--- /dev/null
+/* -*-c-*-
+ *
+ * Arithmetic modulo 2^255 - 19
+ *
+ * (c) 2017 Straylight/Edgeware
+ */
+
+/*----- Licensing notice --------------------------------------------------*
+ *
+ * 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
+ * published by the Free Software Foundation; either version 2 of the
+ * License, or (at your option) any later version.
+ *
+ * Catacomb 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 Library General Public License for more details.
+ *
+ * You should have received a copy of the GNU Library General Public
+ * License along with Catacomb; if not, write to the Free
+ * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+ * MA 02111-1307, USA.
+ */
+
+/*----- Header files ------------------------------------------------------*/
+
+#include "f25519.h"
+
+/*----- Basic setup -------------------------------------------------------*/
+
+/* 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 uint32 upiece; typedef uint64 udblpiece;
+#define P p26
+#define PIECEWD(i) ((i)%2 ? 25 : 26)
+#define NPIECE 10
+
+#define M26 0x03ffffffu
+#define M25 0x01ffffffu
+#define B26 0x04000000u
+#define B25 0x02000000u
+#define B24 0x01000000u
+
+#define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9
+#define FETCH(v, w) do { \
+ v##0 = (w)->P[0]; v##1 = (w)->P[1]; \
+ v##2 = (w)->P[2]; v##3 = (w)->P[3]; \
+ v##4 = (w)->P[4]; v##5 = (w)->P[5]; \
+ v##6 = (w)->P[6]; v##7 = (w)->P[7]; \
+ v##8 = (w)->P[8]; v##9 = (w)->P[9]; \
+} while (0)
+#define STASH(w, v) do { \
+ (w)->P[0] = v##0; (w)->P[1] = v##1; \
+ (w)->P[2] = v##2; (w)->P[3] = v##3; \
+ (w)->P[4] = v##4; (w)->P[5] = v##5; \
+ (w)->P[6] = v##6; (w)->P[7] = v##7; \
+ (w)->P[8] = v##8; (w)->P[9] = v##9; \
+} while (0)
+
+/*----- Debugging machinery -----------------------------------------------*/
+
+#if defined(F25519_DEBUG)
+
+#include <stdio.h>
+
+#include "mp.h"
+#include "mptext.h"
+
+static mp *get_2p255m91(void)
+{
+ mpw w19 = 19;
+ mp *p = MP_NEW, m19;
+
+ p = mp_setbit(p, MP_ZERO, 255);
+ mp_build(&m19, &w19, &w19 + 1);
+ p = mp_sub(p, p, &m19);
+ return (p);
+}
+
+DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 32, get_2p255m91())
+
+#endif
+
+/*----- Loading and storing -----------------------------------------------*/
+
+/* --- @f25519_load@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to store the result
+ * @const octet xv[32]@ = source to read
+ *
+ * Returns: ---
+ *
+ * Use: Reads an element of %$\gf{2^{255} - 19}$% in external
+ * representation from @xv@ and stores it in @z@.
+ *
+ * External representation is little-endian base-256. Some
+ * elements have multiple encodings, which are not produced by
+ * correct software; use of noncanonical encodings is not an
+ * error, and toleration of them is considered a performance
+ * feature.
+ */
+
+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),
+ xw6 = LOAD32_L(xv + 24), xw7 = LOAD32_L(xv + 28);
+ piece PIECES(x), b, c;
+
+ /* First, split the 32-bit words into the irregularly-sized pieces we need
+ * for the field representation. These pieces are all positive. We'll do
+ * the sign correction afterwards.
+ *
+ * It may be that the top bit of the input is set. Avoid trouble by
+ * folding that back round into the bottom piece of the representation.
+ *
+ * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later.
+ * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25.
+ */
+ x0 = ((xw0 << 0)&0x03ffffff) + 19*((xw7 >> 31)&0x00000001);
+ x1 = ((xw1 << 6)&0x01ffffc0) | ((xw0 >> 26)&0x0000003f);
+ x2 = ((xw2 << 13)&0x03ffe000) | ((xw1 >> 19)&0x00001fff);
+ x3 = ((xw3 << 19)&0x01f80000) | ((xw2 >> 13)&0x0007ffff);
+ x4 = ((xw3 >> 6)&0x03ffffff);
+ x5 = (xw4 << 0)&0x01ffffff;
+ x6 = ((xw5 << 7)&0x03ffff80) | ((xw4 >> 25)&0x0000007f);
+ x7 = ((xw6 << 13)&0x01ffe000) | ((xw5 >> 19)&0x00001fff);
+ x8 = ((xw7 << 20)&0x03f00000) | ((xw6 >> 12)&0x000fffff);
+ x9 = ((xw7 >> 6)&0x01ffffff);
+
+ /* Next, we convert these pieces into a roughly balanced signed
+ * representation. For each piece, check to see if its top bit is set. If
+ * it is, then lend a bit to the next piece over. For x_9, this needs to
+ * be carried around, which is a little fiddly. In particular, we delay
+ * the carry until after all of the pieces have been balanced. If we don't
+ * do this, then we have to do a more expensive test for nonzeroness to
+ * decide whether to lend a bit leftwards rather than just testing a single
+ * bit.
+ *
+ * This fixes up the anomalous size of x_0: the loan of a bit becomes an
+ * actual carry if x_0 >= 2^26. By the end, then, we have:
+ *
+ * { 2^25 if i even
+ * |x_i| <= {
+ * { 2^24 if i odd
+ *
+ * Note that we don't try for a canonical representation here: both upper
+ * and lower bounds are achievable.
+ *
+ * All of the x_i at this point are positive, so we don't need to do
+ * anything wierd when masking them.
+ */
+ b = x9&B24; c = 19&((b >> 19) - (b >> 24)); x9 -= b << 1;
+ b = x8&B25; x9 += b >> 25; x8 -= b << 1;
+ b = x7&B24; x8 += b >> 24; x7 -= b << 1;
+ b = x6&B25; x7 += b >> 25; x6 -= b << 1;
+ b = x5&B24; x6 += b >> 24; x5 -= b << 1;
+ b = x4&B25; x5 += b >> 25; x4 -= b << 1;
+ b = x3&B24; x4 += b >> 24; x3 -= b << 1;
+ b = x2&B25; x3 += b >> 25; x2 -= b << 1;
+ b = x1&B24; x2 += b >> 24; x1 -= b << 1;
+ b = x0&B25; x1 += (b >> 25) + (x0 >> 26); x0 = (x0&M26) - (b << 1);
+ x0 += c;
+
+ /* And with that, we're done. */
+ STASH(z, x);
+}
+
+/* --- @f25519_store@ --- *
+ *
+ * Arguments: @octet zv[32]@ = where to write the result
+ * @const f25519 *x@ = the field element to write
+ *
+ * Returns: ---
+ *
+ * Use: Stores a field element in the given octet vector in external
+ * representation. A canonical encoding is always stored, so,
+ * in particular, the top bit of @xv[31]@ is always left clear.
+ */
+
+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;
+
+ FETCH(x, x);
+
+ /* First, propagate the carries throughout the pieces. By the end of this,
+ * we'll have all of the pieces canonically sized and positive, and maybe
+ * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and
+ * the remaining value will be in the half-open interval [0, 2^255). The
+ * whole represented value is then x + 2^255 c.
+ *
+ * It's worth paying careful attention to the bounds. We assume that we
+ * start out with |x_i| <= 2^30. We start by cutting off and reducing the
+ * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and
+ * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto
+ * x_0 and propagate the carries: but what bounds can we calculate on x
+ * before this?
+ *
+ * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so
+ * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0;
+ * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i}
+ * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for
+ * x_9, so
+ *
+ * -2^235 < x + 19 c_9 < 2^255 + 2^235
+ *
+ * Here, the x_i are signed, so we must be cautious about bithacking them.
+ */
+ c = ASR(piece, x9, 25); x9 = (upiece)x9&M25;
+ x0 += 19*c; c = ASR(piece, x0, 26); x0 = (upiece)x0&M26;
+ x1 += c; c = ASR(piece, x1, 25); x1 = (upiece)x1&M25;
+ x2 += c; c = ASR(piece, x2, 26); x2 = (upiece)x2&M26;
+ x3 += c; c = ASR(piece, x3, 25); x3 = (upiece)x3&M25;
+ x4 += c; c = ASR(piece, x4, 26); x4 = (upiece)x4&M26;
+ x5 += c; c = ASR(piece, x5, 25); x5 = (upiece)x5&M25;
+ x6 += c; c = ASR(piece, x6, 26); x6 = (upiece)x6&M26;
+ x7 += c; c = ASR(piece, x7, 25); x7 = (upiece)x7&M25;
+ x8 += c; c = ASR(piece, x8, 26); x8 = (upiece)x8&M26;
+ x9 += c; c = ASR(piece, x9, 25); x9 = (upiece)x9&M25;
+
+ /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and
+ * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole
+ * value; if c = -1 then we should add 2^255 - 19; and otherwise we should
+ * do nothing.
+ *
+ * But conditional behaviour is bad, m'kay. So here's what we do instead.
+ *
+ * The first job is to sort out what we wanted to do. If c = -1 then we
+ * want to (a) invert the constant addend and (b) feed in a carry-in;
+ * otherwise, we don't.
+ */
+ m = SIGN(c);
+ d = m&1;
+
+ /* Now do the addition/subtraction. Remember that all of the x_i are
+ * nonnegative, so shifting and masking are safe and easy.
+ */
+ d += x0 + (19 ^ (M26&m)); y0 = d&M26; d >>= 26;
+ d += x1 + (M25&m); y1 = d&M25; d >>= 25;
+ d += x2 + (M26&m); y2 = d&M26; d >>= 26;
+ d += x3 + (M25&m); y3 = d&M25; d >>= 25;
+ d += x4 + (M26&m); y4 = d&M26; d >>= 26;
+ d += x5 + (M25&m); y5 = d&M25; d >>= 25;
+ d += x6 + (M26&m); y6 = d&M26; d >>= 26;
+ d += x7 + (M25&m); y7 = d&M25; d >>= 25;
+ d += x8 + (M26&m); y8 = d&M26; d >>= 26;
+ d += x9 + (M25&m); y9 = d&M25; d >>= 25;
+
+ /* The final carry-out is in d; since we only did addition, and the x_i are
+ * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x,
+ * if (a) c /= 0 (in which case we know that the old value was
+ * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that
+ * the subtraction didn't cause a borrow, so we must be in the case where
+ * 2^255 - 19 <= x < 2^255).
+ */
+ m = NONZEROP(c) | ~NONZEROP(d - 1);
+ x0 = (y0&m) | (x0&~m); x1 = (y1&m) | (x1&~m);
+ x2 = (y2&m) | (x2&~m); x3 = (y3&m) | (x3&~m);
+ x4 = (y4&m) | (x4&~m); x5 = (y5&m) | (x5&~m);
+ x6 = (y6&m) | (x6&~m); x7 = (y7&m) | (x7&~m);
+ x8 = (y8&m) | (x8&~m); x9 = (y9&m) | (x9&~m);
+
+ /* Extract 32-bit words from the value. */
+ zw0 = ((x0 >> 0)&0x03ffffff) | (((uint32)x1 << 26)&0xfc000000);
+ zw1 = ((x1 >> 6)&0x0007ffff) | (((uint32)x2 << 19)&0xfff80000);
+ zw2 = ((x2 >> 13)&0x00001fff) | (((uint32)x3 << 13)&0xffffe000);
+ zw3 = ((x3 >> 19)&0x0000003f) | (((uint32)x4 << 6)&0xffffffc0);
+ zw4 = ((x5 >> 0)&0x01ffffff) | (((uint32)x6 << 25)&0xfe000000);
+ zw5 = ((x6 >> 7)&0x0007ffff) | (((uint32)x7 << 19)&0xfff80000);
+ zw6 = ((x7 >> 13)&0x00000fff) | (((uint32)x8 << 12)&0xfffff000);
+ zw7 = ((x8 >> 20)&0x0000003f) | (((uint32)x9 << 6)&0x7fffffc0);
+
+ /* Store the result as an octet string. */
+ STORE32_L(zv + 0, zw0); STORE32_L(zv + 4, zw1);
+ 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);
+}
+
+/* --- @f25519_set@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to write the result
+ * @int a@ = a small-ish constant
+ *
+ * Returns: ---
+ *
+ * Use: Sets @z@ to equal @a@.
+ */
+
+void f25519_set(f25519 *x, int a)
+{
+ unsigned i;
+
+ x->P[0] = a;
+ for (i = 1; i < NPIECE; i++) x->P[i] = 0;
+}
+
+/*----- Basic arithmetic --------------------------------------------------*/
+
+/* --- @f25519_add@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x, *y@ = two operands
+ *
+ * Returns: ---
+ *
+ * Use: Set @z@ to the sum %$x + y$%.
+ */
+
+void f25519_add(f25519 *z, const f25519 *x, const f25519 *y)
+{
+ 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];
+}
+
+/* --- @f25519_sub@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x, *y@ = two operands
+ *
+ * Returns: ---
+ *
+ * Use: Set @z@ to the difference %$x - y$%.
+ */
+
+void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y)
+{
+ 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];
+}
+
+/* --- @f25519_neg@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
+ * @const f25519 *x@ = an operand
+ *
+ * Returns: ---
+ *
+ * Use: Set @z = -x@.
+ */
+
+void f25519_neg(f25519 *z, const f25519 *x)
+{
+ 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];
+}
+
+/*----- Constant-time utilities -------------------------------------------*/
+
+/* --- @f25519_pick2@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x, *y@ = two operands
+ * @uint32 m@ = a mask
+ *
+ * Returns: ---
+ *
+ * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set
+ * @z = x@. If @m@ has some other value, then scramble @z@ in
+ * an unhelpful way.
+ */
+
+void f25519_pick2(f25519 *z, const f25519 *x, const f25519 *y, uint32 m)
+{
+ mask32 mm = FIX_MASK32(m);
+
+ 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[3] = PICK2(x->P[3], y->P[3], mm);
+ z->P[4] = PICK2(x->P[4], y->P[4], mm);
+ z->P[5] = PICK2(x->P[5], y->P[5], mm);
+ z->P[6] = PICK2(x->P[6], y->P[6], 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);
+}
+
+/* --- @f25519_pickn@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result
+ * @const f25519 *v@ = a table of entries
+ * @size_t n@ = the number of entries in @v@
+ * @size_t i@ = an index
+ *
+ * Returns: ---
+ *
+ * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then
+ * do something unhelpful; otherwise, if @i >= n@ then set @z@
+ * to zero.
+ */
+
+void f25519_pickn(f25519 *z, const f25519 *v, size_t n, size_t i)
+{
+ uint32 b = (uint32)1 << (31 - i);
+ mask32 m;
+
+ 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--) {
+ m = SIGN(b);
+ CONDPICK(z->P[0], v->P[0], m);
+ CONDPICK(z->P[1], v->P[1], m);
+ CONDPICK(z->P[2], v->P[2], m);
+ CONDPICK(z->P[3], v->P[3], m);
+ CONDPICK(z->P[4], v->P[4], m);
+ CONDPICK(z->P[5], v->P[5], m);
+ CONDPICK(z->P[6], v->P[6], m);
+ CONDPICK(z->P[7], v->P[7], m);
+ CONDPICK(z->P[8], v->P[8], m);
+ CONDPICK(z->P[9], v->P[9], m);
+ v++; b <<= 1;
+ }
+}
+
+/* --- @f25519_condswap@ --- *
+ *
+ * Arguments: @f25519 *x, *y@ = two operands
+ * @uint32 m@ = a mask
+ *
+ * Returns: ---
+ *
+ * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then
+ * exchange @x@ and @y@. If @m@ has some other value, then
+ * scramble @x@ and @y@ in an unhelpful way.
+ */
+
+void f25519_condswap(f25519 *x, f25519 *y, uint32 m)
+{
+ mask32 mm = FIX_MASK32(m);
+
+ 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[3], y->P[3], mm);
+ CONDSWAP(x->P[4], y->P[4], mm);
+ CONDSWAP(x->P[5], y->P[5], mm);
+ CONDSWAP(x->P[6], y->P[6], 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);
+}
+
+/* --- @f25519_condneg@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
+ * @const f25519 *x@ = an operand
+ * @uint32 m@ = a mask
+ *
+ * Returns: ---
+ *
+ * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set
+ * @z = -x@. If @m@ has some other value then scramble @z@ in
+ * an unhelpful way.
+ */
+
+void f25519_condneg(f25519 *z, const f25519 *x, uint32 m)
+{
+ mask32 m_xor = FIX_MASK32(m);
+ piece m_add = m&1;
+# define CONDNEG(x) (((x) ^ m_xor) + m_add)
+
+ z->P[0] = CONDNEG(x->P[0]);
+ z->P[1] = CONDNEG(x->P[1]);
+ z->P[2] = CONDNEG(x->P[2]);
+ z->P[3] = CONDNEG(x->P[3]);
+ z->P[4] = CONDNEG(x->P[4]);
+ z->P[5] = CONDNEG(x->P[5]);
+ z->P[6] = CONDNEG(x->P[6]);
+ z->P[7] = CONDNEG(x->P[7]);
+ z->P[8] = CONDNEG(x->P[8]);
+ z->P[9] = CONDNEG(x->P[9]);
+
+#undef CONDNEG
+}
+
+/*----- Multiplication ----------------------------------------------------*/
+
+/* 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
+ * exit, we will have |Z_i| <= 2^25 + 19 M/2^25.
+ */
+#define CARRYSTEP(z, x, m, b, f, xx, n) do { \
+ (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \
+ (f)*ASR(dblpiece, (xx), (n)); \
+} while (0)
+#define CARRY_REDUCE(z, x) do { \
+ dblpiece PIECES(_t); \
+ \
+ /* Bias the input pieces. This keeps the carries and so on centred \
+ * around zero rather than biased positive. \
+ */ \
+ _t0 = (x##0) + B25; _t1 = (x##1) + B24; \
+ _t2 = (x##2) + B25; _t3 = (x##3) + B24; \
+ _t4 = (x##4) + B25; _t5 = (x##5) + B24; \
+ _t6 = (x##6) + B25; _t7 = (x##7) + B24; \
+ _t8 = (x##8) + B25; _t9 = (x##9) + B24; \
+ \
+ /* Calculate the reduced pieces. Careful with the bithacking. */ \
+ CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \
+ CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \
+ CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \
+ CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \
+ CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \
+ CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \
+ CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \
+ CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \
+ CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \
+ CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \
+} while (0)
+
+/* --- @f25519_mulconst@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
+ * @const f25519 *x@ = an operand
+ * @long a@ = a small-ish constant; %$|a| < 2^{20}$%.
+ *
+ * Returns: ---
+ *
+ * Use: Set @z@ to the product %$a x$%.
+ */
+
+void f25519_mulconst(f25519 *z, const f25519 *x, long a)
+{
+
+ piece PIECES(x);
+ dblpiece PIECES(z), aa = a;
+
+ FETCH(x, x);
+
+ /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have
+ * |z_i| <= 2^50.
+ */
+ z0 = aa*x0; z1 = aa*x1; z2 = aa*x2; z3 = aa*x3; z4 = aa*x4;
+ z5 = aa*x5; z6 = aa*x6; z7 = aa*x7; z8 = aa*x8; z9 = aa*x9;
+
+ /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */
+ CARRY_REDUCE(z, z);
+ STASH(z, z);
+}
+
+/* --- @f25519_mul@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x, *y@ = two operands
+ *
+ * Returns: ---
+ *
+ * Use: Set @z@ to the product %$x y$%.
+ */
+
+void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y)
+{
+
+ piece PIECES(x), PIECES(y);
+ dblpiece PIECES(z);
+ unsigned i;
+
+ FETCH(x, x); FETCH(y, y);
+
+ /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have
+ *
+ * |z_0| <= 267*2^54
+ * |z_1| <= 154*2^54
+ * |z_2| <= 213*2^54
+ * |z_3| <= 118*2^54
+ * |z_4| <= 159*2^54
+ * |z_5| <= 82*2^54
+ * |z_6| <= 105*2^54
+ * |z_7| <= 46*2^54
+ * |z_8| <= 51*2^54
+ * |z_9| <= 10*2^54
+ *
+ * all of which are less than 2^63 - 2^25.
+ */
+
+#define M(a, b) ((dblpiece)(a)*(b))
+ z0 = M(x0, y0) +
+ 19*(M(x2, y8) + M(x4, y6) + M(x6, y4) + M(x8, y2)) +
+ 38*(M(x1, y9) + M(x3, y7) + M(x5, y5) + M(x7, y3) + M(x9, y1));
+ z1 = M(x0, y1) + M(x1, y0) +
+ 19*(M(x2, y9) + M(x3, y8) + M(x4, y7) + M(x5, y6) +
+ M(x6, y5) + M(x7, y4) + M(x8, y3) + M(x9, y2));
+ z2 = M(x0, y2) + M(x2, y0) +
+ 2* M(x1, y1) +
+ 19*(M(x4, y8) + M(x6, y6) + M(x8, y4)) +
+ 38*(M(x3, y9) + M(x5, y7) + M(x7, y5) + M(x9, y3));
+ z3 = M(x0, y3) + M(x1, y2) + M(x2, y1) + M(x3, y0) +
+ 19*(M(x4, y9) + M(x5, y8) + M(x6, y7) +
+ M(x7, y6) + M(x8, y5) + M(x9, y4));
+ z4 = M(x0, y4) + M(x2, y2) + M(x4, y0) +
+ 2*(M(x1, y3) + M(x3, y1)) +
+ 19*(M(x6, y8) + M(x8, y6)) +
+ 38*(M(x5, y9) + M(x7, y7) + M(x9, y5));
+ z5 = M(x0, y5) + M(x1, y4) + M(x2, y3) +
+ M(x3, y2) + M(x4, y1) + M(x5, y0) +
+ 19*(M(x6, y9) + M(x7, y8) + M(x8, y7) + M(x9, y6));
+ z6 = M(x0, y6) + M(x2, y4) + M(x4, y2) + M(x6, y0) +
+ 2*(M(x1, y5) + M(x3, y3) + M(x5, y1)) +
+ 19* M(x8, y8) +
+ 38*(M(x7, y9) + M(x9, y7));
+ z7 = M(x0, y7) + M(x1, y6) + M(x2, y5) + M(x3, y4) +
+ M(x4, y3) + M(x5, y2) + M(x6, y1) + M(x7, y0) +
+ 19*(M(x8, y9) + M(x9, y8));
+ z8 = M(x0, y8) + M(x2, y6) + M(x4, y4) + M(x6, y2) + M(x8, y0) +
+ 2*(M(x1, y7) + M(x3, y5) + M(x5, y3) + M(x7, y1)) +
+ 38* M(x9, y9);
+ z9 = M(x0, y9) + M(x1, y8) + M(x2, y7) + M(x3, y6) + M(x4, y5) +
+ M(x5, y4) + M(x6, y3) + M(x7, y2) + M(x8, y1) + M(x9, y0);
+#undef M
+
+ /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will
+ * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 +
+ * 2^13, which is comfortable for an addition prior to the next
+ * multiplication.
+ */
+ for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
+ STASH(z, z);
+}
+
+/* --- @f25519_sqr@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x@ = an operand
+ *
+ * Returns: ---
+ *
+ * Use: Set @z@ to the square %$x^2$%.
+ */
+
+void f25519_sqr(f25519 *z, const f25519 *x)
+{
+
+ piece PIECES(x);
+ dblpiece PIECES(z);
+ unsigned i;
+
+ FETCH(x, x);
+
+ /* See `f25519_mul' for bounds. */
+
+#define M(a, b) ((dblpiece)(a)*(b))
+ z0 = M(x0, x0) +
+ 38*(M(x2, x8) + M(x4, x6) + M(x5, x5)) +
+ 76*(M(x1, x9) + M(x3, x7));
+ z1 = 2* M(x0, x1) +
+ 38*(M(x2, x9) + M(x3, x8) + M(x4, x7) + M(x5, x6));
+ z2 = 2*(M(x0, x2) + M(x1, x1)) +
+ 19* M(x6, x6) +
+ 38* M(x4, x8) +
+ 76*(M(x3, x9) + M(x5, x7));
+ z3 = 2*(M(x0, x3) + M(x1, x2)) +
+ 38*(M(x4, x9) + M(x5, x8) + M(x6, x7));
+ z4 = M(x2, x2) +
+ 2* M(x0, x4) +
+ 4* M(x1, x3) +
+ 38*(M(x6, x8) + M(x7, x7)) +
+ 76* M(x5, x9);
+ z5 = 2*(M(x0, x5) + M(x1, x4) + M(x2, x3)) +
+ 38*(M(x6, x9) + M(x7, x8));
+ z6 = 2*(M(x0, x6) + M(x2, x4) + M(x3, x3)) +
+ 4* M(x1, x5) +
+ 19* M(x8, x8) +
+ 76* M(x7, x9);
+ z7 = 2*(M(x0, x7) + M(x1, x6) + M(x2, x5) + M(x3, x4)) +
+ 38* M(x8, x9);
+ z8 = M(x4, x4) +
+ 2*(M(x0, x8) + M(x2, x6)) +
+ 4*(M(x1, x7) + M(x3, x5)) +
+ 38* M(x9, x9);
+ z9 = 2*(M(x0, x9) + M(x1, x8) + M(x2, x7) + M(x3, x6) + M(x4, x5));
+#undef M
+
+ /* See `f25519_mul' for details. */
+ for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
+ STASH(z, z);
+}
+
+/*----- More complicated things -------------------------------------------*/
+
+/* --- @f25519_inv@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
+ * @const f25519 *x@ = an operand
+ *
+ * Returns: ---
+ *
+ * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If
+ * %$x = 0$% then @z@ is set to zero. This is considered a
+ * feature.
+ */
+
+void f25519_inv(f25519 *z, const f25519 *x)
+{
+ f25519 t, u, t2, t11, t2p10m1, t2p50m1;
+ unsigned i;
+
+#define SQRN(z, x, n) do { \
+ f25519_sqr((z), (x)); \
+ for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
+} while (0)
+
+ /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as
+ * intended. The addition chain here is from Bernstein's implementation; I
+ * couldn't find a better one.
+ */ /* step | value */
+ f25519_sqr(&t2, x); /* 1 | 2 */
+ SQRN(&u, &t2, 2); /* 3 | 8 */
+ f25519_mul(&t, &u, x); /* 4 | 9 */
+ f25519_mul(&t11, &t, &t2); /* 5 | 11 = 2^5 - 21 */
+ f25519_sqr(&u, &t11); /* 6 | 22 */
+ f25519_mul(&t, &t, &u); /* 7 | 31 = 2^5 - 1 */
+ SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */
+ f25519_mul(&t2p10m1, &t, &u); /* 13 | 2^10 - 1 */
+ SQRN(&u, &t2p10m1, 10); /* 23 | 2^20 - 2^10 */
+ f25519_mul(&t, &t2p10m1, &u); /* 24 | 2^20 - 1 */
+ SQRN(&u, &t, 20); /* 44 | 2^40 - 2^20 */
+ f25519_mul(&t, &t, &u); /* 45 | 2^40 - 1 */
+ SQRN(&u, &t, 10); /* 55 | 2^50 - 2^10 */
+ f25519_mul(&t2p50m1, &t2p10m1, &u); /* 56 | 2^50 - 1 */
+ SQRN(&u, &t2p50m1, 50); /* 106 | 2^100 - 2^50 */
+ f25519_mul(&t, &t2p50m1, &u); /* 107 | 2^100 - 1 */
+ SQRN(&u, &t, 100); /* 207 | 2^200 - 2^100 */
+ f25519_mul(&t, &t, &u); /* 208 | 2^200 - 1 */
+ SQRN(&u, &t, 50); /* 258 | 2^250 - 2^50 */
+ f25519_mul(&t, &t2p50m1, &u); /* 259 | 2^250 - 1 */
+ SQRN(&u, &t, 5); /* 264 | 2^255 - 2^5 */
+ f25519_mul(z, &u, &t11); /* 265 | 2^255 - 21 */
+
+#undef SQRN
+}
+
+/* --- @f25519_quosqrt@ --- *
+ *
+ * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
+ * @const f25519 *x, *y@ = two operands
+ *
+ * Returns: Zero if successful, @-1@ if %$x/y$% is not a square.
+ *
+ * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%.
+ * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x
+ * \ne 0$% then the operation fails. If you wanted a specific
+ * square root then you'll have to pick it yourself.
+ */
+
+static const piece sqrtm1_pieces[NPIECE] = {
+ -32595792, -7943725, 9377950, 3500415, 12389472,
+ -272473, -25146209, -2005654, 326686, 11406482
+};
+#define SQRTM1 ((const f25519 *)sqrtm1_pieces)
+
+int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y)
+{
+ f25519 t, u, w, beta, xy3, t2p50m1;
+ octet xb[32], b0[32], b1[32];
+ int32 rc = -1;
+ mask32 m;
+ unsigned i;
+
+#define SQRN(z, x, n) do { \
+ f25519_sqr((z), (x)); \
+ 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.
+ *
+ * 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.
+ *
+ * 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(&t, &t, y);
+ f25519_neg(&u, &t);
+ f25519_store(xb, x);
+ f25519_store(b0, &t);
+ f25519_store(b1, &u);
+ f25519_mul(&u, &beta, SQRTM1);
+
+ m = -consttime_memeq(b0, xb, 32);
+ rc = PICK2(0, rc, m);
+ f25519_pick2(z, &beta, &u, m);
+ m = -consttime_memeq(b1, xb, 32);
+ rc = PICK2(0, rc, m);
+
+ /* And we're done. */
+ return (rc);
+}
+
+/*----- That's all, folks -------------------------------------------------*/
~mdw / secnet / blobdiff