/*----- Licensing notice --------------------------------------------------*
*
- * This file is part of Catacomb.
+ * This file is part of secnet.
+ * See README for full list of copyright holders.
+ *
+ * secnet is free software; you can redistribute it and/or modify it
+ * under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version d of the License, or
+ * (at your option) any later version.
+ *
+ * secnet is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ * General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * version 3 along with secnet; if not, see
+ * https://www.gnu.org/licenses/gpl.html.
+ *
+ * This file was originally part of Catacomb, but has been automatically
+ * modified for incorporation into secnet: see `import-catacomb-crypto'
+ * for details.
*
* Catacomb is free software; you can redistribute it and/or modify
* it under the terms of the GNU Library General Public License as
/*----- Header files ------------------------------------------------------*/
-#include "config.h"
-
-#include "ct.h"
#include "f25519.h"
/*----- Basic setup -------------------------------------------------------*/
-#if F25519_IMPL == 26
+typedef f25519_piece piece;
+
/* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x
* = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original
* paper.
*/
-typedef int32 piece; typedef int64 dblpiece;
+ typedef int64 dblpiece;
typedef uint32 upiece; typedef uint64 udblpiece;
#define P p26
#define PIECEWD(i) ((i)%2 ? 25 : 26)
#define M26 0x03ffffffu
#define M25 0x01ffffffu
-#define B26 0x04000000u
#define B25 0x02000000u
#define B24 0x01000000u
(w)->P[8] = v##8; (w)->P[9] = v##9; \
} while (0)
-#elif F25519_IMPL == 10
-/* Elements x of GF(2^255 - 19) are represented by 26 signed integers x_i: x
- * = SUM_{0<=i<26} x_i 2^ceil(255i/26); i.e., most pieces are 10 bits wide,
- * except for pieces 5, 10, 15, 20, and 25 which have 9 bits.
- */
-
-typedef int16 piece; typedef int32 dblpiece;
-typedef uint16 upiece; typedef uint32 udblpiece;
-#define P p10
-#define PIECEWD(i) \
- ((i) == 5 || (i) == 10 || (i) == 15 || (i) == 20 || (i) == 25 ? 9 : 10)
-#define NPIECE 26
-
-#define B10 0x0400
-#define B9 0x200
-#define B8 0x100
-#define M10 0x3ff
-#define M9 0x1ff
-
-#endif
-
/*----- Debugging machinery -----------------------------------------------*/
-#if defined(F25519_DEBUG) || defined(TEST_RIG)
+#if defined(F25519_DEBUG)
#include <stdio.h>
void f25519_load(f25519 *z, const octet xv[32])
{
-#if F25519_IMPL == 26
uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4),
xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12),
/* And with that, we're done. */
STASH(z, x);
-
-#elif F25519_IMPL == 10
-
- piece x[NPIECE];
- unsigned i, j, n, wd;
- uint32 a;
- int b, c;
-
- /* First, just get the content out of the buffer. */
- for (i = j = a = n = 0, wd = 10; j < NPIECE; i++) {
- a |= (uint32)xv[i] << n; n += 8;
- if (n >= wd) {
- x[j++] = a&MASK(wd);
- a >>= wd; n -= wd;
- wd = PIECEWD(j);
- }
- }
-
- /* There's a little bit left over from the top byte. Carry it into the low
- * piece.
- */
- x[0] += 19*(int)(a&MASK(n));
-
- /* Next, convert the pieces into a roughly balanced signed representation.
- * If a piece's top bit is set, lend a bit to the next piece over. For
- * x_25, this needs to be carried around, which is a bit fiddly.
- */
- b = x[NPIECE - 1]&B8;
- c = 19&((b >> 3) - (b >> 8));
- x[NPIECE - 1] -= b << 1;
- for (i = NPIECE - 2; i > 0; i--) {
- wd = PIECEWD(i) - 1;
- b = x[i]&BIT(wd);
- x[i + 1] += b >> wd;
- x[i] -= b << 1;
- }
- b = x[0]&B9;
- x[1] += (b >> 9) + (x[0] >> 10);
- x[0] = (x[0]&M10) - (b << 1) + c;
-
- /* And we're done. */
- for (i = 0; i < NPIECE; i++) z->P[i] = x[i];
-
-#endif
}
/* --- @f25519_store@ --- *
void f25519_store(octet zv[32], const f25519 *x)
{
-#if F25519_IMPL == 26
piece PIECES(x), PIECES(y), c, d;
uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7;
STORE32_L(zv + 8, zw2); STORE32_L(zv + 12, zw3);
STORE32_L(zv + 16, zw4); STORE32_L(zv + 20, zw5);
STORE32_L(zv + 24, zw6); STORE32_L(zv + 28, zw7);
-
-#elif F25519_IMPL == 10
-
- piece y[NPIECE], yy[NPIECE], c, d;
- unsigned i, j, n, wd;
- uint32 m, a;
-
- /* Before we do anything, copy the input so we can hack on it. */
- for (i = 0; i < NPIECE; i++) y[i] = x->P[i];
-
- /* First, propagate the carries throughout the pieces.
- *
- * It's worth paying careful attention to the bounds. We assume that we
- * start out with |y_i| <= 2^14. We start by cutting off and reducing the
- * carry c_25 from the topmost piece, y_25. This leaves 0 <= y_25 < 2^9;
- * and we'll have |c_25| <= 2^5. We multiply this by 19 and we'll ad it
- * onto y_0 and propagte the carries: but what bounds can we calculate on
- * y before this?
- *
- * Let o_i = floor(255 i/26). We have Y_i = SUM_{0<=j<i} y_j 2^{o_i}, so
- * y = Y_26. We see, inductively, that |Y_i| < 2^{31+o_{i-1}}: Y_0 = 0;
- * |y_i| <= 2^14; and |Y_{i+1}| = |Y_i + y_i 2^{o_i}| <= |Y_i| + 2^{14+o_i}
- * < 2^{15+o_i}. Then x = Y_25 + 2^246 y_25, and we have better bounds for
- * y_25, so
- *
- * -2^251 < y + 19 c_25 < 2^255 + 2^251
- *
- * Here, the y_i are signed, so we must be cautious about bithacking them.
- *
- * (Rather closer than the 10-piece case above, but still doable in one
- * pass.)
- */
- c = 19*ASR(piece, y[NPIECE - 1], 9);
- y[NPIECE - 1] = (upiece)y[NPIECE - 1]&M9;
- for (i = 0; i < NPIECE; i++) {
- wd = PIECEWD(i);
- y[i] += c;
- c = ASR(piece, y[i], wd);
- y[i] = (upiece)y[i]&MASK(wd);
- }
-
- /* Now the addition or subtraction. */
- m = SIGN(c);
- d = m&1;
-
- d += y[0] + (19 ^ (M10&m));
- yy[0] = d&M10;
- d >>= 10;
- for (i = 1; i < NPIECE; i++) {
- wd = PIECEWD(i);
- d += y[i] + (MASK(wd)&m);
- yy[i] = d&MASK(wd);
- d >>= wd;
- }
-
- /* Choose which value to keep. */
- m = NONZEROP(c) | ~NONZEROP(d - 1);
- for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m);
-
- /* Store the result as an octet string. */
- for (i = j = a = n = 0; i < NPIECE; i++) {
- a |= (upiece)y[i] << n; n += PIECEWD(i);
- while (n >= 8) {
- zv[j++] = a&0xff;
- a >>= 8; n -= 8;
- }
- }
- zv[j++] = a;
-
-#endif
}
/* --- @f25519_set@ --- *
void f25519_add(f25519 *z, const f25519 *x, const f25519 *y)
{
-#if F25519_IMPL == 26
z->P[0] = x->P[0] + y->P[0]; z->P[1] = x->P[1] + y->P[1];
z->P[2] = x->P[2] + y->P[2]; z->P[3] = x->P[3] + y->P[3];
z->P[4] = x->P[4] + y->P[4]; z->P[5] = x->P[5] + y->P[5];
z->P[6] = x->P[6] + y->P[6]; z->P[7] = x->P[7] + y->P[7];
z->P[8] = x->P[8] + y->P[8]; z->P[9] = x->P[9] + y->P[9];
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i];
-#endif
}
/* --- @f25519_sub@ --- *
void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y)
{
-#if F25519_IMPL == 26
z->P[0] = x->P[0] - y->P[0]; z->P[1] = x->P[1] - y->P[1];
z->P[2] = x->P[2] - y->P[2]; z->P[3] = x->P[3] - y->P[3];
z->P[4] = x->P[4] - y->P[4]; z->P[5] = x->P[5] - y->P[5];
z->P[6] = x->P[6] - y->P[6]; z->P[7] = x->P[7] - y->P[7];
z->P[8] = x->P[8] - y->P[8]; z->P[9] = x->P[9] - y->P[9];
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i];
-#endif
}
/* --- @f25519_neg@ --- *
void f25519_neg(f25519 *z, const f25519 *x)
{
-#if F25519_IMPL == 26
z->P[0] = -x->P[0]; z->P[1] = -x->P[1];
z->P[2] = -x->P[2]; z->P[3] = -x->P[3];
z->P[4] = -x->P[4]; z->P[5] = -x->P[5];
z->P[6] = -x->P[6]; z->P[7] = -x->P[7];
z->P[8] = -x->P[8]; z->P[9] = -x->P[9];
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) z->P[i] = -x->P[i];
-#endif
}
/*----- Constant-time utilities -------------------------------------------*/
{
mask32 mm = FIX_MASK32(m);
-#if F25519_IMPL == 26
z->P[0] = PICK2(x->P[0], y->P[0], mm);
z->P[1] = PICK2(x->P[1], y->P[1], mm);
z->P[2] = PICK2(x->P[2], y->P[2], mm);
z->P[7] = PICK2(x->P[7], y->P[7], mm);
z->P[8] = PICK2(x->P[8], y->P[8], mm);
z->P[9] = PICK2(x->P[9], y->P[9], mm);
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) z->P[i] = PICK2(x->P[i], y->P[i], mm);
-#endif
}
/* --- @f25519_pickn@ --- *
uint32 b = (uint32)1 << (31 - i);
mask32 m;
-#if F25519_IMPL == 26
z->P[0] = z->P[1] = z->P[2] = z->P[3] = z->P[4] =
z->P[5] = z->P[6] = z->P[7] = z->P[8] = z->P[9] = 0;
while (n--) {
CONDPICK(z->P[9], v->P[9], m);
v++; b <<= 1;
}
-#elif F25519_IMPL == 10
- unsigned j;
-
- for (j = 0; j < NPIECE; j++) z->P[j] = 0;
- while (n--) {
- m = SIGN(b);
- for (j = 0; j < NPIECE; j++) CONDPICK(z->P[j], v->P[j], m);
- v++; b <<= 1;
- }
-#endif
}
/* --- @f25519_condswap@ --- *
{
mask32 mm = FIX_MASK32(m);
-#if F25519_IMPL == 26
CONDSWAP(x->P[0], y->P[0], mm);
CONDSWAP(x->P[1], y->P[1], mm);
CONDSWAP(x->P[2], y->P[2], mm);
CONDSWAP(x->P[7], y->P[7], mm);
CONDSWAP(x->P[8], y->P[8], mm);
CONDSWAP(x->P[9], y->P[9], mm);
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm);
-#endif
}
/* --- @f25519_condneg@ --- *
void f25519_condneg(f25519 *z, const f25519 *x, uint32 m)
{
-#ifdef NEG_TWOC
mask32 m_xor = FIX_MASK32(m);
piece m_add = m&1;
# define CONDNEG(x) (((x) ^ m_xor) + m_add)
-#else
- int s = PICK2(-1, +1, m);
-# define CONDNEG(x) (s*(x))
-#endif
-#if F25519_IMPL == 26
z->P[0] = CONDNEG(x->P[0]);
z->P[1] = CONDNEG(x->P[1]);
z->P[2] = CONDNEG(x->P[2]);
z->P[7] = CONDNEG(x->P[7]);
z->P[8] = CONDNEG(x->P[8]);
z->P[9] = CONDNEG(x->P[9]);
-#elif F25519_IMPL == 10
- unsigned i;
- for (i = 0; i < NPIECE; i++) z->P[i] = CONDNEG(x->P[i]);
-#endif
#undef CONDNEG
}
/*----- Multiplication ----------------------------------------------------*/
-#if F25519_IMPL == 26
-
/* Let B = 2^63 - 1 be the largest value such that +B and -B can be
* represented in a double-precision piece. On entry, it must be the case
* that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on
CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \
} while (0)
-#elif F25519_IMPL == 10
-
-/* Perform carry propagation on X. */
-static void carry_reduce(dblpiece x[NPIECE])
-{
- /* Initial bounds: we assume |x_i| < 2^31 - 2^27. */
-
- unsigned i, j;
- dblpiece c;
-
- /* The result is nearly canonical, because we do sequential carry
- * propagation, because smaller processors are more likely to prefer the
- * smaller working set than the instruction-level parallelism.
- *
- * Start at x_23; truncate it to 10 bits, and propagate the carry to x_24.
- * Truncate x_24 to 10 bits, and add the carry onto x_25. Truncate x_25 to
- * 9 bits, and add 19 times the carry onto x_0. And so on.
- *
- * Let c_i be the portion of x_i to be carried onto x_{i+1}. I claim that
- * |c_i| <= 2^22. Then the carry /into/ any x_i has magnitude at most
- * 19*2^22 < 2^27 (allowing for the reduction as we carry from x_25 to
- * x_0), and x_i after carry is bounded above by 2^31. Hence, the carry
- * out is at most 2^22, as claimed.
- *
- * Once we reach x_23 for the second time, we start with |x_23| <= 2^9.
- * The carry into x_23 is at most 2^27 as calculated above; so the carry
- * out into x_24 has magnitude at most 2^17. In turn, |x_24| <= 2^9 before
- * the carry, so is now no more than 2^18 in magnitude, and the carry out
- * into x_25 is at most 2^8. This leaves |x_25| < 2^9 after carry
- * propagation.
- *
- * Be careful with the bit hacking because the quantities involved are
- * signed.
- */
-
- /*For each piece, we bias it so that floor division (as done by an
- * arithmetic right shift) and modulus (as done by bitwise-AND) does the
- * right thing.
- */
-#define CARRY(i, wd, b, m) do { \
- x[i] += (b); \
- c = ASR(dblpiece, x[i], (wd)); \
- x[i] = (dblpiece)((udblpiece)x[i]&(m)) - (b); \
-} while (0)
-
- { CARRY(23, 10, B9, M10); }
- { x[24] += c; CARRY(24, 10, B9, M10); }
- { x[25] += c; CARRY(25, 9, B8, M9); }
- { x[0] += 19*c; CARRY( 0, 10, B9, M10); }
- for (i = 1; i < 21; ) {
- for (j = i + 4; i < j; ) { x[i] += c; CARRY( i, 10, B9, M10); i++; }
- { x[i] += c; CARRY( i, 9, B8, M9); i++; }
- }
- while (i < 25) { x[i] += c; CARRY( i, 10, B9, M10); i++; }
- x[25] += c;
-
-#undef CARRY
-}
-
-#endif
-
/* --- @f25519_mulconst@ --- *
*
* Arguments: @f25519 *z@ = where to put the result (may alias @x@)
void f25519_mulconst(f25519 *z, const f25519 *x, long a)
{
-#if F25519_IMPL == 26
piece PIECES(x);
dblpiece PIECES(z), aa = a;
/* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */
CARRY_REDUCE(z, z);
STASH(z, z);
-
-#elif F25519_IMPL == 10
-
- dblpiece y[NPIECE];
- unsigned i;
-
- for (i = 0; i < NPIECE; i++) y[i] = a*x->P[i];
- carry_reduce(y);
- for (i = 0; i < NPIECE; i++) z->P[i] = y[i];
-
-#endif
}
/* --- @f25519_mul@ --- *
void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y)
{
-#if F25519_IMPL == 26
piece PIECES(x), PIECES(y);
dblpiece PIECES(z);
*/
for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
STASH(z, z);
-
-#elif F25519_IMPL == 10
-
- dblpiece u[NPIECE], t, tt, p;
- unsigned i, j, k;
-
- /* This is unpleasant. Honestly, this table seems to be the best way of
- * doing it.
- */
- static const unsigned short off[NPIECE] = {
- 0, 10, 20, 30, 40, 50, 59, 69, 79, 89, 99, 108, 118,
- 128, 138, 148, 157, 167, 177, 187, 197, 206, 216, 226, 236, 246
- };
-
- /* First pass: things we must multiply by 19 or 38. */
- for (i = 0; i < NPIECE - 1; i++) {
- t = tt = 0;
- for (j = i + 1; j < NPIECE; j++) {
- k = NPIECE + i - j; p = (dblpiece)x->P[j]*y->P[k];
- if (off[i] < off[j] + off[k] - 255) tt += p;
- else t += p;
- }
- u[i] = 19*(t + 2*tt);
- }
- u[NPIECE - 1] = 0;
-
- /* Second pass: things we must multiply by 1 or 2. */
- for (i = 0; i < NPIECE; i++) {
- t = tt = 0;
- for (j = 0; j <= i; j++) {
- k = i - j; p = (dblpiece)x->P[j]*y->P[k];
- if (off[i] < off[j] + off[k]) tt += p;
- else t += p;
- }
- u[i] += t + 2*tt;
- }
-
- /* And we're done. */
- carry_reduce(u);
- for (i = 0; i < NPIECE; i++) z->P[i] = u[i];
-
-#endif
}
/* --- @f25519_sqr@ --- *
void f25519_sqr(f25519 *z, const f25519 *x)
{
-#if F25519_IMPL == 26
piece PIECES(x);
dblpiece PIECES(z);
/* See `f25519_mul' for details. */
for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
STASH(z, z);
-
-#elif F25519_IMPL == 10
- f25519_mul(z, x, x);
-#endif
}
/*----- More complicated things -------------------------------------------*/
*/
static const piece sqrtm1_pieces[NPIECE] = {
-#if F25519_IMPL == 26
-32595792, -7943725, 9377950, 3500415, 12389472,
-272473, -25146209, -2005654, 326686, 11406482
-#elif F25519_IMPL == 10
- 176, -88, 161, 157, -485, -196, -231, -220, -416,
- -169, -255, 50, 189, -89, -266, -32, 202, -511,
- 423, 357, 248, -249, 80, 288, 50, 174
-#endif
};
#define SQRTM1 ((const f25519 *)sqrtm1_pieces)
int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y)
{
- f25519 t, u, w, beta, xy3, t2p50m1;
+ f25519 t, u, v, w, t15;
octet xb[32], b0[32], b1[32];
int32 rc = -1;
mask32 m;
for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
} while (0)
- /* This is a bit tricky; the algorithm is from Bernstein, Duif, Lange,
- * Schwabe, and Yang, `High-speed high-security signatures', 2011-09-26,
- * https://ed25519.cr.yp.to/ed25519-20110926.pdf.
+ /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif,
+ * Lange, Schwabe, and Yang, `High-speed high-security signatures',
+ * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf.
+ */
+ f25519_mul(&v, x, y);
+
+ /* Now for an addition chain. */ /* step | value */
+ f25519_sqr(&u, &v); /* 1 | 2 */
+ f25519_mul(&t, &u, &v); /* 2 | 3 */
+ SQRN(&u, &t, 2); /* 4 | 12 */
+ f25519_mul(&t15, &u, &t); /* 5 | 15 */
+ f25519_sqr(&u, &t15); /* 6 | 30 */
+ f25519_mul(&t, &u, &v); /* 7 | 31 = 2^5 - 1 */
+ SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */
+ f25519_mul(&t, &u, &t); /* 13 | 2^10 - 1 */
+ SQRN(&u, &t, 10); /* 23 | 2^20 - 2^10 */
+ f25519_mul(&u, &u, &t); /* 24 | 2^20 - 1 */
+ SQRN(&u, &u, 10); /* 34 | 2^30 - 2^10 */
+ f25519_mul(&t, &u, &t); /* 35 | 2^30 - 1 */
+ f25519_sqr(&u, &t); /* 36 | 2^31 - 2 */
+ f25519_mul(&t, &u, &v); /* 37 | 2^31 - 1 */
+ SQRN(&u, &t, 31); /* 68 | 2^62 - 2^31 */
+ f25519_mul(&t, &u, &t); /* 69 | 2^62 - 1 */
+ SQRN(&u, &t, 62); /* 131 | 2^124 - 2^62 */
+ f25519_mul(&t, &u, &t); /* 132 | 2^124 - 1 */
+ SQRN(&u, &t, 124); /* 256 | 2^248 - 2^124 */
+ f25519_mul(&t, &u, &t); /* 257 | 2^248 - 1 */
+ f25519_sqr(&u, &t); /* 258 | 2^249 - 2 */
+ f25519_mul(&t, &u, &v); /* 259 | 2^249 - 1 */
+ SQRN(&t, &t, 3); /* 262 | 2^252 - 8 */
+ f25519_sqr(&u, &t); /* 263 | 2^253 - 16 */
+ f25519_mul(&t, &u, &t); /* 264 | 3*2^252 - 24 */
+ f25519_mul(&t, &t, &t15); /* 265 | 3*2^252 - 9 */
+ f25519_mul(&w, &t, &v); /* 266 | 3*2^252 - 8 */
+
+ /* Awesome. Now let me explain. Let v be a square in GF(p), and let w =
+ * v^(3*2^252 - 8). In particular, let's consider
*
- * First of all, a complicated exponentation. The addition chain here is
- * mine. We start with some preliminary values.
- */ /* step | value */
- SQRN(&u, y, 1); /* 1 | 0, 2 */
- f25519_mul(&t, &u, y); /* 2 | 0, 3 */
- f25519_mul(&xy3, &t, x); /* 3 | 1, 3 */
- SQRN(&u, &u, 1); /* 4 | 0, 4 */
- f25519_mul(&w, &u, &xy3); /* 5 | 1, 7 */
-
- /* And now we calculate w^((p - 5)/8) = w^(252 - 3). */
- SQRN(&u, &w, 1); /* 6 | 2 */
- f25519_mul(&t, &w, &u); /* 7 | 3 */
- SQRN(&u, &t, 1); /* 8 | 6 */
- f25519_mul(&t, &u, &w); /* 9 | 7 */
- SQRN(&u, &t, 3); /* 12 | 56 */
- f25519_mul(&t, &t, &u); /* 13 | 63 = 2^6 - 1 */
- SQRN(&u, &t, 6); /* 19 | 2^12 - 2^6 */
- f25519_mul(&t, &t, &u); /* 20 | 2^12 - 1 */
- SQRN(&u, &t, 12); /* 32 | 2^24 - 2^12 */
- f25519_mul(&t, &t, &u); /* 33 | 2^24 - 1 */
- SQRN(&u, &t, 1); /* 34 | 2^25 - 2 */
- f25519_mul(&t, &u, &w); /* 35 | 2^25 - 1 */
- SQRN(&u, &t, 25); /* 60 | 2^50 - 2^25 */
- f25519_mul(&t2p50m1, &t, &u); /* 61 | 2^50 - 1 */
- SQRN(&u, &t2p50m1, 50); /* 111 | 2^100 - 2^50 */
- f25519_mul(&t, &t2p50m1, &u); /* 112 | 2^100 - 1 */
- SQRN(&u, &t, 100); /* 212 | 2^200 - 2^100 */
- f25519_mul(&t, &t, &u); /* 213 | 2^200 - 1 */
- SQRN(&u, &t, 50); /* 263 | 2^250 - 2^50 */
- f25519_mul(&t, &t2p50m1, &u); /* 264 | 2^250 - 1 */
- SQRN(&u, &t, 2); /* 266 | 2^252 - 4 */
- f25519_mul(&t, &u, &w); /* 267 | 2^252 - 3 */
-
- /* And finally... */
- f25519_mul(&beta, &t, &xy3); /* 268 | ... */
-
- /* Now we have beta = (x y^3) (x y^7)^((p - 5)/8) = (x/y)^((p + 3)/8), and
- * we're ready to finish the computation. Suppose that alpha^2 = u/w.
- * Then beta^4 = (x/y)^((p + 3)/2) = alpha^(p + 3) = alpha^4 = (x/y)^2, so
- * we have beta^2 = ±x/y. If y beta^2 = x then beta is the one we wanted;
- * if -y beta^2 = x, then we want beta sqrt(-1), which we already know. Of
- * course, it might not match either, in which case we fail.
+ * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3
+ *
+ * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square,
+ * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and
+ *
+ * w^4 = 1/v^2
+ *
+ * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let
+ * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set
+ * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1,
+ * so z^2 = -w^2 = x/y, and we're done.
*
* The easiest way to compare is to encode. This isn't as wasteful as it
* sounds: the hard part is normalizing the representations, which we have
* to do anyway.
*/
- f25519_sqr(&t, &beta);
+ f25519_mul(&w, &w, x);
+ f25519_sqr(&t, &w);
f25519_mul(&t, &t, y);
f25519_neg(&u, &t);
f25519_store(xb, x);
f25519_store(b0, &t);
f25519_store(b1, &u);
- f25519_mul(&u, &beta, SQRTM1);
+ f25519_mul(&u, &w, SQRTM1);
- m = -ct_memeq(b0, xb, 32);
+ m = -consttime_memeq(b0, xb, 32);
rc = PICK2(0, rc, m);
- f25519_pick2(z, &beta, &u, m);
- m = -ct_memeq(b1, xb, 32);
+ f25519_pick2(z, &w, &u, m);
+ m = -consttime_memeq(b1, xb, 32);
rc = PICK2(0, rc, m);
/* And we're done. */
return (rc);
}
-/*----- Test rig ----------------------------------------------------------*/
-
-#ifdef TEST_RIG
-
-#include <mLib/report.h>
-#include <mLib/str.h>
-#include <mLib/testrig.h>
-
-static void fixdstr(dstr *d)
-{
- if (d->len > 32)
- die(1, "invalid length for f25519");
- else if (d->len < 32) {
- dstr_ensure(d, 32);
- memset(d->buf + d->len, 0, 32 - d->len);
- d->len = 32;
- }
-}
-
-static void cvt_f25519(const char *buf, dstr *d)
-{
- dstr dd = DSTR_INIT;
-
- type_hex.cvt(buf, &dd); fixdstr(&dd);
- dstr_ensure(d, sizeof(f25519)); d->len = sizeof(f25519);
- f25519_load((f25519 *)d->buf, (const octet *)dd.buf);
- dstr_destroy(&dd);
-}
-
-static void dump_f25519(dstr *d, FILE *fp)
- { fdump(stderr, "???", (const piece *)d->buf); }
-
-static void cvt_f25519_ref(const char *buf, dstr *d)
- { type_hex.cvt(buf, d); fixdstr(d); }
-
-static void dump_f25519_ref(dstr *d, FILE *fp)
-{
- f25519 x;
-
- f25519_load(&x, (const octet *)d->buf);
- fdump(stderr, "???", x.P);
-}
-
-static int eq(const f25519 *x, dstr *d)
- { octet b[32]; f25519_store(b, x); return (memcmp(b, d->buf, 32) == 0); }
-
-static const test_type
- type_f25519 = { cvt_f25519, dump_f25519 },
- type_f25519_ref = { cvt_f25519_ref, dump_f25519_ref };
-
-#define TEST_UNOP(op) \
- static int vrf_##op(dstr dv[]) \
- { \
- f25519 *x = (f25519 *)dv[0].buf; \
- f25519 z, zz; \
- int ok = 1; \
- \
- f25519_##op(&z, x); \
- if (!eq(&z, &dv[1])) { \
- ok = 0; \
- fprintf(stderr, "failed!\n"); \
- fdump(stderr, "x", x->P); \
- fdump(stderr, "calc", z.P); \
- f25519_load(&zz, (const octet *)dv[1].buf); \
- fdump(stderr, "z", zz.P); \
- } \
- \
- return (ok); \
- }
-
-TEST_UNOP(neg)
-TEST_UNOP(sqr)
-TEST_UNOP(inv)
-
-#define TEST_BINOP(op) \
- static int vrf_##op(dstr dv[]) \
- { \
- f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf; \
- f25519 z, zz; \
- int ok = 1; \
- \
- f25519_##op(&z, x, y); \
- if (!eq(&z, &dv[2])) { \
- ok = 0; \
- fprintf(stderr, "failed!\n"); \
- fdump(stderr, "x", x->P); \
- fdump(stderr, "y", y->P); \
- fdump(stderr, "calc", z.P); \
- f25519_load(&zz, (const octet *)dv[2].buf); \
- fdump(stderr, "z", zz.P); \
- } \
- \
- return (ok); \
- }
-
-TEST_BINOP(add)
-TEST_BINOP(sub)
-TEST_BINOP(mul)
-
-static int vrf_mulc(dstr dv[])
-{
- f25519 *x = (f25519 *)dv[0].buf;
- long a = *(const long *)dv[1].buf;
- f25519 z, zz;
- int ok = 1;
-
- f25519_mulconst(&z, x, a);
- if (!eq(&z, &dv[2])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "x", x->P);
- fprintf(stderr, "a = %ld\n", a);
- fdump(stderr, "calc", z.P);
- f25519_load(&zz, (const octet *)dv[2].buf);
- fdump(stderr, "z", zz.P);
- }
-
- return (ok);
-}
-
-static int vrf_condneg(dstr dv[])
-{
- f25519 *x = (f25519 *)dv[0].buf;
- uint32 m = *(uint32 *)dv[1].buf;
- f25519 z;
- int ok = 1;
-
- f25519_condneg(&z, x, m);
- if (!eq(&z, &dv[2])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "x", x->P);
- fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
- fdump(stderr, "calc z", z.P);
- f25519_load(&z, (const octet *)dv[1].buf);
- fdump(stderr, "want z", z.P);
- }
-
- return (ok);
-}
-
-static int vrf_pick2(dstr dv[])
-{
- f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
- uint32 m = *(uint32 *)dv[2].buf;
- f25519 z;
- int ok = 1;
-
- f25519_pick2(&z, x, y, m);
- if (!eq(&z, &dv[3])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "x", x->P);
- fdump(stderr, "y", y->P);
- fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
- fdump(stderr, "calc z", z.P);
- f25519_load(&z, (const octet *)dv[3].buf);
- fdump(stderr, "want z", z.P);
- }
-
- return (ok);
-}
-
-static int vrf_pickn(dstr dv[])
-{
- dstr d = DSTR_INIT;
- f25519 v[32], z;
- size_t i = *(uint32 *)dv[1].buf, j, n;
- const char *p;
- char *q;
- int ok = 1;
-
- for (q = dv[0].buf, n = 0; (p = str_qword(&q, 0)) != 0; n++)
- { cvt_f25519(p, &d); v[n] = *(f25519 *)d.buf; }
-
- f25519_pickn(&z, v, n, i);
- if (!eq(&z, &dv[2])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- for (j = 0; j < n; j++) {
- fprintf(stderr, "v[%2u]", (unsigned)j);
- fdump(stderr, "", v[j].P);
- }
- fprintf(stderr, "i = %u\n", (unsigned)i);
- fdump(stderr, "calc z", z.P);
- f25519_load(&z, (const octet *)dv[2].buf);
- fdump(stderr, "want z", z.P);
- }
-
- dstr_destroy(&d);
- return (ok);
-}
-
-static int vrf_condswap(dstr dv[])
-{
- f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
- f25519 xx = *x, yy = *y;
- uint32 m = *(uint32 *)dv[2].buf;
- int ok = 1;
-
- f25519_condswap(&xx, &yy, m);
- if (!eq(&xx, &dv[3]) || !eq(&yy, &dv[4])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "x", x->P);
- fdump(stderr, "y", y->P);
- fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
- fdump(stderr, "calc xx", xx.P);
- fdump(stderr, "calc yy", yy.P);
- f25519_load(&xx, (const octet *)dv[3].buf);
- f25519_load(&yy, (const octet *)dv[4].buf);
- fdump(stderr, "want xx", xx.P);
- fdump(stderr, "want yy", yy.P);
- }
-
- return (ok);
-}
-
-static int vrf_quosqrt(dstr dv[])
-{
- f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
- f25519 z, zz;
- int rc;
- int ok = 1;
-
- if (dv[2].len) { fixdstr(&dv[2]); fixdstr(&dv[3]); }
- rc = f25519_quosqrt(&z, x, y);
- if (!dv[2].len ? !rc : (rc || (!eq(&z, &dv[2]) && !eq(&z, &dv[3])))) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "x", x->P);
- fdump(stderr, "y", y->P);
- if (rc) fprintf(stderr, "calc: FAIL\n");
- else fdump(stderr, "calc", z.P);
- if (!dv[2].len)
- fprintf(stderr, "exp: FAIL\n");
- else {
- f25519_load(&zz, (const octet *)dv[2].buf);
- fdump(stderr, "z", zz.P);
- f25519_load(&zz, (const octet *)dv[3].buf);
- fdump(stderr, "z'", zz.P);
- }
- }
-
- return (ok);
-}
-
-static int vrf_sub_mulc_add_sub_mul(dstr dv[])
-{
- f25519 *u = (f25519 *)dv[0].buf, *v = (f25519 *)dv[1].buf,
- *w = (f25519 *)dv[3].buf, *x = (f25519 *)dv[4].buf,
- *y = (f25519 *)dv[5].buf;
- long a = *(const long *)dv[2].buf;
- f25519 umv, aumv, wpaumv, xmy, z, zz;
- int ok = 1;
-
- f25519_sub(&umv, u, v);
- f25519_mulconst(&aumv, &umv, a);
- f25519_add(&wpaumv, w, &aumv);
- f25519_sub(&xmy, x, y);
- f25519_mul(&z, &wpaumv, &xmy);
-
- if (!eq(&z, &dv[6])) {
- ok = 0;
- fprintf(stderr, "failed!\n");
- fdump(stderr, "u", u->P);
- fdump(stderr, "v", v->P);
- fdump(stderr, "u - v", umv.P);
- fprintf(stderr, "a = %ld\n", a);
- fdump(stderr, "a (u - v)", aumv.P);
- fdump(stderr, "w + a (u - v)", wpaumv.P);
- fdump(stderr, "x", x->P);
- fdump(stderr, "y", y->P);
- fdump(stderr, "x - y", xmy.P);
- fdump(stderr, "(x - y) (w + a (u - v))", z.P);
- f25519_load(&zz, (const octet *)dv[6].buf); fdump(stderr, "z", zz.P);
- }
-
- return (ok);
-}
-
-static test_chunk tests[] = {
- { "add", vrf_add, { &type_f25519, &type_f25519, &type_f25519_ref } },
- { "sub", vrf_sub, { &type_f25519, &type_f25519, &type_f25519_ref } },
- { "neg", vrf_neg, { &type_f25519, &type_f25519_ref } },
- { "condneg", vrf_condneg,
- { &type_f25519, &type_uint32, &type_f25519_ref } },
- { "mul", vrf_mul, { &type_f25519, &type_f25519, &type_f25519_ref } },
- { "mulconst", vrf_mulc, { &type_f25519, &type_long, &type_f25519_ref } },
- { "pick2", vrf_pick2,
- { &type_f25519, &type_f25519, &type_uint32, &type_f25519_ref } },
- { "pickn", vrf_pickn,
- { &type_string, &type_uint32, &type_f25519_ref } },
- { "condswap", vrf_condswap,
- { &type_f25519, &type_f25519, &type_uint32,
- &type_f25519_ref, &type_f25519_ref } },
- { "sqr", vrf_sqr, { &type_f25519, &type_f25519_ref } },
- { "inv", vrf_inv, { &type_f25519, &type_f25519_ref } },
- { "quosqrt", vrf_quosqrt,
- { &type_f25519, &type_f25519, &type_hex, &type_hex } },
- { "sub-mulc-add-sub-mul", vrf_sub_mulc_add_sub_mul,
- { &type_f25519, &type_f25519, &type_long, &type_f25519,
- &type_f25519, &type_f25519, &type_f25519_ref } },
- { 0, 0, { 0 } }
-};
-
-int main(int argc, char *argv[])
-{
- test_run(argc, argv, tests, SRCDIR "/t/f25519");
- return (0);
-}
-
-#endif
-
/*----- That's all, folks -------------------------------------------------*/
~mdw / secnet / blobdiff