--- /dev/null
+/* -*-c-*-
+ *
+ * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1)
+ *
+ * (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 "config.h"
+
+#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).
+ */
+
+#if FGOLDI_IMPL == 28
+/* 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
+
+#elif FGOLDI_IMPL == 12
+/* We represent an element of GF(p) as 40 signed integer pieces x_i: x =
+ * SUM_{0<=i<40} x_i 2^ceil(224i/20). Pieces i with i == 0 (mod 5) are 12
+ * bits wide; the others are 11 bits wide, so they form eight groups of 56
+ * bits.
+ */
+
+typedef int16 piece; typedef int32 dblpiece;
+typedef uint16 upiece; typedef uint32 udblpiece;
+#define PIECEWD(i) ((i)%5 ? 11 : 12)
+#define NPIECE 40
+#define P p12
+
+#define B12 0x1000u
+#define B11 0x0800u
+#define B10 0x0400u
+#define M12 0xfffu
+#define M11 0x7ffu
+#define M10 0x3ffu
+#define M8 0xffu
+
+#endif
+
+/*----- Debugging machinery -----------------------------------------------*/
+
+#if defined(FGOLDI_DEBUG) || defined(TEST_RIG)
+
+#include <stdio.h>
+
+#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])
+{
+#if FGOLDI_IMPL == 28
+
+ 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;
+
+#elif FGOLDI_IMPL == 12
+
+ unsigned i, j, n, w, b;
+ uint32 a;
+ int c;
+
+ /* First, convert the bytes into nonnegative pieces. */
+ for (i = j = a = n = 0, w = PIECEWD(0); i < 56; i++) {
+ a |= (uint32)xv[i] << n; n += 8;
+ if (n >= w) {
+ z->P[j++] = a&MASK(w);
+ a >>= w; n -= w; w = PIECEWD(j);
+ }
+ }
+
+ /* Convert the nonnegative pieces into a balanced signed representation, so
+ * each piece ends up in the interval |z_i| <= 2^11 + 1.
+ */
+ b = z->P[39]&B10; z->P[39] -= b << 1; c = b >> 10;
+ for (i = NPIECE - 1; i--; ) {
+ w = PIECEWD(i) - 1;
+ b = z->P[i]&BIT(w);
+ z->P[i] -= b << 1;
+ z->P[i + 1] += b >> w;
+ }
+ z->P[0] += c; z->P[20] += c;
+
+#endif
+}
+
+/* --- @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)
+{
+#if FGOLDI_IMPL == 28
+
+ 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);
+ }
+
+#elif FGOLDI_IMPL == 12
+
+ piece y[NPIECE], yy[NPIECE], c, d;
+ uint32 a;
+ mask32 m, mm;
+ unsigned i, j, n, w;
+
+ 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| <= 2^14. We start off by cutting
+ * off and reducing the carry c_39 from the topmost piece, y_39. This
+ * leaves 0 <= y_39 < 2^11; and we'll have |c_39| <= 16. We'll add this
+ * onto y_0 and y_20, and propagate the carries. It's very clear that
+ * we'll end up with |y + (φ + 1) c_39 - φ^2/2| << φ^2.
+ *
+ * Here, the y_i are signed, so we must be cautious about bithacking them.
+ */
+ c = ASR(piece, y[39], 11); y[39] = (piece)y[39]&M11; y[20] += c;
+ for (i = 0; i < NPIECE; i++) {
+ w = PIECEWD(i); m = (1 << w) - 1;
+ y[i] += c; c = ASR(piece, y[i], w); y[i] = (upiece)y[i]&m;
+ }
+
+ /* 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.
+ */
+ mm = SIGN(c);
+ 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 ^ (mm&M12)); yy[ 0] = d&M12; d >>= 12;
+ for (i = 1; i < 20; i++) {
+ w = PIECEWD(i); m = MASK(w);
+ d += y[ i] + (mm&m); yy[ i] = d&m; d >>= w;
+ }
+ d += y[20] + (1 ^ (mm&M12)); yy[20] = d&M12; d >>= 12;
+ for (i = 21; i < 40; i++) {
+ w = PIECEWD(i); m = MASK(w);
+ d += y[ i] + (mm&m); yy[ i] = d&m; d >>= w;
+ }
+
+ /* 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);
+
+ /* Convert that back into octets. */
+ for (i = j = a = n = 0; i < NPIECE; i++) {
+ a |= (uint32)y[i] << n; n += PIECEWD(i);
+ while (n >= 8) { zv[j++] = a&M8; a >>= 8; n -= 8; }
+ }
+
+#endif
+}
+
+/* --- @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 ----------------------------------------------------*/
+
+#if FGOLDI_IMPL == 28
+
+/* 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)
+
+#elif FGOLDI_IMPL == 12
+
+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_37; truncate it to 10 bits, and propagate the carry to x_38.
+ * Truncate x_38 to 10 bits, and add the carry onto x_39. Truncate x_39 to
+ * 10 bits, and add the carry onto x_0 and x_20. And so on.
+ *
+ * Once we reach x_37 for the second time, we start with |x_37| <= 2^10.
+ * The carry into x_37 is at most 2^21; so the carry out into x_38 has
+ * magnitude at most 2^10. In turn, |x_38| <= 2^10 before the carry, so is
+ * now no more than 2^11 in magnitude, and the carry out into x_39 is at
+ * most 1. This leaves |x_39| <= 2^10 + 1 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(37, 11, B10, M11); }
+ { x[38] += c; CARRY(38, 11, B10, M11); }
+ { x[39] += c; CARRY(39, 11, B10, M11); }
+ x[20] += c;
+ for (i = 0; i < 35; ) {
+ { x[i] += c; CARRY( i, 12, B11, M12); i++; }
+ for (j = i + 4; i < j; ) { x[i] += c; CARRY( i, 11, B10, M11); i++; }
+ }
+ { x[i] += c; CARRY( i, 12, B11, M12); i++; }
+ while (i < 39) { x[i] += c; CARRY( i, 11, B10, M11); i++; }
+ x[39] += c;
+}
+
+#endif
+
+/* --- @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];
+#if FGOLDI_IMPL == 28
+ CARRY_REDUCE(z->P, zz);
+#elif FGOLDI_IMPL == 12
+ carry_reduce(zz);
+ for (i = 0; i < NPIECE; i++) z->P[i] = zz[i];
+#endif
+}
+
+/* --- @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;
+
+#if FGOLDI_IMPL == 28
+
+# define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j])
+
+#elif FGOLDI_IMPL == 12
+
+ static const unsigned short off[39] = {
+ 0, 12, 23, 34, 45, 56, 68, 79, 90, 101,
+ 112, 124, 135, 146, 157, 168, 180, 191, 202, 213,
+ 224, 236, 247, 258, 269, 280, 292, 303, 314, 325,
+ 336, 348, 359, 370, 381, 392, 404, 415, 426
+ };
+
+#define M(x,i, y,j) \
+ (((dblpiece)(x)[i]*(y)[j]) << (off[i] + off[j] - off[(i) + (j)]))
+
+#endif
+
+ /* 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
+
+#if FGOLDI_IMPL == 28
+ /* 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];
+#elif FGOLDI_IMPL == 12
+ carry_reduce(zz);
+ for (i = 0; i < NPIECE; i++) z->P[i] = zz[i];
+#endif
+}
+
+/* --- @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)
+{
+#if FGOLDI_IMPL == 28
+
+ 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];
+
+#elif FGOLDI_IMPL == 12
+ fgoldi_mul(z, x, x);
+#endif
+}
+
+/*----- 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
+}
+
+/*----- 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 > 56)
+ die(1, "invalid length for fgoldi");
+ else if (d->len < 56) {
+ dstr_ensure(d, 56);
+ memset(d->buf + d->len, 0, 56 - d->len);
+ d->len = 56;
+ }
+}
+
+static void cvt_fgoldi(const char *buf, dstr *d)
+{
+ dstr dd = DSTR_INIT;
+
+ type_hex.cvt(buf, &dd); fixdstr(&dd);
+ dstr_ensure(d, sizeof(fgoldi)); d->len = sizeof(fgoldi);
+ fgoldi_load((fgoldi *)d->buf, (const octet *)dd.buf);
+ dstr_destroy(&dd);
+}
+
+static void dump_fgoldi(dstr *d, FILE *fp)
+ { fdump(stderr, "???", (const piece *)d->buf); }
+
+static void cvt_fgoldi_ref(const char *buf, dstr *d)
+ { type_hex.cvt(buf, d); fixdstr(d); }
+
+static void dump_fgoldi_ref(dstr *d, FILE *fp)
+{
+ fgoldi x;
+
+ fgoldi_load(&x, (const octet *)d->buf);
+ fdump(stderr, "???", x.P);
+}
+
+static int eq(const fgoldi *x, dstr *d)
+ { octet b[56]; fgoldi_store(b, x); return (memcmp(b, d->buf, 56) == 0); }
+
+static const test_type
+ type_fgoldi = { cvt_fgoldi, dump_fgoldi },
+ type_fgoldi_ref = { cvt_fgoldi_ref, dump_fgoldi_ref };
+
+#define TEST_UNOP(op) \
+ static int vrf_##op(dstr dv[]) \
+ { \
+ fgoldi *x = (fgoldi *)dv[0].buf; \
+ fgoldi z, zz; \
+ int ok = 1; \
+ \
+ fgoldi_##op(&z, x); \
+ if (!eq(&z, &dv[1])) { \
+ ok = 0; \
+ fprintf(stderr, "failed!\n"); \
+ fdump(stderr, "x", x->P); \
+ fdump(stderr, "calc", z.P); \
+ fgoldi_load(&zz, (const octet *)dv[1].buf); \
+ fdump(stderr, "z", zz.P); \
+ } \
+ \
+ return (ok); \
+ }
+
+TEST_UNOP(sqr)
+TEST_UNOP(inv)
+
+#define TEST_BINOP(op) \
+ static int vrf_##op(dstr dv[]) \
+ { \
+ fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf; \
+ fgoldi z, zz; \
+ int ok = 1; \
+ \
+ fgoldi_##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); \
+ fgoldi_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[])
+{
+ fgoldi *x = (fgoldi *)dv[0].buf;
+ long a = *(const long *)dv[1].buf;
+ fgoldi z, zz;
+ int ok = 1;
+
+ fgoldi_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);
+ fgoldi_load(&zz, (const octet *)dv[2].buf);
+ fdump(stderr, "z", zz.P);
+ }
+
+ return (ok);
+}
+
+static int vrf_condswap(dstr dv[])
+{
+ fgoldi *x = (fgoldi *)dv[0].buf, *y = (fgoldi *)dv[1].buf;
+ fgoldi xx = *x, yy = *y;
+ uint32 m = *(uint32 *)dv[2].buf;
+ int ok = 1;
+
+ fgoldi_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);
+ fgoldi_load(&xx, (const octet *)dv[3].buf);
+ fgoldi_load(&yy, (const octet *)dv[4].buf);
+ fdump(stderr, "want xx", xx.P);
+ fdump(stderr, "want yy", yy.P);
+ }
+
+ return (ok);
+}
+
+static int vrf_sub_mulc_add_sub_mul(dstr dv[])
+{
+ fgoldi *u = (fgoldi *)dv[0].buf, *v = (fgoldi *)dv[1].buf,
+ *w = (fgoldi *)dv[3].buf, *x = (fgoldi *)dv[4].buf,
+ *y = (fgoldi *)dv[5].buf;
+ long a = *(const long *)dv[2].buf;
+ fgoldi umv, aumv, wpaumv, xmy, z, zz;
+ int ok = 1;
+
+ fgoldi_sub(&umv, u, v);
+ fgoldi_mulconst(&aumv, &umv, a);
+ fgoldi_add(&wpaumv, w, &aumv);
+ fgoldi_sub(&xmy, x, y);
+ fgoldi_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);
+ fgoldi_load(&zz, (const octet *)dv[6].buf); fdump(stderr, "z", zz.P);
+ }
+
+ return (ok);
+}
+
+static test_chunk tests[] = {
+ { "add", vrf_add, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } },
+ { "sub", vrf_sub, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } },
+ { "mul", vrf_mul, { &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } },
+ { "mulconst", vrf_mulc, { &type_fgoldi, &type_long, &type_fgoldi_ref } },
+ { "condswap", vrf_condswap,
+ { &type_fgoldi, &type_fgoldi, &type_uint32,
+ &type_fgoldi_ref, &type_fgoldi_ref } },
+ { "sqr", vrf_sqr, { &type_fgoldi, &type_fgoldi_ref } },
+ { "inv", vrf_inv, { &type_fgoldi, &type_fgoldi_ref } },
+ { "sub-mulc-add-sub-mul", vrf_sub_mulc_add_sub_mul,
+ { &type_fgoldi, &type_fgoldi, &type_long, &type_fgoldi,
+ &type_fgoldi, &type_fgoldi, &type_fgoldi_ref } },
+ { 0, 0, { 0 } }
+};
+
+int main(int argc, char *argv[])
+{
+ test_run(argc, argv, tests, SRCDIR "/t/fgoldi");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
~mdw / secnet / blobdiff