ((limb) ((s32) in2[6])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[6]);
- output[15] = ((limb) ((s32) in2[7])) * ((s32) in[8]) +
- ((limb) ((s32) in2[8])) * ((s32) in[7]) +
- ((limb) ((s32) in2[6])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[6]);
- output[16] = ((limb) ((s32) in2[8])) * ((s32) in[8]) +
- 2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[7]));
- output[17] = ((limb) ((s32) in2[8])) * ((s32) in[9]) +
- ((limb) ((s32) in2[9])) * ((s32) in[8]);
- output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]);
-}
-
-/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
- *
- * On entry: |output[i]| < 14*2^54
- * On exit: |output[0..8]| < 280*2^54 */
-static void freduce_degree(limb *output) {
- /* Each of these shifts and adds ends up multiplying the value by 19.
- *
- * For output[0..8], the absolute entry value is < 14*2^54 and we add, at
- * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
- output[8] += output[18] << 4;
- output[8] += output[18] << 1;
- output[8] += output[18];
- output[7] += output[17] << 4;
- output[7] += output[17] << 1;
- output[7] += output[17];
- output[6] += output[16] << 4;
- output[6] += output[16] << 1;
- output[6] += output[16];
- output[5] += output[15] << 4;
- output[5] += output[15] << 1;
- output[5] += output[15];
- output[4] += output[14] << 4;
- output[4] += output[14] << 1;
- output[4] += output[14];
- output[3] += output[13] << 4;
- output[3] += output[13] << 1;
- output[3] += output[13];
- output[2] += output[12] << 4;
- output[2] += output[12] << 1;
- output[2] += output[12];
- output[1] += output[11] << 4;
- output[1] += output[11] << 1;
- output[1] += output[11];
- output[0] += output[10] << 4;
- output[0] += output[10] << 1;
- output[0] += output[10];
-}
-
-#if (-1 & 3) != 3
-#error "This code only works on a two's complement system"
-#endif
-
-/* return v / 2^26, using only shifts and adds.
- *
- * On entry: v can take any value. */
-static inline limb
-div_by_2_26(const limb v)
-{
- /* High word of v; no shift needed. */
- const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
- /* Set to all 1s if v was negative; else set to 0s. */
- const int32_t sign = ((int32_t) highword) >> 31;
- /* Set to 0x3ffffff if v was negative; else set to 0. */
- const int32_t roundoff = ((uint32_t) sign) >> 6;
- /* Should return v / (1<<26) */
- return (v + roundoff) >> 26;
-}
-
-/* return v / (2^25), using only shifts and adds.
- *
- * On entry: v can take any value. */
-static inline limb
-div_by_2_25(const limb v)
-{
- /* High word of v; no shift needed*/
- const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
- /* Set to all 1s if v was negative; else set to 0s. */
- const int32_t sign = ((int32_t) highword) >> 31;
- /* Set to 0x1ffffff if v was negative; else set to 0. */
- const int32_t roundoff = ((uint32_t) sign) >> 7;
- /* Should return v / (1<<25) */
- return (v + roundoff) >> 25;
-}
-
-/* Reduce all coefficients of the short form input so that |x| < 2^26.
- *
- * On entry: |output[i]| < 280*2^54 */
-static void freduce_coefficients(limb *output) {
- unsigned i;
-
- output[10] = 0;
-
- for (i = 0; i < 10; i += 2) {
- limb over = div_by_2_26(output[i]);
- /* The entry condition (that |output[i]| < 280*2^54) means that over is, at
- * most, 280*2^28 in the first iteration of this loop. This is added to the
- * next limb and we can approximate the resulting bound of that limb by
- * 281*2^54. */
- output[i] -= over << 26;
- output[i+1] += over;
-
- /* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
- * 281*2^29. When this is added to the next limb, the resulting bound can
- * be approximated as 281*2^54.
- *
- * For subsequent iterations of the loop, 281*2^54 remains a conservative
- * bound and no overflow occurs. */
- over = div_by_2_25(output[i+1]);
- output[i+1] -= over << 25;
- output[i+2] += over;
- }
- /* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
- output[0] += output[10] << 4;
- output[0] += output[10] << 1;
- output[0] += output[10];
-
- output[10] = 0;
-
- /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
- * So |over| will be no more than 2^16. */
- {
- limb over = div_by_2_26(output[0]);
- output[0] -= over << 26;
- output[1] += over;
- }
-
- /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
- * bound on |output[1]| is sufficient to meet our needs. */
-}
-
-/* A helpful wrapper around fproduct: output = in * in2.
- *
- * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
- *
- * output must be distinct to both inputs. The output is reduced degree
- * (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
-static void
-fmul(limb *output, const limb *in, const limb *in2) {
- limb t[19];
- fproduct(t, in, in2);
- /* |t[i]| < 14*2^54 */
- freduce_degree(t);
- freduce_coefficients(t);
- /* |t[i]| < 2^26 */
- memcpy(output, t, sizeof(limb) * 10);
-}
-
-/* Square a number: output = in**2
- *
- * output must be distinct from the input. The inputs are reduced coefficient
- * form, the output is not.
- *
- * output[x] <= 14 * the largest product of the input limbs. */
-static void fsquare_inner(limb *output, const limb *in) {
- output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]);
- output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]);
- output[2] = 2 * (((limb) ((s32) in[1])) * ((s32) in[1]) +
- ((limb) ((s32) in[0])) * ((s32) in[2]));
- output[3] = 2 * (((limb) ((s32) in[1])) * ((s32) in[2]) +
- ((limb) ((s32) in[0])) * ((s32) in[3]));
- output[4] = ((limb) ((s32) in[2])) * ((s32) in[2]) +
- 4 * ((limb) ((s32) in[1])) * ((s32) in[3]) +
- 2 * ((limb) ((s32) in[0])) * ((s32) in[4]);
- output[5] = 2 * (((limb) ((s32) in[2])) * ((s32) in[3]) +
- ((limb) ((s32) in[1])) * ((s32) in[4]) +
- ((limb) ((s32) in[0])) * ((s32) in[5]));
- output[6] = 2 * (((limb) ((s32) in[3])) * ((s32) in[3]) +
- ((limb) ((s32) in[2])) * ((s32) in[4]) +
- ((limb) ((s32) in[0])) * ((s32) in[6]) +
- 2 * ((limb) ((s32) in[1])) * ((s32) in[5]));
- output[7] = 2 * (((limb) ((s32) in[3])) * ((s32) in[4]) +
- ((limb) ((s32) in[2])) * ((s32) in[5]) +
- ((limb) ((s32) in[1])) * ((s32) in[6]) +
- ((limb) ((s32) in[0])) * ((s32) in[7]));
- output[8] = ((limb) ((s32) in[4])) * ((s32) in[4]) +
- 2 * (((limb) ((s32) in[2])) * ((s32) in[6]) +
- ((limb) ((s32) in[0])) * ((s32) in[8]) +
- 2 * (((limb) ((s32) in[1])) * ((s32) in[7]) +
- ((limb) ((s32) in[3])) * ((s32) in[5])));
- output[9] = 2 * (((limb) ((s32) in[4])) * ((s32) in[5]) +
- ((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]);
-}
-
-/* fsquare sets output = in^2.
- *
- * On entry: The |in| argument is in reduced coefficients form and |in[i]| <
- * 2^27.
- *
- * On exit: The |output| argument is in reduced coefficients form (indeed, one
- * need only provide storage for 10 limbs) and |out[i]| < 2^26. */
-static void
-fsquare(limb *output, const limb *in) {
- limb t[19];
- fsquare_inner(t, in);
- /* |t[i]| < 14*2^54 because the largest product of two limbs will be <
- * 2^(27+27) and fsquare_inner adds together, at most, 14 of those
- * products. */
- freduce_degree(t);
- freduce_coefficients(t);
- /* |t[i]| < 2^26 */
- 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, 0x1ffffff);
-#undef F
-}
-
-#if (-32 >> 1) != -16
-#error "This code only works when >> does sign-extension on negative numbers"
-#endif
-
-/* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
-static s32 s32_eq(s32 a, s32 b) {
- a = ~(a ^ b);
- a &= a << 16;
- a &= a << 8;
- a &= a << 4;
- a &= a << 2;
- a &= a << 1;
- return a >> 31;
-}
-
-/* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
- * both non-negative. */
-static s32 s32_gte(s32 a, s32 b) {
- a -= b;
- /* a >= 0 iff a >= b. */
- return ~(a >> 31);
-}
-
-/* Take a fully reduced polynomial form number and contract it into a
- * little-endian, 32-byte array.
- *
- * On entry: |input_limbs[i]| < 2^26 */
-static void
-fcontract(u8 *output, limb *input_limbs) {
- int i;
- int j;
- s32 input[10];
- s32 mask;
-
- /* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
- for (i = 0; i < 10; i++) {
- input[i] = input_limbs[i];
- }
-
- 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]
- * non-negative by borrowing from the next-larger limb. */
- const s32 mask = input[i] >> 31;
- const s32 carry = -((input[i] & mask) >> 25);
- input[i] = input[i] + (carry << 25);
- input[i+1] = input[i+1] - carry;
- } else {
- const s32 mask = input[i] >> 31;
- const s32 carry = -((input[i] & mask) >> 26);
- input[i] = input[i] + (carry << 26);
- input[i+1] = input[i+1] - carry;
- }
- }
-
- /* There's no greater limb for input[9] to borrow from, but we can multiply
- * by 19 and borrow from input[0], which is valid mod 2^255-19. */
- {
- const s32 mask = input[9] >> 31;
- const s32 carry = -((input[9] & mask) >> 25);
- input[9] = input[9] + (carry << 25);
- input[0] = input[0] - (carry * 19);
- }
-
- /* After the first iteration, input[1..9] are non-negative and fit within
- * 25 or 26 bits, depending on position. However, input[0] may be
- * negative. */
- }
-
- /* The first borrow-propagation pass above ended with every limb
- except (possibly) input[0] non-negative.
-
- If input[0] was negative after the first pass, then it was because of a
- carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
- one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
-
- In the second pass, each limb is decreased by at most one. Thus 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 = input[0] >> 31;
- const s32 carry = -((input[0] & mask) >> 26);
- input[0] = input[0] + (carry << 26);
- input[1] = input[1] - carry;
- }
-
- /* All input[i] are now non-negative. However, there might be values between
- * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
- for (j = 0; j < 2; j++) {
- for (i = 0; i < 9; i++) {
- if ((i & 1) == 1) {
- const s32 carry = input[i] >> 25;
- input[i] &= 0x1ffffff;
- input[i+1] += carry;
- } else {
- const s32 carry = input[i] >> 26;
- input[i] &= 0x3ffffff;
- input[i+1] += carry;
- }
- }
-
- {
- const s32 carry = input[9] >> 25;
- input[9] &= 0x1ffffff;
- input[0] += 19*carry;
- }
- }
-
- /* If the first carry-chain pass, just above, ended up with a carry from
- * input[9], and that caused input[0] to be out-of-bounds, then input[0] was
- * < 2^26 + 2*19, because the carry was, at most, two.
- *
- * If the second pass carried from input[9] again then input[0] is < 2*19 and
- * the input[9] -> input[0] carry didn't push input[0] out of bounds. */
-
- /* It still remains the case that input might be between 2^255-19 and 2^255.
- * In this case, input[1..9] must take their maximum value and input[0] must
- * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
- mask = s32_gte(input[0], 0x3ffffed);
- for (i = 1; i < 10; i++) {
- if ((i & 1) == 1) {
- mask &= s32_eq(input[i], 0x1ffffff);
- } else {
- mask &= s32_eq(input[i], 0x3ffffff);
- }
- }
-
- /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
- * this conditionally subtracts 2^255-19. */
- input[0] -= mask & 0x3ffffed;
-
- for (i = 1; i < 10; i++) {
- if ((i & 1) == 1) {
- input[i] -= mask & 0x1ffffff;
- } else {
- input[i] -= mask & 0x3ffffff;
- }
- }
-
- 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
- *
- * On entry and exit, the absolute value of the limbs of all inputs and outputs
- * are < 2^26. */
-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);
- /* |x[i]| < 2^27 */
- fdifference(z, origx); /* does x - z */
- /* |z[i]| < 2^27 */
-
- memcpy(origxprime, xprime, sizeof(limb) * 10);
- fsum(xprime, zprime);
- /* |xprime[i]| < 2^27 */
- fdifference(zprime, origxprime);
- /* |zprime[i]| < 2^27 */
- fproduct(xxprime, xprime, z);
- /* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
- * 2^(27+27) and fproduct adds together, at most, 14 of those products.
- * (Approximating that to 2^58 doesn't work out.) */
- fproduct(zzprime, x, zprime);
- /* |zzprime[i]| < 14*2^54 */
- freduce_degree(xxprime);
- freduce_coefficients(xxprime);
- /* |xxprime[i]| < 2^26 */
- freduce_degree(zzprime);
- freduce_coefficients(zzprime);
- /* |zzprime[i]| < 2^26 */
- memcpy(origxprime, xxprime, sizeof(limb) * 10);
- fsum(xxprime, zzprime);
- /* |xxprime[i]| < 2^27 */
- fdifference(zzprime, origxprime);
- /* |zzprime[i]| < 2^27 */
- fsquare(xxxprime, xxprime);
- /* |xxxprime[i]| < 2^26 */
- fsquare(zzzprime, zzprime);
- /* |zzzprime[i]| < 2^26 */
- fproduct(zzprime, zzzprime, qmqp);
- /* |zzprime[i]| < 14*2^52 */
- freduce_degree(zzprime);
- freduce_coefficients(zzprime);
- /* |zzprime[i]| < 2^26 */
- memcpy(x3, xxxprime, sizeof(limb) * 10);
- memcpy(z3, zzprime, sizeof(limb) * 10);
-
- fsquare(xx, x);
- /* |xx[i]| < 2^26 */
- fsquare(zz, z);
- /* |zz[i]| < 2^26 */
- fproduct(x2, xx, zz);
- /* |x2[i]| < 14*2^52 */
- freduce_degree(x2);
- freduce_coefficients(x2);
- /* |x2[i]| < 2^26 */
- fdifference(zz, xx); /* does zz = xx - zz */
- /* |zz[i]| < 2^27 */
- memset(zzz + 10, 0, sizeof(limb) * 9);
- fscalar_product(zzz, zz, 121665);
- /* |zzz[i]| < 2^(27+17) */
- /* No need to call freduce_degree here:
- fscalar_product doesn't increase the degree of its input. */
- freduce_coefficients(zzz);
- /* |zzz[i]| < 2^26 */
- fsum(zzz, xx);
- /* |zzz[i]| < 2^27 */
- fproduct(z2, zz, zzz);
- /* |z2[i]| < 14*2^(26+27) */
- freduce_degree(z2);
- freduce_coefficients(z2);
- /* |z2|i| < 2^26 */
-}
-
-/* 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 *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);
- fcontract(mypublic, z);
- return 0;
-}
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