X-Git-Url: https://git.distorted.org.uk/~mdw/secnet/blobdiff_plain/ae8cd973b817f81c075ab20e84d3239125146f24..8a7654a695e8ecdc8c14dced445839a7a7e6a8ca:/fgoldi.c diff --git a/fgoldi.c b/fgoldi.c new file mode 100644 index 0000000..14b98ed --- /dev/null +++ b/fgoldi.c @@ -0,0 +1,589 @@ +/* -*-c-*- + * + * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1) + * + * (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 "fgoldi.h" + +/*----- Basic setup -------------------------------------------------------* + * + * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1 + * (hence the name). + */ + +/* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i: + * x = SUM_{0<=i<16} x_i 2^(28i). + */ + +typedef int32 piece; typedef int64 dblpiece; +typedef uint32 upiece; typedef uint64 udblpiece; +#define PIECEWD(i) 28 +#define NPIECE 16 +#define P p28 + +#define B28 0x10000000u +#define B27 0x08000000u +#define M28 0x0fffffffu +#define M27 0x07ffffffu +#define M32 0xffffffffu + +/*----- Debugging machinery -----------------------------------------------*/ + +#if defined(FGOLDI_DEBUG) + +#include + +#include "mp.h" +#include "mptext.h" + +static mp *get_pgoldi(void) +{ + mp *p = MP_NEW, *t = MP_NEW; + + p = mp_setbit(p, MP_ZERO, 448); + t = mp_setbit(t, MP_ZERO, 224); + p = mp_sub(p, p, t); + p = mp_sub(p, p, MP_ONE); + mp_drop(t); + return (p); +} + +DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 56, get_pgoldi()) + +#endif + +/*----- Loading and storing -----------------------------------------------*/ + +/* --- @fgoldi_load@ --- * + * + * Arguments: @fgoldi *z@ = where to store the result + * @const octet xv[56]@ = source to read + * + * Returns: --- + * + * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% 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 fgoldi_load(fgoldi *z, const octet xv[56]) +{ + + unsigned i; + uint32 xw[14]; + piece b, c; + + /* First, read the input value as words. */ + for (i = 0; i < 14; i++) xw[i] = LOAD32_L(xv + 4*i); + + /* Extract unsigned 28-bit pieces from the words. */ + z->P[ 0] = (xw[ 0] >> 0)&M28; + z->P[ 7] = (xw[ 6] >> 4)&M28; + z->P[ 8] = (xw[ 7] >> 0)&M28; + z->P[15] = (xw[13] >> 4)&M28; + for (i = 1; i < 7; i++) { + z->P[i + 0] = ((xw[i + 0] << (4*i)) | (xw[i - 1] >> (32 - 4*i)))&M28; + z->P[i + 8] = ((xw[i + 7] << (4*i)) | (xw[i + 6] >> (32 - 4*i)))&M28; + } + + /* Convert the nonnegative pieces into a balanced signed representation, so + * each piece ends up in the interval |z_i| <= 2^27. For each piece, if + * its top bit is set, lend a bit leftwards; in the case of z_15, reduce + * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and + * φ^2 = φ + 1. We delay this 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. + * + * Note that we don't try for a canonical representation here: both upper + * and lower bounds are achievable. + */ + b = z->P[15]&B27; z->P[15] -= b << 1; c = b >> 27; + for (i = NPIECE - 1; i--; ) + { b = z->P[i]&B27; z->P[i] -= b << 1; z->P[i + 1] += b >> 27; } + z->P[0] += c; z->P[8] += c; +} + +/* --- @fgoldi_store@ --- * + * + * Arguments: @octet zv[56]@ = where to write the result + * @const fgoldi *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. + */ + +void fgoldi_store(octet zv[56], const fgoldi *x) +{ + + piece y[NPIECE], yy[NPIECE], c, d; + uint32 u, v; + mask32 m; + unsigned i; + + for (i = 0; i < NPIECE; i++) y[i] = x->P[i]; + + /* First, propagate the carries. By the end of this, we'll have all of the + * 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, φ^2). The whole represented + * value is then y + φ^2 c. + * + * Assume that we start out with |y_i| <= 2^30. We start off by cutting + * off and reducing the carry c_15 from the topmost piece, y_15. This + * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this + * onto y_0 and y_8, and propagate the carries. It's very clear that we'll + * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2. + * + * Here, the y_i are signed, so we must be cautious about bithacking them. + */ + c = ASR(piece, y[15], 28); y[15] = (upiece)y[15]&M28; y[8] += c; + for (i = 0; i < NPIECE; i++) + { y[i] += c; c = ASR(piece, y[i], 28); y[i] = (upiece)y[i]&M28; } + + /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and + * y >= p, then we should subtract p from the whole value; if c = -1 then + * we should add p; 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)&M28; + d = m&1; + + /* Now do the addition/subtraction. Remember that all of the y_i are + * nonnegative, so shifting and masking are safe and easy. + */ + d += y[0] + (1 ^ m); yy[0] = d&M28; d >>= 28; + for (i = 1; i < 8; i++) + { d += y[i] + m; yy[i] = d&M28; d >>= 28; } + d += y[8] + (1 ^ m); yy[8] = d&M28; d >>= 28; + for (i = 9; i < 16; i++) + { d += y[i] + m; yy[i] = d&M28; d >>= 28; } + + /* The final carry-out is in d; since we only did addition, and the y_i are + * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y, + * 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 + * p <= y < φ^2. + */ + m = NONZEROP(c) | ~NONZEROP(d - 1); + for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m); + + /* Extract 32-bit words from the value. */ + for (i = 0; i < 7; i++) { + u = ((y[i + 0] >> (4*i)) | ((uint32)y[i + 1] << (28 - 4*i)))&M32; + v = ((y[i + 8] >> (4*i)) | ((uint32)y[i + 9] << (28 - 4*i)))&M32; + STORE32_L(zv + 4*i, u); + STORE32_L(zv + 4*i + 28, v); + } +} + +/* --- @fgoldi_set@ --- * + * + * Arguments: @fgoldi *z@ = where to write the result + * @int a@ = a small-ish constant + * + * Returns: --- + * + * Use: Sets @z@ to equal @a@. + */ + +void fgoldi_set(fgoldi *x, int a) +{ + unsigned i; + + x->P[0] = a; + for (i = 1; i < NPIECE; i++) x->P[i] = 0; +} + +/*----- Basic arithmetic --------------------------------------------------*/ + +/* --- @fgoldi_add@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) + * @const fgoldi *x, *y@ = two operands + * + * Returns: --- + * + * Use: Set @z@ to the sum %$x + y$%. + */ + +void fgoldi_add(fgoldi *z, const fgoldi *x, const fgoldi *y) +{ + unsigned i; + for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i]; +} + +/* --- @fgoldi_sub@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) + * @const fgoldi *x, *y@ = two operands + * + * Returns: --- + * + * Use: Set @z@ to the difference %$x - y$%. + */ + +void fgoldi_sub(fgoldi *z, const fgoldi *x, const fgoldi *y) +{ + unsigned i; + for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i]; +} + +/*----- Constant-time utilities -------------------------------------------*/ + +/* --- @fgoldi_condswap@ --- * + * + * Arguments: @fgoldi *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 fgoldi_condswap(fgoldi *x, fgoldi *y, uint32 m) +{ + unsigned i; + mask32 mm = FIX_MASK32(m); + + for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm); +} + +/*----- 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 - 2^27 for some M. If this is the case, then, on + * exit, we will have |Z_i| <= 2^27 + M/2^27. + */ +#define CARRY_REDUCE(z, x) do { \ + dblpiece _t[NPIECE], _c; \ + unsigned _i; \ + \ + /* Bias the input pieces. This keeps the carries and so on centred \ + * around zero rather than biased positive. \ + */ \ + for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \ + \ + /* Calculate the reduced pieces. Careful with the bithacking. */ \ + _c = ASR(dblpiece, _t[15], 28); \ + (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \ + for (_i = 1; _i < NPIECE; _i++) { \ + (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \ + ASR(dblpiece, _t[_i - 1], 28); \ + } \ + (z)[8] += _c; \ +} while (0) + +/* --- @fgoldi_mulconst@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) + * @const fgoldi *x@ = an operand + * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. + * + * Returns: --- + * + * Use: Set @z@ to the product %$a x$%. + */ + +void fgoldi_mulconst(fgoldi *z, const fgoldi *x, long a) +{ + unsigned i; + dblpiece zz[NPIECE], aa = a; + + for (i = 0; i < NPIECE; i++) zz[i] = aa*x->P[i]; + CARRY_REDUCE(z->P, zz); +} + +/* --- @fgoldi_mul@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) + * @const fgoldi *x, *y@ = two operands + * + * Returns: --- + * + * Use: Set @z@ to the product %$x y$%. + */ + +void fgoldi_mul(fgoldi *z, const fgoldi *x, const fgoldi *y) +{ + dblpiece zz[NPIECE], u[NPIECE]; + piece ab[NPIECE/2], cd[NPIECE/2]; + const piece + *a = x->P + NPIECE/2, *b = x->P, + *c = y->P + NPIECE/2, *d = y->P; + unsigned i, j; + +# define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) + + /* Behold the magic. + * + * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 + + * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c = + * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose + * the prime p so that φ^2 = φ + 1. So + * + * x y = ((a + b) (c + d) - b d) φ + a c + b d + */ + + for (i = 0; i < NPIECE; i++) zz[i] = 0; + + /* Our first job will be to calculate (1 - φ) b d, and write the result + * into z. As we do this, an interesting thing will happen. Write + * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u. + * So, what we do is to write the product end-swapped and negated, and then + * we'll subtract the (negated, remember) high half from the low half. + */ + for (i = 0; i < NPIECE/2; i++) { + for (j = 0; j < NPIECE/2 - i; j++) + zz[i + j + NPIECE/2] -= M(b,i, d,j); + for (; j < NPIECE/2; j++) + zz[i + j - NPIECE/2] -= M(b,i, d,j); + } + for (i = 0; i < NPIECE/2; i++) + zz[i] -= zz[i + NPIECE/2]; + + /* Next, we add on a c. There are no surprises here. */ + for (i = 0; i < NPIECE/2; i++) + for (j = 0; j < NPIECE/2; j++) + zz[i + j] += M(a,i, c,j); + + /* Now, calculate a + b and c + d. */ + for (i = 0; i < NPIECE/2; i++) + { ab[i] = a[i] + b[i]; cd[i] = c[i] + d[i]; } + + /* Finally (for the multiplication) we must add on (a + b) (c + d) φ. + * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ = + * v φ + (1 + φ) u. We'll store u in a temporary place and add it on + * twice. + */ + for (i = 0; i < NPIECE; i++) u[i] = 0; + for (i = 0; i < NPIECE/2; i++) { + for (j = 0; j < NPIECE/2 - i; j++) + zz[i + j + NPIECE/2] += M(ab,i, cd,j); + for (; j < NPIECE/2; j++) + u[i + j - NPIECE/2] += M(ab,i, cd,j); + } + for (i = 0; i < NPIECE/2; i++) + { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } + +#undef M + + /* That wraps it up for the multiplication. Let's figure out some bounds. + * Fortunately, Karatsuba is a polynomial identity, so all of the pieces + * end up the way they'd be if we'd done the thing the easy way, which + * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5 + * 2^28. The overheads in the result are given by the coefficients of + * + * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1 + * + * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63. + * + * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 + + * 2^36; and a second round will leave us with |z_i| < 2^27 + 512. + */ + for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); + for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; +} + +/* --- @fgoldi_sqr@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) + * @const fgoldi *x@ = an operand + * + * Returns: --- + * + * Use: Set @z@ to the square %$x^2$%. + */ + +void fgoldi_sqr(fgoldi *z, const fgoldi *x) +{ + + dblpiece zz[NPIECE], u[NPIECE]; + piece ab[NPIECE]; + const piece *a = x->P + NPIECE/2, *b = x->P; + unsigned i, j; + +# define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) + + /* The magic is basically the same as `fgoldi_mul' above. We write + * x = a φ + b and use Karatsuba and the special prime shape. This time, + * we have + * + * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2 + */ + + for (i = 0; i < NPIECE; i++) zz[i] = 0; + + /* Our first job will be to calculate (1 - φ) b^2, and write the result + * into z. Again, this interacts pleasantly with the prime shape. + */ + for (i = 0; i < NPIECE/4; i++) { + zz[2*i + NPIECE/2] -= M(b,i, b,i); + for (j = i + 1; j < NPIECE/2 - i; j++) + zz[i + j + NPIECE/2] -= 2*M(b,i, b,j); + for (; j < NPIECE/2; j++) + zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); + } + for (; i < NPIECE/2; i++) { + zz[2*i - NPIECE/2] -= M(b,i, b,i); + for (j = i + 1; j < NPIECE/2; j++) + zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); + } + for (i = 0; i < NPIECE/2; i++) + zz[i] -= zz[i + NPIECE/2]; + + /* Next, we add on a^2. There are no surprises here. */ + for (i = 0; i < NPIECE/2; i++) { + zz[2*i] += M(a,i, a,i); + for (j = i + 1; j < NPIECE/2; j++) + zz[i + j] += 2*M(a,i, a,j); + } + + /* Now, calculate a + b. */ + for (i = 0; i < NPIECE/2; i++) + ab[i] = a[i] + b[i]; + + /* Finally (for the multiplication) we must add on (a + b)^2 φ. + * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll + * store u in a temporary place and add it on twice. + */ + for (i = 0; i < NPIECE; i++) u[i] = 0; + for (i = 0; i < NPIECE/4; i++) { + zz[2*i + NPIECE/2] += M(ab,i, ab,i); + for (j = i + 1; j < NPIECE/2 - i; j++) + zz[i + j + NPIECE/2] += 2*M(ab,i, ab,j); + for (; j < NPIECE/2; j++) + u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); + } + for (; i < NPIECE/2; i++) { + u[2*i - NPIECE/2] += M(ab,i, ab,i); + for (j = i + 1; j < NPIECE/2; j++) + u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); + } + for (i = 0; i < NPIECE/2; i++) + { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } + +#undef M + + /* Finally, carrying. */ + for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); + for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; +} + +/*----- More advanced operations ------------------------------------------*/ + +/* --- @fgoldi_inv@ --- * + * + * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) + * @const fgoldi *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 fgoldi_inv(fgoldi *z, const fgoldi *x) +{ + fgoldi t, u; + unsigned i; + +#define SQRN(z, x, n) do { \ + fgoldi_sqr((z), (x)); \ + for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ +} while (0) + + /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles + * x = 0 as intended. The addition chain is home-made. + */ /* step | value */ + fgoldi_sqr(&u, x); /* 1 | 2 */ + fgoldi_mul(&t, &u, x); /* 2 | 3 */ + SQRN(&u, &t, 2); /* 4 | 12 */ + fgoldi_mul(&t, &u, &t); /* 5 | 15 */ + SQRN(&u, &t, 4); /* 9 | 240 */ + fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ + SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ + fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ + SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ + fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ + SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ + fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ + fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ + fgoldi_mul(&t, &t, x); /* 43 | 2^37 - 1 */ + SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ + fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ + SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ + fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ + SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ + fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ + fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ + fgoldi_mul(&u, &u, x); /* 233 | 2^223 - 1 */ + SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ + fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ + SQRN(&t, &t, 2); /* 459 | 2^448 - 2^224 - 4 */ + fgoldi_mul(z, &t, x); /* 460 | 2^448 - 2^224 - 3 */ + +#undef SQRN +} + +/*----- That's all, folks -------------------------------------------------*/