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3584 lines
148 KiB
C
3584 lines
148 KiB
C
/* ====================================================================
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* Copyright (c) 2016 The OpenSSL Project. 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
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* are met:
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*
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* 1. 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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*
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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*
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* 3. All advertising materials mentioning features or use of this
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* software must display the following acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit. (http://www.OpenSSL.org/)"
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*
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* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
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* endorse or promote products derived from this software without
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* prior written permission. For written permission, please contact
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* licensing@OpenSSL.org.
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*
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* 5. Products derived from this software may not be called "OpenSSL"
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* nor may "OpenSSL" appear in their names without prior written
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* permission of the OpenSSL Project.
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*
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* 6. Redistributions of any form whatsoever must retain the following
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* acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit (http://www.OpenSSL.org/)"
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*
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* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
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* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
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* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
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* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
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* OF THE POSSIBILITY OF SUCH DAMAGE.
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* ====================================================================
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*/
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/* This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
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* 20141124 (http://bench.cr.yp.to/supercop.html).
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*
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* The field functions are shared by Ed25519 and X25519 where possible. */
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#include <string.h>
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#include "ec_lcl.h"
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/* fe means field element. Here the field is \Z/(2^255-19). An element t,
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* entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
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* t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
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* context. */
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typedef int32_t fe[10];
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static uint64_t load_3(const uint8_t *in) {
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uint64_t result;
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result = (uint64_t)in[0];
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result |= ((uint64_t)in[1]) << 8;
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result |= ((uint64_t)in[2]) << 16;
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return result;
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}
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static uint64_t load_4(const uint8_t *in) {
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uint64_t result;
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result = (uint64_t)in[0];
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result |= ((uint64_t)in[1]) << 8;
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result |= ((uint64_t)in[2]) << 16;
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result |= ((uint64_t)in[3]) << 24;
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return result;
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}
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static void fe_frombytes(fe h, const uint8_t *s) {
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/* Ignores top bit of h. */
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int64_t h0 = load_4(s);
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int64_t h1 = load_3(s + 4) << 6;
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int64_t h2 = load_3(s + 7) << 5;
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int64_t h3 = load_3(s + 10) << 3;
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int64_t h4 = load_3(s + 13) << 2;
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int64_t h5 = load_4(s + 16);
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int64_t h6 = load_3(s + 20) << 7;
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int64_t h7 = load_3(s + 23) << 5;
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int64_t h8 = load_3(s + 26) << 4;
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int64_t h9 = (load_3(s + 29) & 8388607) << 2;
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int64_t carry0;
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int64_t carry1;
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int64_t carry2;
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int64_t carry3;
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int64_t carry4;
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int64_t carry5;
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int64_t carry6;
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int64_t carry7;
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int64_t carry8;
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int64_t carry9;
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carry9 = (h9 + (int64_t) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
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carry1 = (h1 + (int64_t) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
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carry3 = (h3 + (int64_t) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
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carry5 = (h5 + (int64_t) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
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carry7 = (h7 + (int64_t) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
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carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
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carry2 = (h2 + (int64_t) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
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carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
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carry6 = (h6 + (int64_t) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
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carry8 = (h8 + (int64_t) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
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h[0] = h0;
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h[1] = h1;
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h[2] = h2;
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h[3] = h3;
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h[4] = h4;
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h[5] = h5;
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h[6] = h6;
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h[7] = h7;
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h[8] = h8;
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h[9] = h9;
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}
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/* Preconditions:
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* |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
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*
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* Write p=2^255-19; q=floor(h/p).
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* Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
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*
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* Proof:
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* Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
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* Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
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*
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* Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
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* Then 0<y<1.
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*
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* Write r=h-pq.
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* Have 0<=r<=p-1=2^255-20.
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* Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
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*
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* Write x=r+19(2^-255)r+y.
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* Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
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*
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* Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
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* so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. */
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static void fe_tobytes(uint8_t *s, const fe h) {
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int32_t h0 = h[0];
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int32_t h1 = h[1];
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int32_t h2 = h[2];
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int32_t h3 = h[3];
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int32_t h4 = h[4];
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int32_t h5 = h[5];
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int32_t h6 = h[6];
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int32_t h7 = h[7];
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int32_t h8 = h[8];
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int32_t h9 = h[9];
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int32_t q;
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int32_t carry0;
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int32_t carry1;
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int32_t carry2;
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int32_t carry3;
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int32_t carry4;
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int32_t carry5;
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int32_t carry6;
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int32_t carry7;
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int32_t carry8;
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int32_t carry9;
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q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
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q = (h0 + q) >> 26;
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q = (h1 + q) >> 25;
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q = (h2 + q) >> 26;
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q = (h3 + q) >> 25;
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q = (h4 + q) >> 26;
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q = (h5 + q) >> 25;
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q = (h6 + q) >> 26;
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q = (h7 + q) >> 25;
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q = (h8 + q) >> 26;
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q = (h9 + q) >> 25;
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/* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
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h0 += 19 * q;
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/* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
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carry0 = h0 >> 26; h1 += carry0; h0 -= carry0 << 26;
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carry1 = h1 >> 25; h2 += carry1; h1 -= carry1 << 25;
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carry2 = h2 >> 26; h3 += carry2; h2 -= carry2 << 26;
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carry3 = h3 >> 25; h4 += carry3; h3 -= carry3 << 25;
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carry4 = h4 >> 26; h5 += carry4; h4 -= carry4 << 26;
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carry5 = h5 >> 25; h6 += carry5; h5 -= carry5 << 25;
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carry6 = h6 >> 26; h7 += carry6; h6 -= carry6 << 26;
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carry7 = h7 >> 25; h8 += carry7; h7 -= carry7 << 25;
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carry8 = h8 >> 26; h9 += carry8; h8 -= carry8 << 26;
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carry9 = h9 >> 25; h9 -= carry9 << 25;
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/* h10 = carry9 */
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/* Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
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* Have h0+...+2^230 h9 between 0 and 2^255-1;
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* evidently 2^255 h10-2^255 q = 0.
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* Goal: Output h0+...+2^230 h9. */
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s[0] = h0 >> 0;
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s[1] = h0 >> 8;
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s[2] = h0 >> 16;
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s[3] = (h0 >> 24) | (h1 << 2);
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s[4] = h1 >> 6;
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s[5] = h1 >> 14;
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s[6] = (h1 >> 22) | (h2 << 3);
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s[7] = h2 >> 5;
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s[8] = h2 >> 13;
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s[9] = (h2 >> 21) | (h3 << 5);
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s[10] = h3 >> 3;
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s[11] = h3 >> 11;
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s[12] = (h3 >> 19) | (h4 << 6);
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s[13] = h4 >> 2;
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s[14] = h4 >> 10;
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s[15] = h4 >> 18;
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s[16] = h5 >> 0;
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s[17] = h5 >> 8;
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s[18] = h5 >> 16;
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s[19] = (h5 >> 24) | (h6 << 1);
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s[20] = h6 >> 7;
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s[21] = h6 >> 15;
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s[22] = (h6 >> 23) | (h7 << 3);
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s[23] = h7 >> 5;
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s[24] = h7 >> 13;
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s[25] = (h7 >> 21) | (h8 << 4);
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s[26] = h8 >> 4;
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s[27] = h8 >> 12;
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s[28] = (h8 >> 20) | (h9 << 6);
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s[29] = h9 >> 2;
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s[30] = h9 >> 10;
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s[31] = h9 >> 18;
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}
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/* h = f */
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static void fe_copy(fe h, const fe f) {
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memmove(h, f, sizeof(int32_t) * 10);
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}
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/* h = 0 */
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static void fe_0(fe h) { memset(h, 0, sizeof(int32_t) * 10); }
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/* h = 1 */
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static void fe_1(fe h) {
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memset(h, 0, sizeof(int32_t) * 10);
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h[0] = 1;
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}
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/* h = f + g
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* Can overlap h with f or g.
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*
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* Preconditions:
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* |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
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* |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
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*
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* Postconditions:
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* |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
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static void fe_add(fe h, const fe f, const fe g) {
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unsigned i;
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for (i = 0; i < 10; i++) {
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h[i] = f[i] + g[i];
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}
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}
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/* h = f - g
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* Can overlap h with f or g.
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*
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* Preconditions:
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* |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
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* |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
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*
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* Postconditions:
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* |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
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static void fe_sub(fe h, const fe f, const fe g) {
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unsigned i;
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for (i = 0; i < 10; i++) {
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h[i] = f[i] - g[i];
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}
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}
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/* h = f * g
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* Can overlap h with f or g.
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*
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* Preconditions:
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* |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
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* |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
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*
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* Postconditions:
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* |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
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*
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* Notes on implementation strategy:
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*
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* Using schoolbook multiplication.
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* Karatsuba would save a little in some cost models.
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*
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* Most multiplications by 2 and 19 are 32-bit precomputations;
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* cheaper than 64-bit postcomputations.
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*
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* There is one remaining multiplication by 19 in the carry chain;
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* one *19 precomputation can be merged into this,
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* but the resulting data flow is considerably less clean.
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*
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* There are 12 carries below.
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* 10 of them are 2-way parallelizable and vectorizable.
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* Can get away with 11 carries, but then data flow is much deeper.
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*
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* With tighter constraints on inputs can squeeze carries into int32. */
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static void fe_mul(fe h, const fe f, const fe g) {
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int32_t f0 = f[0];
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int32_t f1 = f[1];
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int32_t f2 = f[2];
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int32_t f3 = f[3];
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int32_t f4 = f[4];
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int32_t f5 = f[5];
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int32_t f6 = f[6];
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int32_t f7 = f[7];
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int32_t f8 = f[8];
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int32_t f9 = f[9];
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int32_t g0 = g[0];
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int32_t g1 = g[1];
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int32_t g2 = g[2];
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int32_t g3 = g[3];
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int32_t g4 = g[4];
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int32_t g5 = g[5];
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int32_t g6 = g[6];
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int32_t g7 = g[7];
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int32_t g8 = g[8];
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int32_t g9 = g[9];
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int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
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int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
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int32_t g3_19 = 19 * g3;
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int32_t g4_19 = 19 * g4;
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int32_t g5_19 = 19 * g5;
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int32_t g6_19 = 19 * g6;
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int32_t g7_19 = 19 * g7;
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int32_t g8_19 = 19 * g8;
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int32_t g9_19 = 19 * g9;
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int32_t f1_2 = 2 * f1;
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int32_t f3_2 = 2 * f3;
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int32_t f5_2 = 2 * f5;
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int32_t f7_2 = 2 * f7;
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int32_t f9_2 = 2 * f9;
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int64_t f0g0 = f0 * (int64_t) g0;
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int64_t f0g1 = f0 * (int64_t) g1;
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int64_t f0g2 = f0 * (int64_t) g2;
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int64_t f0g3 = f0 * (int64_t) g3;
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int64_t f0g4 = f0 * (int64_t) g4;
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int64_t f0g5 = f0 * (int64_t) g5;
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int64_t f0g6 = f0 * (int64_t) g6;
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int64_t f0g7 = f0 * (int64_t) g7;
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int64_t f0g8 = f0 * (int64_t) g8;
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int64_t f0g9 = f0 * (int64_t) g9;
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int64_t f1g0 = f1 * (int64_t) g0;
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int64_t f1g1_2 = f1_2 * (int64_t) g1;
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int64_t f1g2 = f1 * (int64_t) g2;
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int64_t f1g3_2 = f1_2 * (int64_t) g3;
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int64_t f1g4 = f1 * (int64_t) g4;
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int64_t f1g5_2 = f1_2 * (int64_t) g5;
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int64_t f1g6 = f1 * (int64_t) g6;
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int64_t f1g7_2 = f1_2 * (int64_t) g7;
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int64_t f1g8 = f1 * (int64_t) g8;
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int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
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int64_t f2g0 = f2 * (int64_t) g0;
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int64_t f2g1 = f2 * (int64_t) g1;
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int64_t f2g2 = f2 * (int64_t) g2;
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int64_t f2g3 = f2 * (int64_t) g3;
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int64_t f2g4 = f2 * (int64_t) g4;
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int64_t f2g5 = f2 * (int64_t) g5;
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int64_t f2g6 = f2 * (int64_t) g6;
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int64_t f2g7 = f2 * (int64_t) g7;
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int64_t f2g8_19 = f2 * (int64_t) g8_19;
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int64_t f2g9_19 = f2 * (int64_t) g9_19;
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int64_t f3g0 = f3 * (int64_t) g0;
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int64_t f3g1_2 = f3_2 * (int64_t) g1;
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int64_t f3g2 = f3 * (int64_t) g2;
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int64_t f3g3_2 = f3_2 * (int64_t) g3;
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int64_t f3g4 = f3 * (int64_t) g4;
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int64_t f3g5_2 = f3_2 * (int64_t) g5;
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int64_t f3g6 = f3 * (int64_t) g6;
|
||
int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
|
||
int64_t f3g8_19 = f3 * (int64_t) g8_19;
|
||
int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
|
||
int64_t f4g0 = f4 * (int64_t) g0;
|
||
int64_t f4g1 = f4 * (int64_t) g1;
|
||
int64_t f4g2 = f4 * (int64_t) g2;
|
||
int64_t f4g3 = f4 * (int64_t) g3;
|
||
int64_t f4g4 = f4 * (int64_t) g4;
|
||
int64_t f4g5 = f4 * (int64_t) g5;
|
||
int64_t f4g6_19 = f4 * (int64_t) g6_19;
|
||
int64_t f4g7_19 = f4 * (int64_t) g7_19;
|
||
int64_t f4g8_19 = f4 * (int64_t) g8_19;
|
||
int64_t f4g9_19 = f4 * (int64_t) g9_19;
|
||
int64_t f5g0 = f5 * (int64_t) g0;
|
||
int64_t f5g1_2 = f5_2 * (int64_t) g1;
|
||
int64_t f5g2 = f5 * (int64_t) g2;
|
||
int64_t f5g3_2 = f5_2 * (int64_t) g3;
|
||
int64_t f5g4 = f5 * (int64_t) g4;
|
||
int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
|
||
int64_t f5g6_19 = f5 * (int64_t) g6_19;
|
||
int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
|
||
int64_t f5g8_19 = f5 * (int64_t) g8_19;
|
||
int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
|
||
int64_t f6g0 = f6 * (int64_t) g0;
|
||
int64_t f6g1 = f6 * (int64_t) g1;
|
||
int64_t f6g2 = f6 * (int64_t) g2;
|
||
int64_t f6g3 = f6 * (int64_t) g3;
|
||
int64_t f6g4_19 = f6 * (int64_t) g4_19;
|
||
int64_t f6g5_19 = f6 * (int64_t) g5_19;
|
||
int64_t f6g6_19 = f6 * (int64_t) g6_19;
|
||
int64_t f6g7_19 = f6 * (int64_t) g7_19;
|
||
int64_t f6g8_19 = f6 * (int64_t) g8_19;
|
||
int64_t f6g9_19 = f6 * (int64_t) g9_19;
|
||
int64_t f7g0 = f7 * (int64_t) g0;
|
||
int64_t f7g1_2 = f7_2 * (int64_t) g1;
|
||
int64_t f7g2 = f7 * (int64_t) g2;
|
||
int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
|
||
int64_t f7g4_19 = f7 * (int64_t) g4_19;
|
||
int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
|
||
int64_t f7g6_19 = f7 * (int64_t) g6_19;
|
||
int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
|
||
int64_t f7g8_19 = f7 * (int64_t) g8_19;
|
||
int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
|
||
int64_t f8g0 = f8 * (int64_t) g0;
|
||
int64_t f8g1 = f8 * (int64_t) g1;
|
||
int64_t f8g2_19 = f8 * (int64_t) g2_19;
|
||
int64_t f8g3_19 = f8 * (int64_t) g3_19;
|
||
int64_t f8g4_19 = f8 * (int64_t) g4_19;
|
||
int64_t f8g5_19 = f8 * (int64_t) g5_19;
|
||
int64_t f8g6_19 = f8 * (int64_t) g6_19;
|
||
int64_t f8g7_19 = f8 * (int64_t) g7_19;
|
||
int64_t f8g8_19 = f8 * (int64_t) g8_19;
|
||
int64_t f8g9_19 = f8 * (int64_t) g9_19;
|
||
int64_t f9g0 = f9 * (int64_t) g0;
|
||
int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
|
||
int64_t f9g2_19 = f9 * (int64_t) g2_19;
|
||
int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
|
||
int64_t f9g4_19 = f9 * (int64_t) g4_19;
|
||
int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
|
||
int64_t f9g6_19 = f9 * (int64_t) g6_19;
|
||
int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
|
||
int64_t f9g8_19 = f9 * (int64_t) g8_19;
|
||
int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
|
||
int64_t h0 = f0g0+f1g9_38+f2g8_19+f3g7_38+f4g6_19+f5g5_38+f6g4_19+f7g3_38+f8g2_19+f9g1_38;
|
||
int64_t h1 = f0g1+f1g0 +f2g9_19+f3g8_19+f4g7_19+f5g6_19+f6g5_19+f7g4_19+f8g3_19+f9g2_19;
|
||
int64_t h2 = f0g2+f1g1_2 +f2g0 +f3g9_38+f4g8_19+f5g7_38+f6g6_19+f7g5_38+f8g4_19+f9g3_38;
|
||
int64_t h3 = f0g3+f1g2 +f2g1 +f3g0 +f4g9_19+f5g8_19+f6g7_19+f7g6_19+f8g5_19+f9g4_19;
|
||
int64_t h4 = f0g4+f1g3_2 +f2g2 +f3g1_2 +f4g0 +f5g9_38+f6g8_19+f7g7_38+f8g6_19+f9g5_38;
|
||
int64_t h5 = f0g5+f1g4 +f2g3 +f3g2 +f4g1 +f5g0 +f6g9_19+f7g8_19+f8g7_19+f9g6_19;
|
||
int64_t h6 = f0g6+f1g5_2 +f2g4 +f3g3_2 +f4g2 +f5g1_2 +f6g0 +f7g9_38+f8g8_19+f9g7_38;
|
||
int64_t h7 = f0g7+f1g6 +f2g5 +f3g4 +f4g3 +f5g2 +f6g1 +f7g0 +f8g9_19+f9g8_19;
|
||
int64_t h8 = f0g8+f1g7_2 +f2g6 +f3g5_2 +f4g4 +f5g3_2 +f6g2 +f7g1_2 +f8g0 +f9g9_38;
|
||
int64_t h9 = f0g9+f1g8 +f2g7 +f3g6 +f4g5 +f5g4 +f6g3 +f7g2 +f8g1 +f9g0 ;
|
||
int64_t carry0;
|
||
int64_t carry1;
|
||
int64_t carry2;
|
||
int64_t carry3;
|
||
int64_t carry4;
|
||
int64_t carry5;
|
||
int64_t carry6;
|
||
int64_t carry7;
|
||
int64_t carry8;
|
||
int64_t carry9;
|
||
|
||
/* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
|
||
* i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
|
||
* |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
|
||
* i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
/* |h0| <= 2^25 */
|
||
/* |h4| <= 2^25 */
|
||
/* |h1| <= 1.71*2^59 */
|
||
/* |h5| <= 1.71*2^59 */
|
||
|
||
carry1 = (h1 + (int64_t) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
|
||
carry5 = (h5 + (int64_t) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
|
||
/* |h1| <= 2^24; from now on fits into int32 */
|
||
/* |h5| <= 2^24; from now on fits into int32 */
|
||
/* |h2| <= 1.41*2^60 */
|
||
/* |h6| <= 1.41*2^60 */
|
||
|
||
carry2 = (h2 + (int64_t) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
|
||
carry6 = (h6 + (int64_t) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
|
||
/* |h2| <= 2^25; from now on fits into int32 unchanged */
|
||
/* |h6| <= 2^25; from now on fits into int32 unchanged */
|
||
/* |h3| <= 1.71*2^59 */
|
||
/* |h7| <= 1.71*2^59 */
|
||
|
||
carry3 = (h3 + (int64_t) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
|
||
carry7 = (h7 + (int64_t) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
|
||
/* |h3| <= 2^24; from now on fits into int32 unchanged */
|
||
/* |h7| <= 2^24; from now on fits into int32 unchanged */
|
||
/* |h4| <= 1.72*2^34 */
|
||
/* |h8| <= 1.41*2^60 */
|
||
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
carry8 = (h8 + (int64_t) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
|
||
/* |h4| <= 2^25; from now on fits into int32 unchanged */
|
||
/* |h8| <= 2^25; from now on fits into int32 unchanged */
|
||
/* |h5| <= 1.01*2^24 */
|
||
/* |h9| <= 1.71*2^59 */
|
||
|
||
carry9 = (h9 + (int64_t) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
|
||
/* |h9| <= 2^24; from now on fits into int32 unchanged */
|
||
/* |h0| <= 1.1*2^39 */
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
/* |h0| <= 2^25; from now on fits into int32 unchanged */
|
||
/* |h1| <= 1.01*2^24 */
|
||
|
||
h[0] = h0;
|
||
h[1] = h1;
|
||
h[2] = h2;
|
||
h[3] = h3;
|
||
h[4] = h4;
|
||
h[5] = h5;
|
||
h[6] = h6;
|
||
h[7] = h7;
|
||
h[8] = h8;
|
||
h[9] = h9;
|
||
}
|
||
|
||
/* h = f * f
|
||
* Can overlap h with f.
|
||
*
|
||
* Preconditions:
|
||
* |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
|
||
*
|
||
* Postconditions:
|
||
* |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
|
||
*
|
||
* See fe_mul.c for discussion of implementation strategy. */
|
||
static void fe_sq(fe h, const fe f) {
|
||
int32_t f0 = f[0];
|
||
int32_t f1 = f[1];
|
||
int32_t f2 = f[2];
|
||
int32_t f3 = f[3];
|
||
int32_t f4 = f[4];
|
||
int32_t f5 = f[5];
|
||
int32_t f6 = f[6];
|
||
int32_t f7 = f[7];
|
||
int32_t f8 = f[8];
|
||
int32_t f9 = f[9];
|
||
int32_t f0_2 = 2 * f0;
|
||
int32_t f1_2 = 2 * f1;
|
||
int32_t f2_2 = 2 * f2;
|
||
int32_t f3_2 = 2 * f3;
|
||
int32_t f4_2 = 2 * f4;
|
||
int32_t f5_2 = 2 * f5;
|
||
int32_t f6_2 = 2 * f6;
|
||
int32_t f7_2 = 2 * f7;
|
||
int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
|
||
int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
|
||
int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
|
||
int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
|
||
int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
|
||
int64_t f0f0 = f0 * (int64_t) f0;
|
||
int64_t f0f1_2 = f0_2 * (int64_t) f1;
|
||
int64_t f0f2_2 = f0_2 * (int64_t) f2;
|
||
int64_t f0f3_2 = f0_2 * (int64_t) f3;
|
||
int64_t f0f4_2 = f0_2 * (int64_t) f4;
|
||
int64_t f0f5_2 = f0_2 * (int64_t) f5;
|
||
int64_t f0f6_2 = f0_2 * (int64_t) f6;
|
||
int64_t f0f7_2 = f0_2 * (int64_t) f7;
|
||
int64_t f0f8_2 = f0_2 * (int64_t) f8;
|
||
int64_t f0f9_2 = f0_2 * (int64_t) f9;
|
||
int64_t f1f1_2 = f1_2 * (int64_t) f1;
|
||
int64_t f1f2_2 = f1_2 * (int64_t) f2;
|
||
int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
|
||
int64_t f1f4_2 = f1_2 * (int64_t) f4;
|
||
int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
|
||
int64_t f1f6_2 = f1_2 * (int64_t) f6;
|
||
int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
|
||
int64_t f1f8_2 = f1_2 * (int64_t) f8;
|
||
int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
|
||
int64_t f2f2 = f2 * (int64_t) f2;
|
||
int64_t f2f3_2 = f2_2 * (int64_t) f3;
|
||
int64_t f2f4_2 = f2_2 * (int64_t) f4;
|
||
int64_t f2f5_2 = f2_2 * (int64_t) f5;
|
||
int64_t f2f6_2 = f2_2 * (int64_t) f6;
|
||
int64_t f2f7_2 = f2_2 * (int64_t) f7;
|
||
int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
|
||
int64_t f2f9_38 = f2 * (int64_t) f9_38;
|
||
int64_t f3f3_2 = f3_2 * (int64_t) f3;
|
||
int64_t f3f4_2 = f3_2 * (int64_t) f4;
|
||
int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
|
||
int64_t f3f6_2 = f3_2 * (int64_t) f6;
|
||
int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
|
||
int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
|
||
int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
|
||
int64_t f4f4 = f4 * (int64_t) f4;
|
||
int64_t f4f5_2 = f4_2 * (int64_t) f5;
|
||
int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
|
||
int64_t f4f7_38 = f4 * (int64_t) f7_38;
|
||
int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
|
||
int64_t f4f9_38 = f4 * (int64_t) f9_38;
|
||
int64_t f5f5_38 = f5 * (int64_t) f5_38;
|
||
int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
|
||
int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
|
||
int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
|
||
int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
|
||
int64_t f6f6_19 = f6 * (int64_t) f6_19;
|
||
int64_t f6f7_38 = f6 * (int64_t) f7_38;
|
||
int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
|
||
int64_t f6f9_38 = f6 * (int64_t) f9_38;
|
||
int64_t f7f7_38 = f7 * (int64_t) f7_38;
|
||
int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
|
||
int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
|
||
int64_t f8f8_19 = f8 * (int64_t) f8_19;
|
||
int64_t f8f9_38 = f8 * (int64_t) f9_38;
|
||
int64_t f9f9_38 = f9 * (int64_t) f9_38;
|
||
int64_t h0 = f0f0 +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
|
||
int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
|
||
int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
|
||
int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
|
||
int64_t h4 = f0f4_2+f1f3_4 +f2f2 +f5f9_76+f6f8_38+f7f7_38;
|
||
int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
|
||
int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
|
||
int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
|
||
int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4 +f9f9_38;
|
||
int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;
|
||
int64_t carry0;
|
||
int64_t carry1;
|
||
int64_t carry2;
|
||
int64_t carry3;
|
||
int64_t carry4;
|
||
int64_t carry5;
|
||
int64_t carry6;
|
||
int64_t carry7;
|
||
int64_t carry8;
|
||
int64_t carry9;
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
|
||
carry1 = (h1 + (int64_t) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
|
||
carry5 = (h5 + (int64_t) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
|
||
|
||
carry2 = (h2 + (int64_t) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
|
||
carry6 = (h6 + (int64_t) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
|
||
|
||
carry3 = (h3 + (int64_t) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
|
||
carry7 = (h7 + (int64_t) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
|
||
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
carry8 = (h8 + (int64_t) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
|
||
|
||
carry9 = (h9 + (int64_t) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
|
||
h[0] = h0;
|
||
h[1] = h1;
|
||
h[2] = h2;
|
||
h[3] = h3;
|
||
h[4] = h4;
|
||
h[5] = h5;
|
||
h[6] = h6;
|
||
h[7] = h7;
|
||
h[8] = h8;
|
||
h[9] = h9;
|
||
}
|
||
|
||
static void fe_invert(fe out, const fe z) {
|
||
fe t0;
|
||
fe t1;
|
||
fe t2;
|
||
fe t3;
|
||
int i;
|
||
|
||
fe_sq(t0, z);
|
||
for (i = 1; i < 1; ++i) {
|
||
fe_sq(t0, t0);
|
||
}
|
||
fe_sq(t1, t0);
|
||
for (i = 1; i < 2; ++i) {
|
||
fe_sq(t1, t1);
|
||
}
|
||
fe_mul(t1, z, t1);
|
||
fe_mul(t0, t0, t1);
|
||
fe_sq(t2, t0);
|
||
for (i = 1; i < 1; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t1, t1, t2);
|
||
fe_sq(t2, t1);
|
||
for (i = 1; i < 5; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t1, t2, t1);
|
||
fe_sq(t2, t1);
|
||
for (i = 1; i < 10; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t2, t2, t1);
|
||
fe_sq(t3, t2);
|
||
for (i = 1; i < 20; ++i) {
|
||
fe_sq(t3, t3);
|
||
}
|
||
fe_mul(t2, t3, t2);
|
||
fe_sq(t2, t2);
|
||
for (i = 1; i < 10; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t1, t2, t1);
|
||
fe_sq(t2, t1);
|
||
for (i = 1; i < 50; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t2, t2, t1);
|
||
fe_sq(t3, t2);
|
||
for (i = 1; i < 100; ++i) {
|
||
fe_sq(t3, t3);
|
||
}
|
||
fe_mul(t2, t3, t2);
|
||
fe_sq(t2, t2);
|
||
for (i = 1; i < 50; ++i) {
|
||
fe_sq(t2, t2);
|
||
}
|
||
fe_mul(t1, t2, t1);
|
||
fe_sq(t1, t1);
|
||
for (i = 1; i < 5; ++i) {
|
||
fe_sq(t1, t1);
|
||
}
|
||
fe_mul(out, t1, t0);
|
||
}
|
||
|
||
/* h = -f
|
||
*
|
||
* Preconditions:
|
||
* |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
|
||
*
|
||
* Postconditions:
|
||
* |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. */
|
||
static void fe_neg(fe h, const fe f) {
|
||
unsigned i;
|
||
for (i = 0; i < 10; i++) {
|
||
h[i] = -f[i];
|
||
}
|
||
}
|
||
|
||
/* Replace (f,g) with (g,g) if b == 1;
|
||
* replace (f,g) with (f,g) if b == 0.
|
||
*
|
||
* Preconditions: b in {0,1}. */
|
||
static void fe_cmov(fe f, const fe g, unsigned b) {
|
||
size_t i;
|
||
b = 0-b;
|
||
for (i = 0; i < 10; i++) {
|
||
int32_t x = f[i] ^ g[i];
|
||
x &= b;
|
||
f[i] ^= x;
|
||
}
|
||
}
|
||
|
||
/* h = 2 * f * f
|
||
* Can overlap h with f.
|
||
*
|
||
* Preconditions:
|
||
* |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
|
||
*
|
||
* Postconditions:
|
||
* |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
|
||
*
|
||
* See fe_mul.c for discussion of implementation strategy. */
|
||
static void fe_sq2(fe h, const fe f) {
|
||
int32_t f0 = f[0];
|
||
int32_t f1 = f[1];
|
||
int32_t f2 = f[2];
|
||
int32_t f3 = f[3];
|
||
int32_t f4 = f[4];
|
||
int32_t f5 = f[5];
|
||
int32_t f6 = f[6];
|
||
int32_t f7 = f[7];
|
||
int32_t f8 = f[8];
|
||
int32_t f9 = f[9];
|
||
int32_t f0_2 = 2 * f0;
|
||
int32_t f1_2 = 2 * f1;
|
||
int32_t f2_2 = 2 * f2;
|
||
int32_t f3_2 = 2 * f3;
|
||
int32_t f4_2 = 2 * f4;
|
||
int32_t f5_2 = 2 * f5;
|
||
int32_t f6_2 = 2 * f6;
|
||
int32_t f7_2 = 2 * f7;
|
||
int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
|
||
int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
|
||
int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
|
||
int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
|
||
int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
|
||
int64_t f0f0 = f0 * (int64_t) f0;
|
||
int64_t f0f1_2 = f0_2 * (int64_t) f1;
|
||
int64_t f0f2_2 = f0_2 * (int64_t) f2;
|
||
int64_t f0f3_2 = f0_2 * (int64_t) f3;
|
||
int64_t f0f4_2 = f0_2 * (int64_t) f4;
|
||
int64_t f0f5_2 = f0_2 * (int64_t) f5;
|
||
int64_t f0f6_2 = f0_2 * (int64_t) f6;
|
||
int64_t f0f7_2 = f0_2 * (int64_t) f7;
|
||
int64_t f0f8_2 = f0_2 * (int64_t) f8;
|
||
int64_t f0f9_2 = f0_2 * (int64_t) f9;
|
||
int64_t f1f1_2 = f1_2 * (int64_t) f1;
|
||
int64_t f1f2_2 = f1_2 * (int64_t) f2;
|
||
int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
|
||
int64_t f1f4_2 = f1_2 * (int64_t) f4;
|
||
int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
|
||
int64_t f1f6_2 = f1_2 * (int64_t) f6;
|
||
int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
|
||
int64_t f1f8_2 = f1_2 * (int64_t) f8;
|
||
int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
|
||
int64_t f2f2 = f2 * (int64_t) f2;
|
||
int64_t f2f3_2 = f2_2 * (int64_t) f3;
|
||
int64_t f2f4_2 = f2_2 * (int64_t) f4;
|
||
int64_t f2f5_2 = f2_2 * (int64_t) f5;
|
||
int64_t f2f6_2 = f2_2 * (int64_t) f6;
|
||
int64_t f2f7_2 = f2_2 * (int64_t) f7;
|
||
int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
|
||
int64_t f2f9_38 = f2 * (int64_t) f9_38;
|
||
int64_t f3f3_2 = f3_2 * (int64_t) f3;
|
||
int64_t f3f4_2 = f3_2 * (int64_t) f4;
|
||
int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
|
||
int64_t f3f6_2 = f3_2 * (int64_t) f6;
|
||
int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
|
||
int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
|
||
int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
|
||
int64_t f4f4 = f4 * (int64_t) f4;
|
||
int64_t f4f5_2 = f4_2 * (int64_t) f5;
|
||
int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
|
||
int64_t f4f7_38 = f4 * (int64_t) f7_38;
|
||
int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
|
||
int64_t f4f9_38 = f4 * (int64_t) f9_38;
|
||
int64_t f5f5_38 = f5 * (int64_t) f5_38;
|
||
int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
|
||
int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
|
||
int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
|
||
int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
|
||
int64_t f6f6_19 = f6 * (int64_t) f6_19;
|
||
int64_t f6f7_38 = f6 * (int64_t) f7_38;
|
||
int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
|
||
int64_t f6f9_38 = f6 * (int64_t) f9_38;
|
||
int64_t f7f7_38 = f7 * (int64_t) f7_38;
|
||
int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
|
||
int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
|
||
int64_t f8f8_19 = f8 * (int64_t) f8_19;
|
||
int64_t f8f9_38 = f8 * (int64_t) f9_38;
|
||
int64_t f9f9_38 = f9 * (int64_t) f9_38;
|
||
int64_t h0 = f0f0 +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
|
||
int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
|
||
int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
|
||
int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
|
||
int64_t h4 = f0f4_2+f1f3_4 +f2f2 +f5f9_76+f6f8_38+f7f7_38;
|
||
int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
|
||
int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
|
||
int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
|
||
int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4 +f9f9_38;
|
||
int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;
|
||
int64_t carry0;
|
||
int64_t carry1;
|
||
int64_t carry2;
|
||
int64_t carry3;
|
||
int64_t carry4;
|
||
int64_t carry5;
|
||
int64_t carry6;
|
||
int64_t carry7;
|
||
int64_t carry8;
|
||
int64_t carry9;
|
||
|
||
h0 += h0;
|
||
h1 += h1;
|
||
h2 += h2;
|
||
h3 += h3;
|
||
h4 += h4;
|
||
h5 += h5;
|
||
h6 += h6;
|
||
h7 += h7;
|
||
h8 += h8;
|
||
h9 += h9;
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
|
||
carry1 = (h1 + (int64_t) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
|
||
carry5 = (h5 + (int64_t) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
|
||
|
||
carry2 = (h2 + (int64_t) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
|
||
carry6 = (h6 + (int64_t) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
|
||
|
||
carry3 = (h3 + (int64_t) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
|
||
carry7 = (h7 + (int64_t) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
|
||
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
carry8 = (h8 + (int64_t) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
|
||
|
||
carry9 = (h9 + (int64_t) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
|
||
h[0] = h0;
|
||
h[1] = h1;
|
||
h[2] = h2;
|
||
h[3] = h3;
|
||
h[4] = h4;
|
||
h[5] = h5;
|
||
h[6] = h6;
|
||
h[7] = h7;
|
||
h[8] = h8;
|
||
h[9] = h9;
|
||
}
|
||
|
||
/* ge means group element.
|
||
|
||
* Here the group is the set of pairs (x,y) of field elements (see fe.h)
|
||
* satisfying -x^2 + y^2 = 1 + d x^2y^2
|
||
* where d = -121665/121666.
|
||
*
|
||
* Representations:
|
||
* ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
|
||
* ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
|
||
* ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
|
||
* ge_precomp (Duif): (y+x,y-x,2dxy) */
|
||
|
||
typedef struct {
|
||
fe X;
|
||
fe Y;
|
||
fe Z;
|
||
} ge_p2;
|
||
|
||
typedef struct {
|
||
fe X;
|
||
fe Y;
|
||
fe Z;
|
||
fe T;
|
||
} ge_p3;
|
||
|
||
typedef struct {
|
||
fe X;
|
||
fe Y;
|
||
fe Z;
|
||
fe T;
|
||
} ge_p1p1;
|
||
|
||
typedef struct {
|
||
fe yplusx;
|
||
fe yminusx;
|
||
fe xy2d;
|
||
} ge_precomp;
|
||
|
||
typedef struct {
|
||
fe YplusX;
|
||
fe YminusX;
|
||
fe Z;
|
||
fe T2d;
|
||
} ge_cached;
|
||
|
||
static void ge_p3_0(ge_p3 *h) {
|
||
fe_0(h->X);
|
||
fe_1(h->Y);
|
||
fe_1(h->Z);
|
||
fe_0(h->T);
|
||
}
|
||
|
||
static void ge_precomp_0(ge_precomp *h) {
|
||
fe_1(h->yplusx);
|
||
fe_1(h->yminusx);
|
||
fe_0(h->xy2d);
|
||
}
|
||
|
||
/* r = p */
|
||
static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) {
|
||
fe_copy(r->X, p->X);
|
||
fe_copy(r->Y, p->Y);
|
||
fe_copy(r->Z, p->Z);
|
||
}
|
||
|
||
/* r = p */
|
||
static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) {
|
||
fe_mul(r->X, p->X, p->T);
|
||
fe_mul(r->Y, p->Y, p->Z);
|
||
fe_mul(r->Z, p->Z, p->T);
|
||
}
|
||
|
||
/* r = p */
|
||
static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) {
|
||
fe_mul(r->X, p->X, p->T);
|
||
fe_mul(r->Y, p->Y, p->Z);
|
||
fe_mul(r->Z, p->Z, p->T);
|
||
fe_mul(r->T, p->X, p->Y);
|
||
}
|
||
|
||
/* r = 2 * p */
|
||
static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) {
|
||
fe t0;
|
||
|
||
fe_sq(r->X, p->X);
|
||
fe_sq(r->Z, p->Y);
|
||
fe_sq2(r->T, p->Z);
|
||
fe_add(r->Y, p->X, p->Y);
|
||
fe_sq(t0, r->Y);
|
||
fe_add(r->Y, r->Z, r->X);
|
||
fe_sub(r->Z, r->Z, r->X);
|
||
fe_sub(r->X, t0, r->Y);
|
||
fe_sub(r->T, r->T, r->Z);
|
||
}
|
||
|
||
/* r = 2 * p */
|
||
static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) {
|
||
ge_p2 q;
|
||
ge_p3_to_p2(&q, p);
|
||
ge_p2_dbl(r, &q);
|
||
}
|
||
|
||
/* r = p + q */
|
||
static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
|
||
fe t0;
|
||
|
||
fe_add(r->X, p->Y, p->X);
|
||
fe_sub(r->Y, p->Y, p->X);
|
||
fe_mul(r->Z, r->X, q->yplusx);
|
||
fe_mul(r->Y, r->Y, q->yminusx);
|
||
fe_mul(r->T, q->xy2d, p->T);
|
||
fe_add(t0, p->Z, p->Z);
|
||
fe_sub(r->X, r->Z, r->Y);
|
||
fe_add(r->Y, r->Z, r->Y);
|
||
fe_add(r->Z, t0, r->T);
|
||
fe_sub(r->T, t0, r->T);
|
||
}
|
||
|
||
static uint8_t equal(signed char b, signed char c) {
|
||
uint8_t ub = b;
|
||
uint8_t uc = c;
|
||
uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
|
||
uint32_t y = x; /* 0: yes; 1..255: no */
|
||
y -= 1; /* 4294967295: yes; 0..254: no */
|
||
y >>= 31; /* 1: yes; 0: no */
|
||
return y;
|
||
}
|
||
|
||
static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) {
|
||
fe_cmov(t->yplusx, u->yplusx, b);
|
||
fe_cmov(t->yminusx, u->yminusx, b);
|
||
fe_cmov(t->xy2d, u->xy2d, b);
|
||
}
|
||
|
||
#if defined(OPENSSL_SMALL)
|
||
|
||
/* This block of code replaces the standard base-point table with a much smaller
|
||
* one. The standard table is 30,720 bytes while this one is just 960.
|
||
*
|
||
* This table contains 15 pairs of group elements, (x, y), where each field
|
||
* element is serialised with |fe_tobytes|. If |i| is the index of the group
|
||
* element then consider i+1 as a four-bit number: (i₀, i₁, i₂, i₃) (where i₀
|
||
* is the most significant bit). The value of the group element is then:
|
||
* (i₀×2^192 + i₁×2^128 + i₂×2^64 + i₃)G, where G is the generator. */
|
||
static const uint8_t k25519SmallPrecomp[15 * 2 * 32] = {
|
||
0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95,
|
||
0x60, 0xc7, 0x2c, 0x69, 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0,
|
||
0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21, 0x58, 0x66, 0x66, 0x66,
|
||
0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
|
||
0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
|
||
0x66, 0x66, 0x66, 0x66, 0x02, 0xa2, 0xed, 0xf4, 0x8f, 0x6b, 0x0b, 0x3e,
|
||
0xeb, 0x35, 0x1a, 0xd5, 0x7e, 0xdb, 0x78, 0x00, 0x96, 0x8a, 0xa0, 0xb4,
|
||
0xcf, 0x60, 0x4b, 0xd4, 0xd5, 0xf9, 0x2d, 0xbf, 0x88, 0xbd, 0x22, 0x62,
|
||
0x13, 0x53, 0xe4, 0x82, 0x57, 0xfa, 0x1e, 0x8f, 0x06, 0x2b, 0x90, 0xba,
|
||
0x08, 0xb6, 0x10, 0x54, 0x4f, 0x7c, 0x1b, 0x26, 0xed, 0xda, 0x6b, 0xdd,
|
||
0x25, 0xd0, 0x4e, 0xea, 0x42, 0xbb, 0x25, 0x03, 0xa2, 0xfb, 0xcc, 0x61,
|
||
0x67, 0x06, 0x70, 0x1a, 0xc4, 0x78, 0x3a, 0xff, 0x32, 0x62, 0xdd, 0x2c,
|
||
0xab, 0x50, 0x19, 0x3b, 0xf2, 0x9b, 0x7d, 0xb8, 0xfd, 0x4f, 0x29, 0x9c,
|
||
0xa7, 0x91, 0xba, 0x0e, 0x46, 0x5e, 0x51, 0xfe, 0x1d, 0xbf, 0xe5, 0xe5,
|
||
0x9b, 0x95, 0x0d, 0x67, 0xf8, 0xd1, 0xb5, 0x5a, 0xa1, 0x93, 0x2c, 0xc3,
|
||
0xde, 0x0e, 0x97, 0x85, 0x2d, 0x7f, 0xea, 0xab, 0x3e, 0x47, 0x30, 0x18,
|
||
0x24, 0xe8, 0xb7, 0x60, 0xae, 0x47, 0x80, 0xfc, 0xe5, 0x23, 0xe7, 0xc2,
|
||
0xc9, 0x85, 0xe6, 0x98, 0xa0, 0x29, 0x4e, 0xe1, 0x84, 0x39, 0x2d, 0x95,
|
||
0x2c, 0xf3, 0x45, 0x3c, 0xff, 0xaf, 0x27, 0x4c, 0x6b, 0xa6, 0xf5, 0x4b,
|
||
0x11, 0xbd, 0xba, 0x5b, 0x9e, 0xc4, 0xa4, 0x51, 0x1e, 0xbe, 0xd0, 0x90,
|
||
0x3a, 0x9c, 0xc2, 0x26, 0xb6, 0x1e, 0xf1, 0x95, 0x7d, 0xc8, 0x6d, 0x52,
|
||
0xe6, 0x99, 0x2c, 0x5f, 0x9a, 0x96, 0x0c, 0x68, 0x29, 0xfd, 0xe2, 0xfb,
|
||
0xe6, 0xbc, 0xec, 0x31, 0x08, 0xec, 0xe6, 0xb0, 0x53, 0x60, 0xc3, 0x8c,
|
||
0xbe, 0xc1, 0xb3, 0x8a, 0x8f, 0xe4, 0x88, 0x2b, 0x55, 0xe5, 0x64, 0x6e,
|
||
0x9b, 0xd0, 0xaf, 0x7b, 0x64, 0x2a, 0x35, 0x25, 0x10, 0x52, 0xc5, 0x9e,
|
||
0x58, 0x11, 0x39, 0x36, 0x45, 0x51, 0xb8, 0x39, 0x93, 0xfc, 0x9d, 0x6a,
|
||
0xbe, 0x58, 0xcb, 0xa4, 0x0f, 0x51, 0x3c, 0x38, 0x05, 0xca, 0xab, 0x43,
|
||
0x63, 0x0e, 0xf3, 0x8b, 0x41, 0xa6, 0xf8, 0x9b, 0x53, 0x70, 0x80, 0x53,
|
||
0x86, 0x5e, 0x8f, 0xe3, 0xc3, 0x0d, 0x18, 0xc8, 0x4b, 0x34, 0x1f, 0xd8,
|
||
0x1d, 0xbc, 0xf2, 0x6d, 0x34, 0x3a, 0xbe, 0xdf, 0xd9, 0xf6, 0xf3, 0x89,
|
||
0xa1, 0xe1, 0x94, 0x9f, 0x5d, 0x4c, 0x5d, 0xe9, 0xa1, 0x49, 0x92, 0xef,
|
||
0x0e, 0x53, 0x81, 0x89, 0x58, 0x87, 0xa6, 0x37, 0xf1, 0xdd, 0x62, 0x60,
|
||
0x63, 0x5a, 0x9d, 0x1b, 0x8c, 0xc6, 0x7d, 0x52, 0xea, 0x70, 0x09, 0x6a,
|
||
0xe1, 0x32, 0xf3, 0x73, 0x21, 0x1f, 0x07, 0x7b, 0x7c, 0x9b, 0x49, 0xd8,
|
||
0xc0, 0xf3, 0x25, 0x72, 0x6f, 0x9d, 0xed, 0x31, 0x67, 0x36, 0x36, 0x54,
|
||
0x40, 0x92, 0x71, 0xe6, 0x11, 0x28, 0x11, 0xad, 0x93, 0x32, 0x85, 0x7b,
|
||
0x3e, 0xb7, 0x3b, 0x49, 0x13, 0x1c, 0x07, 0xb0, 0x2e, 0x93, 0xaa, 0xfd,
|
||
0xfd, 0x28, 0x47, 0x3d, 0x8d, 0xd2, 0xda, 0xc7, 0x44, 0xd6, 0x7a, 0xdb,
|
||
0x26, 0x7d, 0x1d, 0xb8, 0xe1, 0xde, 0x9d, 0x7a, 0x7d, 0x17, 0x7e, 0x1c,
|
||
0x37, 0x04, 0x8d, 0x2d, 0x7c, 0x5e, 0x18, 0x38, 0x1e, 0xaf, 0xc7, 0x1b,
|
||
0x33, 0x48, 0x31, 0x00, 0x59, 0xf6, 0xf2, 0xca, 0x0f, 0x27, 0x1b, 0x63,
|
||
0x12, 0x7e, 0x02, 0x1d, 0x49, 0xc0, 0x5d, 0x79, 0x87, 0xef, 0x5e, 0x7a,
|
||
0x2f, 0x1f, 0x66, 0x55, 0xd8, 0x09, 0xd9, 0x61, 0x38, 0x68, 0xb0, 0x07,
|
||
0xa3, 0xfc, 0xcc, 0x85, 0x10, 0x7f, 0x4c, 0x65, 0x65, 0xb3, 0xfa, 0xfa,
|
||
0xa5, 0x53, 0x6f, 0xdb, 0x74, 0x4c, 0x56, 0x46, 0x03, 0xe2, 0xd5, 0x7a,
|
||
0x29, 0x1c, 0xc6, 0x02, 0xbc, 0x59, 0xf2, 0x04, 0x75, 0x63, 0xc0, 0x84,
|
||
0x2f, 0x60, 0x1c, 0x67, 0x76, 0xfd, 0x63, 0x86, 0xf3, 0xfa, 0xbf, 0xdc,
|
||
0xd2, 0x2d, 0x90, 0x91, 0xbd, 0x33, 0xa9, 0xe5, 0x66, 0x0c, 0xda, 0x42,
|
||
0x27, 0xca, 0xf4, 0x66, 0xc2, 0xec, 0x92, 0x14, 0x57, 0x06, 0x63, 0xd0,
|
||
0x4d, 0x15, 0x06, 0xeb, 0x69, 0x58, 0x4f, 0x77, 0xc5, 0x8b, 0xc7, 0xf0,
|
||
0x8e, 0xed, 0x64, 0xa0, 0xb3, 0x3c, 0x66, 0x71, 0xc6, 0x2d, 0xda, 0x0a,
|
||
0x0d, 0xfe, 0x70, 0x27, 0x64, 0xf8, 0x27, 0xfa, 0xf6, 0x5f, 0x30, 0xa5,
|
||
0x0d, 0x6c, 0xda, 0xf2, 0x62, 0x5e, 0x78, 0x47, 0xd3, 0x66, 0x00, 0x1c,
|
||
0xfd, 0x56, 0x1f, 0x5d, 0x3f, 0x6f, 0xf4, 0x4c, 0xd8, 0xfd, 0x0e, 0x27,
|
||
0xc9, 0x5c, 0x2b, 0xbc, 0xc0, 0xa4, 0xe7, 0x23, 0x29, 0x02, 0x9f, 0x31,
|
||
0xd6, 0xe9, 0xd7, 0x96, 0xf4, 0xe0, 0x5e, 0x0b, 0x0e, 0x13, 0xee, 0x3c,
|
||
0x09, 0xed, 0xf2, 0x3d, 0x76, 0x91, 0xc3, 0xa4, 0x97, 0xae, 0xd4, 0x87,
|
||
0xd0, 0x5d, 0xf6, 0x18, 0x47, 0x1f, 0x1d, 0x67, 0xf2, 0xcf, 0x63, 0xa0,
|
||
0x91, 0x27, 0xf8, 0x93, 0x45, 0x75, 0x23, 0x3f, 0xd1, 0xf1, 0xad, 0x23,
|
||
0xdd, 0x64, 0x93, 0x96, 0x41, 0x70, 0x7f, 0xf7, 0xf5, 0xa9, 0x89, 0xa2,
|
||
0x34, 0xb0, 0x8d, 0x1b, 0xae, 0x19, 0x15, 0x49, 0x58, 0x23, 0x6d, 0x87,
|
||
0x15, 0x4f, 0x81, 0x76, 0xfb, 0x23, 0xb5, 0xea, 0xcf, 0xac, 0x54, 0x8d,
|
||
0x4e, 0x42, 0x2f, 0xeb, 0x0f, 0x63, 0xdb, 0x68, 0x37, 0xa8, 0xcf, 0x8b,
|
||
0xab, 0xf5, 0xa4, 0x6e, 0x96, 0x2a, 0xb2, 0xd6, 0xbe, 0x9e, 0xbd, 0x0d,
|
||
0xb4, 0x42, 0xa9, 0xcf, 0x01, 0x83, 0x8a, 0x17, 0x47, 0x76, 0xc4, 0xc6,
|
||
0x83, 0x04, 0x95, 0x0b, 0xfc, 0x11, 0xc9, 0x62, 0xb8, 0x0c, 0x76, 0x84,
|
||
0xd9, 0xb9, 0x37, 0xfa, 0xfc, 0x7c, 0xc2, 0x6d, 0x58, 0x3e, 0xb3, 0x04,
|
||
0xbb, 0x8c, 0x8f, 0x48, 0xbc, 0x91, 0x27, 0xcc, 0xf9, 0xb7, 0x22, 0x19,
|
||
0x83, 0x2e, 0x09, 0xb5, 0x72, 0xd9, 0x54, 0x1c, 0x4d, 0xa1, 0xea, 0x0b,
|
||
0xf1, 0xc6, 0x08, 0x72, 0x46, 0x87, 0x7a, 0x6e, 0x80, 0x56, 0x0a, 0x8a,
|
||
0xc0, 0xdd, 0x11, 0x6b, 0xd6, 0xdd, 0x47, 0xdf, 0x10, 0xd9, 0xd8, 0xea,
|
||
0x7c, 0xb0, 0x8f, 0x03, 0x00, 0x2e, 0xc1, 0x8f, 0x44, 0xa8, 0xd3, 0x30,
|
||
0x06, 0x89, 0xa2, 0xf9, 0x34, 0xad, 0xdc, 0x03, 0x85, 0xed, 0x51, 0xa7,
|
||
0x82, 0x9c, 0xe7, 0x5d, 0x52, 0x93, 0x0c, 0x32, 0x9a, 0x5b, 0xe1, 0xaa,
|
||
0xca, 0xb8, 0x02, 0x6d, 0x3a, 0xd4, 0xb1, 0x3a, 0xf0, 0x5f, 0xbe, 0xb5,
|
||
0x0d, 0x10, 0x6b, 0x38, 0x32, 0xac, 0x76, 0x80, 0xbd, 0xca, 0x94, 0x71,
|
||
0x7a, 0xf2, 0xc9, 0x35, 0x2a, 0xde, 0x9f, 0x42, 0x49, 0x18, 0x01, 0xab,
|
||
0xbc, 0xef, 0x7c, 0x64, 0x3f, 0x58, 0x3d, 0x92, 0x59, 0xdb, 0x13, 0xdb,
|
||
0x58, 0x6e, 0x0a, 0xe0, 0xb7, 0x91, 0x4a, 0x08, 0x20, 0xd6, 0x2e, 0x3c,
|
||
0x45, 0xc9, 0x8b, 0x17, 0x79, 0xe7, 0xc7, 0x90, 0x99, 0x3a, 0x18, 0x25,
|
||
};
|
||
|
||
static void ge_scalarmult_base(ge_p3 *h, const uint8_t a[32]) {
|
||
/* k25519SmallPrecomp is first expanded into matching |ge_precomp|
|
||
* elements. */
|
||
ge_precomp multiples[15];
|
||
|
||
unsigned i;
|
||
for (i = 0; i < 15; i++) {
|
||
const uint8_t *bytes = &k25519SmallPrecomp[i*(2 * 32)];
|
||
fe x, y;
|
||
fe_frombytes(x, bytes);
|
||
fe_frombytes(y, bytes + 32);
|
||
|
||
ge_precomp *out = &multiples[i];
|
||
fe_add(out->yplusx, y, x);
|
||
fe_sub(out->yminusx, y, x);
|
||
fe_mul(out->xy2d, x, y);
|
||
fe_mul(out->xy2d, out->xy2d, d2);
|
||
}
|
||
|
||
/* See the comment above |k25519SmallPrecomp| about the structure of the
|
||
* precomputed elements. This loop does 64 additions and 64 doublings to
|
||
* calculate the result. */
|
||
ge_p3_0(h);
|
||
|
||
for (i = 63; i < 64; i--) {
|
||
unsigned j;
|
||
signed char index = 0;
|
||
|
||
for (j = 0; j < 4; j++) {
|
||
const uint8_t bit = 1 & (a[(8 * j) + (i / 8)] >> (i & 7));
|
||
index |= (bit << j);
|
||
}
|
||
|
||
ge_precomp e;
|
||
ge_precomp_0(&e);
|
||
|
||
for (j = 1; j < 16; j++) {
|
||
cmov(&e, &multiples[j-1], equal(index, j));
|
||
}
|
||
|
||
ge_cached cached;
|
||
ge_p1p1 r;
|
||
ge_p3_to_cached(&cached, h);
|
||
ge_add(&r, h, &cached);
|
||
ge_p1p1_to_p3(h, &r);
|
||
|
||
ge_madd(&r, h, &e);
|
||
ge_p1p1_to_p3(h, &r);
|
||
}
|
||
}
|
||
|
||
#else
|
||
|
||
/* k25519Precomp[i][j] = (j+1)*256^i*B */
|
||
static const ge_precomp k25519Precomp[32][8] = {
|
||
{
|
||
{
|
||
{25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
|
||
27544626, -11754271, -6079156, 2047605},
|
||
{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
|
||
5043384, 19500929, -15469378},
|
||
{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
|
||
29287919, 11864899, -24514362, -4438546},
|
||
},
|
||
{
|
||
{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
|
||
-11717903, -3814571, -358445, -10211303},
|
||
{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
|
||
-15616551, 11189268, -26829678, -5319081},
|
||
{26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
|
||
-3128826, -9541118, -15472047, -4166697},
|
||
},
|
||
{
|
||
{15636291, -9688557, 24204773, -7912398, 616977, -16685262,
|
||
27787600, -14772189, 28944400, -1550024},
|
||
{16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
|
||
16354577, -11775962, 7689662, 11199574},
|
||
{30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
|
||
7512774, 10017326, -17749093, -9920357},
|
||
},
|
||
{
|
||
{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
|
||
-28926210, 15006023, 3284568, -6276540},
|
||
{23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
|
||
7464579, 9656445, 13059162, 10374397},
|
||
{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
|
||
-3839045, -641708, -101325},
|
||
},
|
||
{
|
||
{10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
|
||
32867885, 14515107, -15438304, 10819380},
|
||
{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
|
||
12483688, -12668491, 5581306},
|
||
{19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
|
||
28542350, 13850243, -23678021, -15815942},
|
||
},
|
||
{
|
||
{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
|
||
-19188627, -15224819, -9818940, -12085777},
|
||
{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
|
||
-15689887, 1762328, 14866737},
|
||
{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
|
||
-28236412, 3959421, 27914454, 4383652},
|
||
},
|
||
{
|
||
{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
|
||
5230134, -23952439, -15175766},
|
||
{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
|
||
20654025, 16520125, 30598449, 7715701},
|
||
{28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
|
||
-31400660, 1370708, 29794553, -1409300},
|
||
},
|
||
{
|
||
{14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
|
||
30290735, 10876454, -33154098, 2381726},
|
||
{-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
|
||
-6236617, 3696005, -32300832, 15351955},
|
||
{27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
|
||
-32961401, -2970515, 29551813, 10109425},
|
||
},
|
||
},
|
||
{
|
||
{
|
||
{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
|
||
-2378284, -1627556, 10092783, -4764171},
|
||
{27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
|
||
-32508754, 12005538, -17810127, 12803510},
|
||
{17228999, -15661624, -1233527, 300140, -1224870, -11714777,
|
||
30364213, -9038194, 18016357, 4397660},
|
||
},
|
||
{
|
||
{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
|
||
-26619106, 14544525, -17477504, 982639},
|
||
{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
|
||
-4120128, -21047696, 9934963},
|
||
{5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
|
||
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|
||
{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
|
||
-3763210, 26224235, -3297458},
|
||
{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
|
||
21728352, 9493610, 18620611, -16428628},
|
||
{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
|
||
-5269471, -9725556, -30701573, -16479657},
|
||
},
|
||
{
|
||
{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
|
||
12248509, -5240639, 13735342, 1934062},
|
||
{25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
|
||
-15136294, -3765346, -21277997, 5473616},
|
||
{31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
|
||
-7125085, 12469656, 29111212, -5451014},
|
||
},
|
||
{
|
||
{24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
|
||
24367466, 6388839, -10295587, 452383},
|
||
{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
|
||
-24236251, -5915248, 15766062, 8407814},
|
||
{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
|
||
-8917023, -4388953, -8067909, 2276718},
|
||
},
|
||
{
|
||
{30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
|
||
-23827587, 5096219, 22740376, -7303417},
|
||
{2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
|
||
24051124, 13742383, -15637599, 13295222},
|
||
{33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
|
||
-17720195, -4612972, -4451357, -14669444},
|
||
},
|
||
{
|
||
{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
|
||
-2469266, -4141880, 7770569, 9620597},
|
||
{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
|
||
-1694323, -33502340, -14767970},
|
||
{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
|
||
1220118, 30494170, -11440799},
|
||
},
|
||
{
|
||
{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
|
||
-26739026, 926050, -1684339, -13333647},
|
||
{13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
|
||
9021034, 9078865, 3353509, 4033511},
|
||
{-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
|
||
23161163, 8839127, 27485041, 7356032},
|
||
},
|
||
},
|
||
{
|
||
{
|
||
{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
|
||
2625015, 28431036, -16771834},
|
||
{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
|
||
-22545972, 14150565, 15970762, 4099461},
|
||
{29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
|
||
13617293, -9937143, 11465739, 8317062},
|
||
},
|
||
{
|
||
{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
|
||
14898637, 3848455, 20969334, -5157516},
|
||
{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
|
||
-21610826, -3649888, 11177095, 14989547},
|
||
{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
|
||
13515641, 2581286, -28487508, 9930240},
|
||
},
|
||
{
|
||
{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
|
||
18345767, -13403753, 16291481, -5314038},
|
||
{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
|
||
6957617, 4368891, 9788741},
|
||
{16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
|
||
-21722536, -8613148, 16250552, -11111103},
|
||
},
|
||
{
|
||
{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
|
||
10604807, -30190403, 4782747},
|
||
{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
|
||
-9981571, 4383045, 22546403, 437323},
|
||
{31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
|
||
27343084, 2786261, -30633590, -14097016},
|
||
},
|
||
{
|
||
{-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
|
||
-19690631, 2355319, -19284671, -6114373},
|
||
{15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
|
||
18952177, 15496498, -29380133, 11754228},
|
||
{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
|
||
7141596, 11724556, 22761615, -10134141},
|
||
},
|
||
{
|
||
{16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
|
||
-28741185, -12227393, 32851222, 11717399},
|
||
{11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
|
||
31474879, 3483633, -1193175, -4030831},
|
||
{-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
|
||
33142652, 6546660, -19985279, -3948376},
|
||
},
|
||
{
|
||
{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
|
||
-8537131, -12833048, -30772034, -15486313},
|
||
{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
|
||
-31135347, -16049879, 10928917, 3011958},
|
||
{-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
|
||
-30831056, -12805180, 18008031, 10258577},
|
||
},
|
||
{
|
||
{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
|
||
-1853465, 1367120, 25127874, 6671743},
|
||
{29701166, -14373934, -10878120, 9279288, -17568, 13127210,
|
||
21382910, 11042292, 25838796, 4642684},
|
||
{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
|
||
30468147, -13900640, 18423289, 4177476},
|
||
},
|
||
},
|
||
};
|
||
|
||
static uint8_t negative(signed char b) {
|
||
uint32_t x = b;
|
||
x >>= 31; /* 1: yes; 0: no */
|
||
return x;
|
||
}
|
||
|
||
static void table_select(ge_precomp *t, int pos, signed char b) {
|
||
ge_precomp minust;
|
||
uint8_t bnegative = negative(b);
|
||
uint8_t babs = b - (((-bnegative) & b) << 1);
|
||
|
||
ge_precomp_0(t);
|
||
cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
|
||
cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
|
||
cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
|
||
cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
|
||
cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
|
||
cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
|
||
cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
|
||
cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
|
||
fe_copy(minust.yplusx, t->yminusx);
|
||
fe_copy(minust.yminusx, t->yplusx);
|
||
fe_neg(minust.xy2d, t->xy2d);
|
||
cmov(t, &minust, bnegative);
|
||
}
|
||
|
||
/* h = a * B
|
||
* where a = a[0]+256*a[1]+...+256^31 a[31]
|
||
* B is the Ed25519 base point (x,4/5) with x positive.
|
||
*
|
||
* Preconditions:
|
||
* a[31] <= 127 */
|
||
static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a) {
|
||
signed char e[64];
|
||
signed char carry;
|
||
ge_p1p1 r;
|
||
ge_p2 s;
|
||
ge_precomp t;
|
||
int i;
|
||
|
||
for (i = 0; i < 32; ++i) {
|
||
e[2 * i + 0] = (a[i] >> 0) & 15;
|
||
e[2 * i + 1] = (a[i] >> 4) & 15;
|
||
}
|
||
/* each e[i] is between 0 and 15 */
|
||
/* e[63] is between 0 and 7 */
|
||
|
||
carry = 0;
|
||
for (i = 0; i < 63; ++i) {
|
||
e[i] += carry;
|
||
carry = e[i] + 8;
|
||
carry >>= 4;
|
||
e[i] -= carry << 4;
|
||
}
|
||
e[63] += carry;
|
||
/* each e[i] is between -8 and 8 */
|
||
|
||
ge_p3_0(h);
|
||
for (i = 1; i < 64; i += 2) {
|
||
table_select(&t, i / 2, e[i]);
|
||
ge_madd(&r, h, &t);
|
||
ge_p1p1_to_p3(h, &r);
|
||
}
|
||
|
||
ge_p3_dbl(&r, h);
|
||
ge_p1p1_to_p2(&s, &r);
|
||
ge_p2_dbl(&r, &s);
|
||
ge_p1p1_to_p2(&s, &r);
|
||
ge_p2_dbl(&r, &s);
|
||
ge_p1p1_to_p2(&s, &r);
|
||
ge_p2_dbl(&r, &s);
|
||
ge_p1p1_to_p3(h, &r);
|
||
|
||
for (i = 0; i < 64; i += 2) {
|
||
table_select(&t, i / 2, e[i]);
|
||
ge_madd(&r, h, &t);
|
||
ge_p1p1_to_p3(h, &r);
|
||
}
|
||
}
|
||
|
||
#endif
|
||
|
||
#if defined(OPENSSL_X25519_X86_64)
|
||
|
||
static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
|
||
const uint8_t point[32]) {
|
||
x25519_x86_64(out, scalar, point);
|
||
}
|
||
|
||
#else
|
||
|
||
/* Replace (f,g) with (g,f) if b == 1;
|
||
* replace (f,g) with (f,g) if b == 0.
|
||
*
|
||
* Preconditions: b in {0,1}. */
|
||
static void fe_cswap(fe f, fe g, unsigned int b) {
|
||
size_t i;
|
||
b = 0-b;
|
||
for (i = 0; i < 10; i++) {
|
||
int32_t x = f[i] ^ g[i];
|
||
x &= b;
|
||
f[i] ^= x;
|
||
g[i] ^= x;
|
||
}
|
||
}
|
||
|
||
/* h = f * 121666
|
||
* Can overlap h with f.
|
||
*
|
||
* Preconditions:
|
||
* |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
|
||
*
|
||
* Postconditions:
|
||
* |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. */
|
||
static void fe_mul121666(fe h, fe f) {
|
||
int32_t f0 = f[0];
|
||
int32_t f1 = f[1];
|
||
int32_t f2 = f[2];
|
||
int32_t f3 = f[3];
|
||
int32_t f4 = f[4];
|
||
int32_t f5 = f[5];
|
||
int32_t f6 = f[6];
|
||
int32_t f7 = f[7];
|
||
int32_t f8 = f[8];
|
||
int32_t f9 = f[9];
|
||
int64_t h0 = f0 * (int64_t) 121666;
|
||
int64_t h1 = f1 * (int64_t) 121666;
|
||
int64_t h2 = f2 * (int64_t) 121666;
|
||
int64_t h3 = f3 * (int64_t) 121666;
|
||
int64_t h4 = f4 * (int64_t) 121666;
|
||
int64_t h5 = f5 * (int64_t) 121666;
|
||
int64_t h6 = f6 * (int64_t) 121666;
|
||
int64_t h7 = f7 * (int64_t) 121666;
|
||
int64_t h8 = f8 * (int64_t) 121666;
|
||
int64_t h9 = f9 * (int64_t) 121666;
|
||
int64_t carry0;
|
||
int64_t carry1;
|
||
int64_t carry2;
|
||
int64_t carry3;
|
||
int64_t carry4;
|
||
int64_t carry5;
|
||
int64_t carry6;
|
||
int64_t carry7;
|
||
int64_t carry8;
|
||
int64_t carry9;
|
||
|
||
carry9 = (h9 + (int64_t) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
|
||
carry1 = (h1 + (int64_t) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
|
||
carry3 = (h3 + (int64_t) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
|
||
carry5 = (h5 + (int64_t) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
|
||
carry7 = (h7 + (int64_t) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
|
||
|
||
carry0 = (h0 + (int64_t) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
|
||
carry2 = (h2 + (int64_t) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
|
||
carry4 = (h4 + (int64_t) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
|
||
carry6 = (h6 + (int64_t) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
|
||
carry8 = (h8 + (int64_t) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
|
||
|
||
h[0] = h0;
|
||
h[1] = h1;
|
||
h[2] = h2;
|
||
h[3] = h3;
|
||
h[4] = h4;
|
||
h[5] = h5;
|
||
h[6] = h6;
|
||
h[7] = h7;
|
||
h[8] = h8;
|
||
h[9] = h9;
|
||
}
|
||
|
||
static void x25519_scalar_mult_generic(uint8_t out[32],
|
||
const uint8_t scalar[32],
|
||
const uint8_t point[32]) {
|
||
fe x1, x2, z2, x3, z3, tmp0, tmp1;
|
||
uint8_t e[32];
|
||
unsigned swap = 0;
|
||
int pos;
|
||
|
||
memcpy(e, scalar, 32);
|
||
e[0] &= 248;
|
||
e[31] &= 127;
|
||
e[31] |= 64;
|
||
fe_frombytes(x1, point);
|
||
fe_1(x2);
|
||
fe_0(z2);
|
||
fe_copy(x3, x1);
|
||
fe_1(z3);
|
||
|
||
for (pos = 254; pos >= 0; --pos) {
|
||
unsigned b = 1 & (e[pos / 8] >> (pos & 7));
|
||
swap ^= b;
|
||
fe_cswap(x2, x3, swap);
|
||
fe_cswap(z2, z3, swap);
|
||
swap = b;
|
||
fe_sub(tmp0, x3, z3);
|
||
fe_sub(tmp1, x2, z2);
|
||
fe_add(x2, x2, z2);
|
||
fe_add(z2, x3, z3);
|
||
fe_mul(z3, tmp0, x2);
|
||
fe_mul(z2, z2, tmp1);
|
||
fe_sq(tmp0, tmp1);
|
||
fe_sq(tmp1, x2);
|
||
fe_add(x3, z3, z2);
|
||
fe_sub(z2, z3, z2);
|
||
fe_mul(x2, tmp1, tmp0);
|
||
fe_sub(tmp1, tmp1, tmp0);
|
||
fe_sq(z2, z2);
|
||
fe_mul121666(z3, tmp1);
|
||
fe_sq(x3, x3);
|
||
fe_add(tmp0, tmp0, z3);
|
||
fe_mul(z3, x1, z2);
|
||
fe_mul(z2, tmp1, tmp0);
|
||
}
|
||
fe_cswap(x2, x3, swap);
|
||
fe_cswap(z2, z3, swap);
|
||
|
||
fe_invert(z2, z2);
|
||
fe_mul(x2, x2, z2);
|
||
fe_tobytes(out, x2);
|
||
}
|
||
|
||
static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
|
||
const uint8_t point[32]) {
|
||
x25519_scalar_mult_generic(out, scalar, point);
|
||
}
|
||
|
||
#endif /* OPENSSL_X25519_X86_64 */
|
||
|
||
int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
|
||
const uint8_t peer_public_value[32]) {
|
||
static const uint8_t kZeros[32] = {0};
|
||
x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
|
||
/* The all-zero output results when the input is a point of small order. */
|
||
return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
|
||
}
|
||
|
||
#if defined(OPENSSL_X25519_X86_64)
|
||
|
||
/* When |OPENSSL_X25519_X86_64| is set, base point multiplication is done with
|
||
* the Montgomery ladder because it's faster. Otherwise it's done using the
|
||
* Ed25519 tables. */
|
||
|
||
void X25519_public_from_private(uint8_t out_public_value[32],
|
||
const uint8_t private_key[32]) {
|
||
static const uint8_t kMongomeryBasePoint[32] = {9};
|
||
x25519_scalar_mult(out_public_value, private_key, kMongomeryBasePoint);
|
||
}
|
||
|
||
#else
|
||
|
||
void X25519_public_from_private(uint8_t out_public_value[32],
|
||
const uint8_t private_key[32]) {
|
||
uint8_t e[32];
|
||
ge_p3 A;
|
||
fe zplusy, zminusy, zminusy_inv;
|
||
|
||
memcpy(e, private_key, 32);
|
||
e[0] &= 248;
|
||
e[31] &= 127;
|
||
e[31] |= 64;
|
||
|
||
ge_scalarmult_base(&A, e);
|
||
|
||
/* We only need the u-coordinate of the curve25519 point. The map is
|
||
* u=(y+1)/(1-y). Since y=Y/Z, this gives u=(Z+Y)/(Z-Y). */
|
||
fe_add(zplusy, A.Z, A.Y);
|
||
fe_sub(zminusy, A.Z, A.Y);
|
||
fe_invert(zminusy_inv, zminusy);
|
||
fe_mul(zplusy, zplusy, zminusy_inv);
|
||
fe_tobytes(out_public_value, zplusy);
|
||
}
|
||
|
||
#endif /* OPENSSL_X25519_X86_64 */
|