ref: f11139d4c918802a87730bc14d094670ee4ce572
dir: /libsec/curve25519.c/
/* 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: 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 "os.h"
#include <libsec.h>
typedef vlong felem;
/* Sum two numbers: output += in */
static void fsum(felem *output, felem *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(felem *output, felem *in) {
unsigned i;
for (i = 0; i < 10; ++i) {
output[i] = (in[i] - output[i]);
}
}
/* Multiply a number my a scalar: output = in * scalar */
static void fscalar_product(felem *output, felem *in, felem 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.
*/
static void fproduct(felem *output, felem *in2, felem *in) {
output[0] = in2[0] * in[0];
output[1] = in2[0] * in[1] +
in2[1] * in[0];
output[2] = 2 * in2[1] * in[1] +
in2[0] * in[2] +
in2[2] * in[0];
output[3] = in2[1] * in[2] +
in2[2] * in[1] +
in2[0] * in[3] +
in2[3] * in[0];
output[4] = in2[2] * in[2] +
2 * (in2[1] * in[3] +
in2[3] * in[1]) +
in2[0] * in[4] +
in2[4] * in[0];
output[5] = in2[2] * in[3] +
in2[3] * in[2] +
in2[1] * in[4] +
in2[4] * in[1] +
in2[0] * in[5] +
in2[5] * in[0];
output[6] = 2 * (in2[3] * in[3] +
in2[1] * in[5] +
in2[5] * in[1]) +
in2[2] * in[4] +
in2[4] * in[2] +
in2[0] * in[6] +
in2[6] * in[0];
output[7] = in2[3] * in[4] +
in2[4] * in[3] +
in2[2] * in[5] +
in2[5] * in[2] +
in2[1] * in[6] +
in2[6] * in[1] +
in2[0] * in[7] +
in2[7] * in[0];
output[8] = in2[4] * in[4] +
2 * (in2[3] * in[5] +
in2[5] * in[3] +
in2[1] * in[7] +
in2[7] * in[1]) +
in2[2] * in[6] +
in2[6] * in[2] +
in2[0] * in[8] +
in2[8] * in[0];
output[9] = in2[4] * in[5] +
in2[5] * in[4] +
in2[3] * in[6] +
in2[6] * in[3] +
in2[2] * in[7] +
in2[7] * in[2] +
in2[1] * in[8] +
in2[8] * in[1] +
in2[0] * in[9] +
in2[9] * in[0];
output[10] = 2 * (in2[5] * in[5] +
in2[3] * in[7] +
in2[7] * in[3] +
in2[1] * in[9] +
in2[9] * in[1]) +
in2[4] * in[6] +
in2[6] * in[4] +
in2[2] * in[8] +
in2[8] * in[2];
output[11] = in2[5] * in[6] +
in2[6] * in[5] +
in2[4] * in[7] +
in2[7] * in[4] +
in2[3] * in[8] +
in2[8] * in[3] +
in2[2] * in[9] +
in2[9] * in[2];
output[12] = in2[6] * in[6] +
2 * (in2[5] * in[7] +
in2[7] * in[5] +
in2[3] * in[9] +
in2[9] * in[3]) +
in2[4] * in[8] +
in2[8] * in[4];
output[13] = in2[6] * in[7] +
in2[7] * in[6] +
in2[5] * in[8] +
in2[8] * in[5] +
in2[4] * in[9] +
in2[9] * in[4];
output[14] = 2 * (in2[7] * in[7] +
in2[5] * in[9] +
in2[9] * in[5]) +
in2[6] * in[8] +
in2[8] * in[6];
output[15] = in2[7] * in[8] +
in2[8] * in[7] +
in2[6] * in[9] +
in2[9] * in[6];
output[16] = in2[8] * in[8] +
2 * (in2[7] * in[9] +
in2[9] * in[7]);
output[17] = in2[8] * in[9] +
in2[9] * in[8];
output[18] = 2 * in2[9] * in[9];
}
/* Reduce a long form to a short form by taking the input mod 2^255 - 19. */
static void freduce_degree(felem *output) {
output[8] += 19 * output[18];
output[7] += 19 * output[17];
output[6] += 19 * output[16];
output[5] += 19 * output[15];
output[4] += 19 * output[14];
output[3] += 19 * output[13];
output[2] += 19 * output[12];
output[1] += 19 * output[11];
output[0] += 19 * output[10];
}
/* Reduce all coefficients of the short form input to be -2**25 <= x <= 2**25
*/
static void freduce_coefficients(felem *output) {
unsigned i;
do {
output[10] = 0;
for (i = 0; i < 10; i += 2) {
felem over = output[i] / 0x2000000l;
felem over2 = (over + ((over >> 63) * 2) + 1) / 2;
output[i+1] += over2;
output[i] -= over2 * 0x4000000l;
over = output[i+1] / 0x2000000;
output[i+2] += over;
output[i+1] -= over * 0x2000000;
}
output[0] += 19 * output[10];
} while (output[10]);
}
/* A helpful wrapper around fproduct: output = in * in2.
*
* output must be distinct to both inputs. The output is reduced degree and
* reduced coefficient.
*/
static void
fmul(felem *output, felem *in, felem *in2) {
felem t[19];
fproduct(t, in, in2);
freduce_degree(t);
freduce_coefficients(t);
memcpy(output, t, sizeof(felem) * 10);
}
static void fsquare_inner(felem *output, felem *in) {
felem tmp;
output[0] = in[0] * in[0];
output[1] = 2 * in[0] * in[1];
output[2] = 2 * (in[1] * in[1] +
in[0] * in[2]);
output[3] = 2 * (in[1] * in[2] +
in[0] * in[3]);
output[4] = in[2] * in[2] +
4 * in[1] * in[3] +
2 * in[0] * in[4];
output[5] = 2 * (in[2] * in[3] +
in[1] * in[4] +
in[0] * in[5]);
output[6] = 2 * (in[3] * in[3] +
in[2] * in[4] +
in[0] * in[6] +
2 * in[1] * in[5]);
output[7] = 2 * (in[3] * in[4] +
in[2] * in[5] +
in[1] * in[6] +
in[0] * in[7]);
tmp = in[1] * in[7] + in[3] * in[5];
output[8] = in[4] * in[4] +
2 * (in[2] * in[6] +
in[0] * in[8] +
2 * tmp);
output[9] = 2 * (in[4] * in[5] +
in[3] * in[6] +
in[2] * in[7] +
in[1] * in[8] +
in[0] * in[9]);
tmp = in[3] * in[7] + in[1] * in[9];
output[10] = 2 * (in[5] * in[5] +
in[4] * in[6] +
in[2] * in[8] +
2 * tmp);
output[11] = 2 * (in[5] * in[6] +
in[4] * in[7] +
in[3] * in[8] +
in[2] * in[9]);
output[12] = in[6] * in[6] +
2 * (in[4] * in[8] +
2 * (in[5] * in[7] +
in[3] * in[9]));
output[13] = 2 * (in[6] * in[7] +
in[5] * in[8] +
in[4] * in[9]);
output[14] = 2 * (in[7] * in[7] +
in[6] * in[8] +
2 * in[5] * in[9]);
output[15] = 2 * (in[7] * in[8] +
in[6] * in[9]);
output[16] = in[8] * in[8] +
4 * in[7] * in[9];
output[17] = 2 * in[8] * in[9];
output[18] = 2 * in[9] * in[9];
}
static void
fsquare(felem *output, felem *in) {
felem t[19];
fsquare_inner(t, in);
freduce_degree(t);
freduce_coefficients(t);
memcpy(output, t, sizeof(felem) * 10);
}
/* Take a little-endian, 32-byte number and expand it into polynomial form */
static void
fexpand(felem *output, uchar *input) {
#define F(n,start,shift,mask) \
output[n] = ((((felem) input[start + 0]) | \
((felem) input[start + 1]) << 8 | \
((felem) input[start + 2]) << 16 | \
((felem) 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
}
/* Take a fully reduced polynomial form number and contract it into a
* little-endian, 32-byte array
*/
static void
fcontract(uchar *output, felem *input) {
int i;
do {
for (i = 0; i < 9; ++i) {
if ((i & 1) == 1) {
while (input[i] < 0) {
input[i] += 0x2000000;
input[i + 1]--;
}
} else {
while (input[i] < 0) {
input[i] += 0x4000000;
input[i + 1]--;
}
}
}
while (input[9] < 0) {
input[9] += 0x2000000;
input[0] -= 19;
}
} while (input[0] < 0);
input[1] <<= 2;
input[2] <<= 3;
input[3] <<= 5;
input[4] <<= 6;
input[6] <<= 1;
input[7] <<= 3;
input[8] <<= 4;
input[9] <<= 6;
#define F(i, s) \
output[s+0] |= input[i] & 0xff; \
output[s+1] = (input[i] >> 8) & 0xff; \
output[s+2] = (input[i] >> 16) & 0xff; \
output[s+3] = (input[i] >> 24) & 0xff;
output[0] = 0;
output[16] = 0;
F(0,0);
F(1,3);
F(2,6);
F(3,9);
F(4,12);
F(5,16);
F(6,19);
F(7,22);
F(8,25);
F(9,28);
#undef F
}
/* Input: Q, Q', Q-Q'
* Output: 2Q, Q+Q'
*
* x2 z3: long form
* x3 z3: long form
* x z: short form, destroyed
* xprime zprime: short form, destroyed
* qmqp: short form, preserved
*/
static void fmonty(felem *x2, felem *z2, /* output 2Q */
felem *x3, felem *z3, /* output Q + Q' */
felem *x, felem *z, /* input Q */
felem *xprime, felem *zprime, /* input Q' */
felem *qmqp /* input Q - Q' */) {
felem origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
zzprime[19], zzzprime[19], xxxprime[19];
memcpy(origx, x, 10 * sizeof(felem));
fsum(x, z);
fdifference(z, origx); // does x - z
memcpy(origxprime, xprime, sizeof(felem) * 10);
fsum(xprime, zprime);
fdifference(zprime, origxprime);
fproduct(xxprime, xprime, z);
fproduct(zzprime, x, zprime);
freduce_degree(xxprime);
freduce_coefficients(xxprime);
freduce_degree(zzprime);
freduce_coefficients(zzprime);
memcpy(origxprime, xxprime, sizeof(felem) * 10);
fsum(xxprime, zzprime);
fdifference(zzprime, origxprime);
fsquare(xxxprime, xxprime);
fsquare(zzzprime, zzprime);
fproduct(zzprime, zzzprime, qmqp);
freduce_degree(zzprime);
freduce_coefficients(zzprime);
memcpy(x3, xxxprime, sizeof(felem) * 10);
memcpy(z3, zzprime, sizeof(felem) * 10);
fsquare(xx, x);
fsquare(zz, z);
fproduct(x2, xx, zz);
freduce_degree(x2);
freduce_coefficients(x2);
fdifference(zz, xx); // does zz = xx - zz
memset(zzz + 10, 0, sizeof(felem) * 9);
fscalar_product(zzz, zz, 121665);
freduce_degree(zzz);
freduce_coefficients(zzz);
fsum(zzz, xx);
fproduct(z2, zz, zzz);
freduce_degree(z2);
freduce_coefficients(z2);
}
/* 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(felem *resultx, felem *resultz, uchar *n, felem *q) {
felem a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
felem *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
felem e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
felem *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
unsigned i, j;
memcpy(nqpqx, q, sizeof(felem) * 10);
for (i = 0; i < 32; ++i) {
uchar byte = n[31 - i];
for (j = 0; j < 8; ++j) {
if (byte & 0x80) {
fmonty(nqpqx2, nqpqz2,
nqx2, nqz2,
nqpqx, nqpqz,
nqx, nqz,
q);
} else {
fmonty(nqx2, nqz2,
nqpqx2, nqpqz2,
nqx, nqz,
nqpqx, nqpqz,
q);
}
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(felem) * 10);
memcpy(resultz, nqz, sizeof(felem) * 10);
}
// -----------------------------------------------------------------------------
// Shamelessly copied from djb's code
// -----------------------------------------------------------------------------
static void
crecip(felem *out, felem *z) {
felem z2[10];
felem z9[10];
felem z11[10];
felem z2_5_0[10];
felem z2_10_0[10];
felem z2_20_0[10];
felem z2_50_0[10];
felem z2_100_0[10];
felem t0[10];
felem 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);
}
void
curve25519(uchar mypublic[32], uchar secret[32], uchar basepoint[32]) {
felem bp[10], x[10], z[10], zmone[10];
fexpand(bp, basepoint);
cmult(x, z, secret, bp);
crecip(zmone, z);
fmul(z, x, zmone);
fcontract(mypublic, z);
}