X-Git-Url: https://git.distorted.org.uk/~mdw/secnet/blobdiff_plain/137d3b36ce3d121855364d3a28383ab86fbb9d7b..ae8cd973b817f81c075ab20e84d3239125146f24:/f25519.c diff --git a/f25519.c b/f25519.c deleted file mode 100644 index 4e0d1cc..0000000 --- a/f25519.c +++ /dev/null @@ -1,915 +0,0 @@ -/* - * 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 . - * - * * 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 - * - * (c) 2017 Straylight/Edgeware - */ - -/*----- Licensing notice --------------------------------------------------* - * - * This file is part of Catacomb. - * - * 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 - -#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= 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]; -} - -#ifndef F25519_TRIM_X25519 - -/* --- @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]; -} - -#endif - -/*----- Constant-time utilities -------------------------------------------*/ - -#ifndef F25519_TRIM_X25519 - -/* --- @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; - } -} - -#endif - -/* --- @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); -} - -#ifndef F25519_TRIM_X25519 - -/* --- @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 -} - -#endif - -/*----- 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 -} - -#ifndef F25519_TRIM_X25519 - -/* --- @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] = { -#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; - 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 = -ct_memeq(b0, xb, 32); - rc = PICK2(0, rc, m); - f25519_pick2(z, &beta, &u, m); - m = -ct_memeq(b1, xb, 32); - rc = PICK2(0, rc, m); - - /* And we're done. */ - return (rc); -} - -#endif - -/*----- That's all, folks -------------------------------------------------*/