-/*
- * f25519.c: arithmetic modulo 2^255 - 19
- */
-/*
- * This file is Free Software. It has been modified to as part of its
- * incorporation into secnet.
- *
- * Copyright 2017 Mark Wooding
- *
- * You may redistribute this file and/or modify it under the terms of
- * the permissive licence shown below.
- *
- * You may redistribute secnet as a whole and/or modify it under the
- * terms of the GNU General Public License as published by the Free
- * Software Foundation; either version 3, or (at your option) any
- * later version.
- *
- * This program 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
- * along with this program; if not, see
- * https://www.gnu.org/licenses/gpl.html.
- */
-/*
- * Imported from Catacomb, and modified for Secnet (2017-04-30):
- *
- * * Use `fake-mLib-bits.h' in place of the real <mLib/bits.h>.
- *
- * * Remove the 16/32-bit implementation, since C99 always has 64-bit
- * arithmetic.
- *
- * * Remove the test rig code: a replacement is in a separate source file.
- *
- * * Disable some of the operations which aren't needed for X25519.
- * (They're used for Ed25519, which we don't need.)
- *
- * The file's original comment headers are preserved below.
- */
/* -*-c-*-
*
* Arithmetic modulo 2^255 - 19
/*----- 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
/*----- Basic setup -------------------------------------------------------*/
+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
void f25519_load(f25519 *z, const octet xv[32])
{
+
uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4),
xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12),
xw4 = LOAD32_L(xv + 16), xw5 = LOAD32_L(xv + 20),
void f25519_store(octet zv[32], const f25519 *x)
{
+
piece PIECES(x), PIECES(y), c, d;
uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7;
mask32 m;
z->P[8] = x->P[8] - y->P[8]; z->P[9] = x->P[9] - y->P[9];
}
-#ifndef F25519_TRIM_X25519
-
/* --- @f25519_neg@ --- *
*
* Arguments: @f25519 *z@ = where to put the result (may alias @x@)
z->P[8] = -x->P[8]; z->P[9] = -x->P[9];
}
-#endif
-
/*----- Constant-time utilities -------------------------------------------*/
-#ifndef F25519_TRIM_X25519
-
/* --- @f25519_pick2@ --- *
*
* Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
}
}
-#endif
-
/* --- @f25519_condswap@ --- *
*
* Arguments: @f25519 *x, *y@ = two operands
CONDSWAP(x->P[9], y->P[9], mm);
}
-#ifndef F25519_TRIM_X25519
-
/* --- @f25519_condneg@ --- *
*
* Arguments: @f25519 *z@ = where to put the result (may alias @x@)
#undef CONDNEG
}
-#endif
-
/*----- Multiplication ----------------------------------------------------*/
/* Let B = 2^63 - 1 be the largest value such that +B and -B can be
void f25519_mulconst(f25519 *z, const f25519 *x, long a)
{
+
piece PIECES(x);
dblpiece PIECES(z), aa = a;
void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y)
{
+
piece PIECES(x), PIECES(y);
dblpiece PIECES(z);
unsigned i;
void f25519_sqr(f25519 *z, const f25519 *x)
{
+
piece PIECES(x);
dblpiece PIECES(z);
unsigned i;
#undef SQRN
}
-#ifndef F25519_TRIM_X25519
-
/* --- @f25519_quosqrt@ --- *
*
* Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
*/
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);
}
-#endif
-
/*----- That's all, folks -------------------------------------------------*/
~mdw / secnet / blobdiff