--- /dev/null
+/*
+ * Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
+ *
+ * Permission is hereby granted, free of charge, to any person obtaining
+ * a copy of this software and associated documentation files (the
+ * "Software"), to deal in the Software without restriction, including
+ * without limitation the rights to use, copy, modify, merge, publish,
+ * distribute, sublicense, and/or sell copies of the Software, and to
+ * permit persons to whom the Software is furnished to do so, subject to
+ * the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be
+ * included in all copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+ * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+ * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+ * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
+ * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+ * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+ * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+ * SOFTWARE.
+ */
+
+#include "inner.h"
+
+#if BR_INT128 || BR_UMUL128
+
+#if BR_UMUL128
+#include <intrin.h>
+#endif
+
+static const unsigned char P256_G[] = {
+ 0x04, 0x6B, 0x17, 0xD1, 0xF2, 0xE1, 0x2C, 0x42, 0x47, 0xF8,
+ 0xBC, 0xE6, 0xE5, 0x63, 0xA4, 0x40, 0xF2, 0x77, 0x03, 0x7D,
+ 0x81, 0x2D, 0xEB, 0x33, 0xA0, 0xF4, 0xA1, 0x39, 0x45, 0xD8,
+ 0x98, 0xC2, 0x96, 0x4F, 0xE3, 0x42, 0xE2, 0xFE, 0x1A, 0x7F,
+ 0x9B, 0x8E, 0xE7, 0xEB, 0x4A, 0x7C, 0x0F, 0x9E, 0x16, 0x2B,
+ 0xCE, 0x33, 0x57, 0x6B, 0x31, 0x5E, 0xCE, 0xCB, 0xB6, 0x40,
+ 0x68, 0x37, 0xBF, 0x51, 0xF5
+};
+
+static const unsigned char P256_N[] = {
+ 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0xFF, 0xFF,
+ 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xBC, 0xE6, 0xFA, 0xAD,
+ 0xA7, 0x17, 0x9E, 0x84, 0xF3, 0xB9, 0xCA, 0xC2, 0xFC, 0x63,
+ 0x25, 0x51
+};
+
+static const unsigned char *
+api_generator(int curve, size_t *len)
+{
+ (void)curve;
+ *len = sizeof P256_G;
+ return P256_G;
+}
+
+static const unsigned char *
+api_order(int curve, size_t *len)
+{
+ (void)curve;
+ *len = sizeof P256_N;
+ return P256_N;
+}
+
+static size_t
+api_xoff(int curve, size_t *len)
+{
+ (void)curve;
+ *len = 32;
+ return 1;
+}
+
+/*
+ * A field element is encoded as four 64-bit integers, in basis 2^64.
+ * Values may reach up to 2^256-1. Montgomery multiplication is used.
+ */
+
+/* R = 2^256 mod p */
+static const uint64_t F256_R[] = {
+ 0x0000000000000001, 0xFFFFFFFF00000000,
+ 0xFFFFFFFFFFFFFFFF, 0x00000000FFFFFFFE
+};
+
+/* Curve equation is y^2 = x^3 - 3*x + B. This constant is B*R mod p
+ (Montgomery representation of B). */
+static const uint64_t P256_B_MONTY[] = {
+ 0xD89CDF6229C4BDDF, 0xACF005CD78843090,
+ 0xE5A220ABF7212ED6, 0xDC30061D04874834
+};
+
+/*
+ * Addition in the field.
+ */
+static inline void
+f256_add(uint64_t *d, const uint64_t *a, const uint64_t *b)
+{
+#if BR_INT128
+ unsigned __int128 w;
+ uint64_t t;
+
+ w = (unsigned __int128)a[0] + b[0];
+ d[0] = (uint64_t)w;
+ w = (unsigned __int128)a[1] + b[1] + (w >> 64);
+ d[1] = (uint64_t)w;
+ w = (unsigned __int128)a[2] + b[2] + (w >> 64);
+ d[2] = (uint64_t)w;
+ w = (unsigned __int128)a[3] + b[3] + (w >> 64);
+ d[3] = (uint64_t)w;
+ t = (uint64_t)(w >> 64);
+
+ /*
+ * 2^256 = 2^224 - 2^192 - 2^96 + 1 in the field.
+ */
+ w = (unsigned __int128)d[0] + t;
+ d[0] = (uint64_t)w;
+ w = (unsigned __int128)d[1] + (w >> 64) - (t << 32);
+ d[1] = (uint64_t)w;
+ /* Here, carry "w >> 64" can only be 0 or -1 */
+ w = (unsigned __int128)d[2] - ((w >> 64) & 1);
+ d[2] = (uint64_t)w;
+ /* Again, carry is 0 or -1 */
+ d[3] += (uint64_t)(w >> 64) + (t << 32) - t;
+
+#elif BR_UMUL128
+
+ unsigned char cc;
+ uint64_t t;
+
+ cc = _addcarry_u64(0, a[0], b[0], &d[0]);
+ cc = _addcarry_u64(cc, a[1], b[1], &d[1]);
+ cc = _addcarry_u64(cc, a[2], b[2], &d[2]);
+ cc = _addcarry_u64(cc, a[3], b[3], &d[3]);
+
+ /*
+ * If there is a carry, then we want to subtract p, which we
+ * do by adding 2^256 - p.
+ */
+ t = cc;
+ cc = _addcarry_u64(cc, d[0], 0, &d[0]);
+ cc = _addcarry_u64(cc, d[1], -(t << 32), &d[1]);
+ cc = _addcarry_u64(cc, d[2], -t, &d[2]);
+ (void)_addcarry_u64(cc, d[3], (t << 32) - (t << 1), &d[3]);
+
+#endif
+}
+
+/*
+ * Subtraction in the field.
+ */
+static inline void
+f256_sub(uint64_t *d, const uint64_t *a, const uint64_t *b)
+{
+#if BR_INT128
+
+ unsigned __int128 w;
+ uint64_t t;
+
+ w = (unsigned __int128)a[0] - b[0];
+ d[0] = (uint64_t)w;
+ w = (unsigned __int128)a[1] - b[1] - ((w >> 64) & 1);
+ d[1] = (uint64_t)w;
+ w = (unsigned __int128)a[2] - b[2] - ((w >> 64) & 1);
+ d[2] = (uint64_t)w;
+ w = (unsigned __int128)a[3] - b[3] - ((w >> 64) & 1);
+ d[3] = (uint64_t)w;
+ t = (uint64_t)(w >> 64) & 1;
+
+ /*
+ * p = 2^256 - 2^224 + 2^192 + 2^96 - 1.
+ */
+ w = (unsigned __int128)d[0] - t;
+ d[0] = (uint64_t)w;
+ w = (unsigned __int128)d[1] + (t << 32) - ((w >> 64) & 1);
+ d[1] = (uint64_t)w;
+ /* Here, carry "w >> 64" can only be 0 or +1 */
+ w = (unsigned __int128)d[2] + (w >> 64);
+ d[2] = (uint64_t)w;
+ /* Again, carry is 0 or +1 */
+ d[3] += (uint64_t)(w >> 64) - (t << 32) + t;
+
+#elif BR_UMUL128
+
+ unsigned char cc;
+ uint64_t t;
+
+ cc = _subborrow_u64(0, a[0], b[0], &d[0]);
+ cc = _subborrow_u64(cc, a[1], b[1], &d[1]);
+ cc = _subborrow_u64(cc, a[2], b[2], &d[2]);
+ cc = _subborrow_u64(cc, a[3], b[3], &d[3]);
+
+ /*
+ * If there is a carry, then we need to add p.
+ */
+ t = cc;
+ cc = _addcarry_u64(0, d[0], -t, &d[0]);
+ cc = _addcarry_u64(cc, d[1], (-t) >> 32, &d[1]);
+ cc = _addcarry_u64(cc, d[2], 0, &d[2]);
+ (void)_addcarry_u64(cc, d[3], t - (t << 32), &d[3]);
+
+#endif
+}
+
+/*
+ * Montgomery multiplication in the field.
+ */
+static void
+f256_montymul(uint64_t *d, const uint64_t *a, const uint64_t *b)
+{
+#if BR_INT128
+
+ uint64_t x, f, t0, t1, t2, t3, t4;
+ unsigned __int128 z, ff;
+ int i;
+
+ /*
+ * When computing d <- d + a[u]*b, we also add f*p such
+ * that d + a[u]*b + f*p is a multiple of 2^64. Since
+ * p = -1 mod 2^64, we can compute f = d[0] + a[u]*b[0] mod 2^64.
+ */
+
+ /*
+ * Step 1: t <- (a[0]*b + f*p) / 2^64
+ * We have f = a[0]*b[0] mod 2^64. Since p = -1 mod 2^64, this
+ * ensures that (a[0]*b + f*p) is a multiple of 2^64.
+ *
+ * We also have: f*p = f*2^256 - f*2^224 + f*2^192 + f*2^96 - f.
+ */
+ x = a[0];
+ z = (unsigned __int128)b[0] * x;
+ f = (uint64_t)z;
+ z = (unsigned __int128)b[1] * x + (z >> 64) + (uint64_t)(f << 32);
+ t0 = (uint64_t)z;
+ z = (unsigned __int128)b[2] * x + (z >> 64) + (uint64_t)(f >> 32);
+ t1 = (uint64_t)z;
+ z = (unsigned __int128)b[3] * x + (z >> 64) + f;
+ t2 = (uint64_t)z;
+ t3 = (uint64_t)(z >> 64);
+ ff = ((unsigned __int128)f << 64) - ((unsigned __int128)f << 32);
+ z = (unsigned __int128)t2 + (uint64_t)ff;
+ t2 = (uint64_t)z;
+ z = (unsigned __int128)t3 + (z >> 64) + (ff >> 64);
+ t3 = (uint64_t)z;
+ t4 = (uint64_t)(z >> 64);
+
+ /*
+ * Steps 2 to 4: t <- (t + a[i]*b + f*p) / 2^64
+ */
+ for (i = 1; i < 4; i ++) {
+ x = a[i];
+
+ /* t <- (t + x*b - f) / 2^64 */
+ z = (unsigned __int128)b[0] * x + t0;
+ f = (uint64_t)z;
+ z = (unsigned __int128)b[1] * x + t1 + (z >> 64);
+ t0 = (uint64_t)z;
+ z = (unsigned __int128)b[2] * x + t2 + (z >> 64);
+ t1 = (uint64_t)z;
+ z = (unsigned __int128)b[3] * x + t3 + (z >> 64);
+ t2 = (uint64_t)z;
+ z = t4 + (z >> 64);
+ t3 = (uint64_t)z;
+ t4 = (uint64_t)(z >> 64);
+
+ /* t <- t + f*2^32, carry in the upper half of z */
+ z = (unsigned __int128)t0 + (uint64_t)(f << 32);
+ t0 = (uint64_t)z;
+ z = (z >> 64) + (unsigned __int128)t1 + (uint64_t)(f >> 32);
+ t1 = (uint64_t)z;
+
+ /* t <- t + f*2^192 - f*2^160 + f*2^128 */
+ ff = ((unsigned __int128)f << 64)
+ - ((unsigned __int128)f << 32) + f;
+ z = (z >> 64) + (unsigned __int128)t2 + (uint64_t)ff;
+ t2 = (uint64_t)z;
+ z = (unsigned __int128)t3 + (z >> 64) + (ff >> 64);
+ t3 = (uint64_t)z;
+ t4 += (uint64_t)(z >> 64);
+ }
+
+ /*
+ * At that point, we have computed t = (a*b + F*p) / 2^256, where
+ * F is a 256-bit integer whose limbs are the "f" coefficients
+ * in the steps above. We have:
+ * a <= 2^256-1
+ * b <= 2^256-1
+ * F <= 2^256-1
+ * Hence:
+ * a*b + F*p <= (2^256-1)*(2^256-1) + p*(2^256-1)
+ * a*b + F*p <= 2^256*(2^256 - 2 + p) + 1 - p
+ * Therefore:
+ * t < 2^256 + p - 2
+ * Since p < 2^256, it follows that:
+ * t4 can be only 0 or 1
+ * t - p < 2^256
+ * We can therefore subtract p from t, conditionally on t4, to
+ * get a nonnegative result that fits on 256 bits.
+ */
+ z = (unsigned __int128)t0 + t4;
+ t0 = (uint64_t)z;
+ z = (unsigned __int128)t1 - (t4 << 32) + (z >> 64);
+ t1 = (uint64_t)z;
+ z = (unsigned __int128)t2 - (z >> 127);
+ t2 = (uint64_t)z;
+ t3 = t3 - (uint64_t)(z >> 127) - t4 + (t4 << 32);
+
+ d[0] = t0;
+ d[1] = t1;
+ d[2] = t2;
+ d[3] = t3;
+
+#elif BR_UMUL128
+
+ uint64_t x, f, t0, t1, t2, t3, t4;
+ uint64_t zl, zh, ffl, ffh;
+ unsigned char k, m;
+ int i;
+
+ /*
+ * When computing d <- d + a[u]*b, we also add f*p such
+ * that d + a[u]*b + f*p is a multiple of 2^64. Since
+ * p = -1 mod 2^64, we can compute f = d[0] + a[u]*b[0] mod 2^64.
+ */
+
+ /*
+ * Step 1: t <- (a[0]*b + f*p) / 2^64
+ * We have f = a[0]*b[0] mod 2^64. Since p = -1 mod 2^64, this
+ * ensures that (a[0]*b + f*p) is a multiple of 2^64.
+ *
+ * We also have: f*p = f*2^256 - f*2^224 + f*2^192 + f*2^96 - f.
+ */
+ x = a[0];
+
+ zl = _umul128(b[0], x, &zh);
+ f = zl;
+ t0 = zh;
+
+ zl = _umul128(b[1], x, &zh);
+ k = _addcarry_u64(0, zl, t0, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, f << 32, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ t0 = zl;
+ t1 = zh;
+
+ zl = _umul128(b[2], x, &zh);
+ k = _addcarry_u64(0, zl, t1, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, f >> 32, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ t1 = zl;
+ t2 = zh;
+
+ zl = _umul128(b[3], x, &zh);
+ k = _addcarry_u64(0, zl, t2, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, f, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ t2 = zl;
+ t3 = zh;
+
+ t4 = _addcarry_u64(0, t3, f, &t3);
+ k = _subborrow_u64(0, t2, f << 32, &t2);
+ k = _subborrow_u64(k, t3, f >> 32, &t3);
+ (void)_subborrow_u64(k, t4, 0, &t4);
+
+ /*
+ * Steps 2 to 4: t <- (t + a[i]*b + f*p) / 2^64
+ */
+ for (i = 1; i < 4; i ++) {
+ x = a[i];
+ /* f = t0 + x * b[0]; -- computed below */
+
+ /* t <- (t + x*b - f) / 2^64 */
+ zl = _umul128(b[0], x, &zh);
+ k = _addcarry_u64(0, zl, t0, &f);
+ (void)_addcarry_u64(k, zh, 0, &t0);
+
+ zl = _umul128(b[1], x, &zh);
+ k = _addcarry_u64(0, zl, t0, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, t1, &t0);
+ (void)_addcarry_u64(k, zh, 0, &t1);
+
+ zl = _umul128(b[2], x, &zh);
+ k = _addcarry_u64(0, zl, t1, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, t2, &t1);
+ (void)_addcarry_u64(k, zh, 0, &t2);
+
+ zl = _umul128(b[3], x, &zh);
+ k = _addcarry_u64(0, zl, t2, &zl);
+ (void)_addcarry_u64(k, zh, 0, &zh);
+ k = _addcarry_u64(0, zl, t3, &t2);
+ (void)_addcarry_u64(k, zh, 0, &t3);
+
+ t4 = _addcarry_u64(0, t3, t4, &t3);
+
+ /* t <- t + f*2^32, carry in k */
+ k = _addcarry_u64(0, t0, f << 32, &t0);
+ k = _addcarry_u64(k, t1, f >> 32, &t1);
+
+ /* t <- t + f*2^192 - f*2^160 + f*2^128 */
+ m = _subborrow_u64(0, f, f << 32, &ffl);
+ (void)_subborrow_u64(m, f, f >> 32, &ffh);
+ k = _addcarry_u64(k, t2, ffl, &t2);
+ k = _addcarry_u64(k, t3, ffh, &t3);
+ (void)_addcarry_u64(k, t4, 0, &t4);
+ }
+
+ /*
+ * At that point, we have computed t = (a*b + F*p) / 2^256, where
+ * F is a 256-bit integer whose limbs are the "f" coefficients
+ * in the steps above. We have:
+ * a <= 2^256-1
+ * b <= 2^256-1
+ * F <= 2^256-1
+ * Hence:
+ * a*b + F*p <= (2^256-1)*(2^256-1) + p*(2^256-1)
+ * a*b + F*p <= 2^256*(2^256 - 2 + p) + 1 - p
+ * Therefore:
+ * t < 2^256 + p - 2
+ * Since p < 2^256, it follows that:
+ * t4 can be only 0 or 1
+ * t - p < 2^256
+ * We can therefore subtract p from t, conditionally on t4, to
+ * get a nonnegative result that fits on 256 bits.
+ */
+ k = _addcarry_u64(0, t0, t4, &t0);
+ k = _addcarry_u64(k, t1, -(t4 << 32), &t1);
+ k = _addcarry_u64(k, t2, -t4, &t2);
+ (void)_addcarry_u64(k, t3, (t4 << 32) - (t4 << 1), &t3);
+
+ d[0] = t0;
+ d[1] = t1;
+ d[2] = t2;
+ d[3] = t3;
+
+#endif
+}
+
+/*
+ * Montgomery squaring in the field; currently a basic wrapper around
+ * multiplication (inline, should be optimized away).
+ * TODO: see if some extra speed can be gained here.
+ */
+static inline void
+f256_montysquare(uint64_t *d, const uint64_t *a)
+{
+ f256_montymul(d, a, a);
+}
+
+/*
+ * Convert to Montgomery representation.
+ */
+static void
+f256_tomonty(uint64_t *d, const uint64_t *a)
+{
+ /*
+ * R2 = 2^512 mod p.
+ * If R = 2^256 mod p, then R2 = R^2 mod p; and the Montgomery
+ * multiplication of a by R2 is: a*R2/R = a*R mod p, i.e. the
+ * conversion to Montgomery representation.
+ */
+ static const uint64_t R2[] = {
+ 0x0000000000000003,
+ 0xFFFFFFFBFFFFFFFF,
+ 0xFFFFFFFFFFFFFFFE,
+ 0x00000004FFFFFFFD
+ };
+
+ f256_montymul(d, a, R2);
+}
+
+/*
+ * Convert from Montgomery representation.
+ */
+static void
+f256_frommonty(uint64_t *d, const uint64_t *a)
+{
+ /*
+ * Montgomery multiplication by 1 is division by 2^256 modulo p.
+ */
+ static const uint64_t one[] = { 1, 0, 0, 0 };
+
+ f256_montymul(d, a, one);
+}
+
+/*
+ * Inversion in the field. If the source value is 0 modulo p, then this
+ * returns 0 or p. This function uses Montgomery representation.
+ */
+static void
+f256_invert(uint64_t *d, const uint64_t *a)
+{
+ /*
+ * We compute a^(p-2) mod p. The exponent pattern (from high to
+ * low) is:
+ * - 32 bits of value 1
+ * - 31 bits of value 0
+ * - 1 bit of value 1
+ * - 96 bits of value 0
+ * - 94 bits of value 1
+ * - 1 bit of value 0
+ * - 1 bit of value 1
+ * To speed up the square-and-multiply algorithm, we precompute
+ * a^(2^31-1).
+ */
+
+ uint64_t r[4], t[4];
+ int i;
+
+ memcpy(t, a, sizeof t);
+ for (i = 0; i < 30; i ++) {
+ f256_montysquare(t, t);
+ f256_montymul(t, t, a);
+ }
+
+ memcpy(r, t, sizeof t);
+ for (i = 224; i >= 0; i --) {
+ f256_montysquare(r, r);
+ switch (i) {
+ case 0:
+ case 2:
+ case 192:
+ case 224:
+ f256_montymul(r, r, a);
+ break;
+ case 3:
+ case 34:
+ case 65:
+ f256_montymul(r, r, t);
+ break;
+ }
+ }
+ memcpy(d, r, sizeof r);
+}
+
+/*
+ * Finalize reduction.
+ * Input value fits on 256 bits. This function subtracts p if and only
+ * if the input is greater than or equal to p.
+ */
+static inline void
+f256_final_reduce(uint64_t *a)
+{
+#if BR_INT128
+
+ uint64_t t0, t1, t2, t3, cc;
+ unsigned __int128 z;
+
+ /*
+ * We add 2^224 - 2^192 - 2^96 + 1 to a. If there is no carry,
+ * then a < p; otherwise, the addition result we computed is
+ * the value we must return.
+ */
+ z = (unsigned __int128)a[0] + 1;
+ t0 = (uint64_t)z;
+ z = (unsigned __int128)a[1] + (z >> 64) - ((uint64_t)1 << 32);
+ t1 = (uint64_t)z;
+ z = (unsigned __int128)a[2] - (z >> 127);
+ t2 = (uint64_t)z;
+ z = (unsigned __int128)a[3] - (z >> 127) + 0xFFFFFFFF;
+ t3 = (uint64_t)z;
+ cc = -(uint64_t)(z >> 64);
+
+ a[0] ^= cc & (a[0] ^ t0);
+ a[1] ^= cc & (a[1] ^ t1);
+ a[2] ^= cc & (a[2] ^ t2);
+ a[3] ^= cc & (a[3] ^ t3);
+
+#elif BR_UMUL128
+
+ uint64_t t0, t1, t2, t3, m;
+ unsigned char k;
+
+ k = _addcarry_u64(0, a[0], (uint64_t)1, &t0);
+ k = _addcarry_u64(k, a[1], -((uint64_t)1 << 32), &t1);
+ k = _addcarry_u64(k, a[2], -(uint64_t)1, &t2);
+ k = _addcarry_u64(k, a[3], ((uint64_t)1 << 32) - 2, &t3);
+ m = -(uint64_t)k;
+
+ a[0] ^= m & (a[0] ^ t0);
+ a[1] ^= m & (a[1] ^ t1);
+ a[2] ^= m & (a[2] ^ t2);
+ a[3] ^= m & (a[3] ^ t3);
+
+#endif
+}
+
+/*
+ * Points in affine and Jacobian coordinates.
+ *
+ * - In affine coordinates, the point-at-infinity cannot be encoded.
+ * - Jacobian coordinates (X,Y,Z) correspond to affine (X/Z^2,Y/Z^3);
+ * if Z = 0 then this is the point-at-infinity.
+ */
+typedef struct {
+ uint64_t x[4];
+ uint64_t y[4];
+} p256_affine;
+
+typedef struct {
+ uint64_t x[4];
+ uint64_t y[4];
+ uint64_t z[4];
+} p256_jacobian;
+
+/*
+ * Decode a point. The returned point is in Jacobian coordinates, but
+ * with z = 1. If the encoding is invalid, or encodes a point which is
+ * not on the curve, or encodes the point at infinity, then this function
+ * returns 0. Otherwise, 1 is returned.
+ *
+ * The buffer is assumed to have length exactly 65 bytes.
+ */
+static uint32_t
+point_decode(p256_jacobian *P, const unsigned char *buf)
+{
+ uint64_t x[4], y[4], t[4], x3[4], tt;
+ uint32_t r;
+
+ /*
+ * Header byte shall be 0x04.
+ */
+ r = EQ(buf[0], 0x04);
+
+ /*
+ * Decode X and Y coordinates, and convert them into
+ * Montgomery representation.
+ */
+ x[3] = br_dec64be(buf + 1);
+ x[2] = br_dec64be(buf + 9);
+ x[1] = br_dec64be(buf + 17);
+ x[0] = br_dec64be(buf + 25);
+ y[3] = br_dec64be(buf + 33);
+ y[2] = br_dec64be(buf + 41);
+ y[1] = br_dec64be(buf + 49);
+ y[0] = br_dec64be(buf + 57);
+ f256_tomonty(x, x);
+ f256_tomonty(y, y);
+
+ /*
+ * Verify y^2 = x^3 + A*x + B. In curve P-256, A = -3.
+ * Note that the Montgomery representation of 0 is 0. We must
+ * take care to apply the final reduction to make sure we have
+ * 0 and not p.
+ */
+ f256_montysquare(t, y);
+ f256_montysquare(x3, x);
+ f256_montymul(x3, x3, x);
+ f256_sub(t, t, x3);
+ f256_add(t, t, x);
+ f256_add(t, t, x);
+ f256_add(t, t, x);
+ f256_sub(t, t, P256_B_MONTY);
+ f256_final_reduce(t);
+ tt = t[0] | t[1] | t[2] | t[3];
+ r &= EQ((uint32_t)(tt | (tt >> 32)), 0);
+
+ /*
+ * Return the point in Jacobian coordinates (and Montgomery
+ * representation).
+ */
+ memcpy(P->x, x, sizeof x);
+ memcpy(P->y, y, sizeof y);
+ memcpy(P->z, F256_R, sizeof F256_R);
+ return r;
+}
+
+/*
+ * Final conversion for a point:
+ * - The point is converted back to affine coordinates.
+ * - Final reduction is performed.
+ * - The point is encoded into the provided buffer.
+ *
+ * If the point is the point-at-infinity, all operations are performed,
+ * but the buffer contents are indeterminate, and 0 is returned. Otherwise,
+ * the encoded point is written in the buffer, and 1 is returned.
+ */
+static uint32_t
+point_encode(unsigned char *buf, const p256_jacobian *P)
+{
+ uint64_t t1[4], t2[4], z;
+
+ /* Set t1 = 1/z^2 and t2 = 1/z^3. */
+ f256_invert(t2, P->z);
+ f256_montysquare(t1, t2);
+ f256_montymul(t2, t2, t1);
+
+ /* Compute affine coordinates x (in t1) and y (in t2). */
+ f256_montymul(t1, P->x, t1);
+ f256_montymul(t2, P->y, t2);
+
+ /* Convert back from Montgomery representation, and finalize
+ reductions. */
+ f256_frommonty(t1, t1);
+ f256_frommonty(t2, t2);
+ f256_final_reduce(t1);
+ f256_final_reduce(t2);
+
+ /* Encode. */
+ buf[0] = 0x04;
+ br_enc64be(buf + 1, t1[3]);
+ br_enc64be(buf + 9, t1[2]);
+ br_enc64be(buf + 17, t1[1]);
+ br_enc64be(buf + 25, t1[0]);
+ br_enc64be(buf + 33, t2[3]);
+ br_enc64be(buf + 41, t2[2]);
+ br_enc64be(buf + 49, t2[1]);
+ br_enc64be(buf + 57, t2[0]);
+
+ /* Return success if and only if P->z != 0. */
+ z = P->z[0] | P->z[1] | P->z[2] | P->z[3];
+ return NEQ((uint32_t)(z | z >> 32), 0);
+}
+
+/*
+ * Point doubling in Jacobian coordinates: point P is doubled.
+ * Note: if the source point is the point-at-infinity, then the result is
+ * still the point-at-infinity, which is correct. Moreover, if the three
+ * coordinates were zero, then they still are zero in the returned value.
+ *
+ * (Note: this is true even without the final reduction: if the three
+ * coordinates are encoded as four words of value zero each, then the
+ * result will also have all-zero coordinate encodings, not the alternate
+ * encoding as the integer p.)
+ */
+static void
+p256_double(p256_jacobian *P)
+{
+ /*
+ * Doubling formulas are:
+ *
+ * s = 4*x*y^2
+ * m = 3*(x + z^2)*(x - z^2)
+ * x' = m^2 - 2*s
+ * y' = m*(s - x') - 8*y^4
+ * z' = 2*y*z
+ *
+ * These formulas work for all points, including points of order 2
+ * and points at infinity:
+ * - If y = 0 then z' = 0. But there is no such point in P-256
+ * anyway.
+ * - If z = 0 then z' = 0.
+ */
+ uint64_t t1[4], t2[4], t3[4], t4[4];
+
+ /*
+ * Compute z^2 in t1.
+ */
+ f256_montysquare(t1, P->z);
+
+ /*
+ * Compute x-z^2 in t2 and x+z^2 in t1.
+ */
+ f256_add(t2, P->x, t1);
+ f256_sub(t1, P->x, t1);
+
+ /*
+ * Compute 3*(x+z^2)*(x-z^2) in t1.
+ */
+ f256_montymul(t3, t1, t2);
+ f256_add(t1, t3, t3);
+ f256_add(t1, t3, t1);
+
+ /*
+ * Compute 4*x*y^2 (in t2) and 2*y^2 (in t3).
+ */
+ f256_montysquare(t3, P->y);
+ f256_add(t3, t3, t3);
+ f256_montymul(t2, P->x, t3);
+ f256_add(t2, t2, t2);
+
+ /*
+ * Compute x' = m^2 - 2*s.
+ */
+ f256_montysquare(P->x, t1);
+ f256_sub(P->x, P->x, t2);
+ f256_sub(P->x, P->x, t2);
+
+ /*
+ * Compute z' = 2*y*z.
+ */
+ f256_montymul(t4, P->y, P->z);
+ f256_add(P->z, t4, t4);
+
+ /*
+ * Compute y' = m*(s - x') - 8*y^4. Note that we already have
+ * 2*y^2 in t3.
+ */
+ f256_sub(t2, t2, P->x);
+ f256_montymul(P->y, t1, t2);
+ f256_montysquare(t4, t3);
+ f256_add(t4, t4, t4);
+ f256_sub(P->y, P->y, t4);
+}
+
+/*
+ * Point addition (Jacobian coordinates): P1 is replaced with P1+P2.
+ * This function computes the wrong result in the following cases:
+ *
+ * - If P1 == 0 but P2 != 0
+ * - If P1 != 0 but P2 == 0
+ * - If P1 == P2
+ *
+ * In all three cases, P1 is set to the point at infinity.
+ *
+ * Returned value is 0 if one of the following occurs:
+ *
+ * - P1 and P2 have the same Y coordinate.
+ * - P1 == 0 and P2 == 0.
+ * - The Y coordinate of one of the points is 0 and the other point is
+ * the point at infinity.
+ *
+ * The third case cannot actually happen with valid points, since a point
+ * with Y == 0 is a point of order 2, and there is no point of order 2 on
+ * curve P-256.
+ *
+ * Therefore, assuming that P1 != 0 and P2 != 0 on input, then the caller
+ * can apply the following:
+ *
+ * - If the result is not the point at infinity, then it is correct.
+ * - Otherwise, if the returned value is 1, then this is a case of
+ * P1+P2 == 0, so the result is indeed the point at infinity.
+ * - Otherwise, P1 == P2, so a "double" operation should have been
+ * performed.
+ *
+ * Note that you can get a returned value of 0 with a correct result,
+ * e.g. if P1 and P2 have the same Y coordinate, but distinct X coordinates.
+ */
+static uint32_t
+p256_add(p256_jacobian *P1, const p256_jacobian *P2)
+{
+ /*
+ * Addtions formulas are:
+ *
+ * u1 = x1 * z2^2
+ * u2 = x2 * z1^2
+ * s1 = y1 * z2^3
+ * s2 = y2 * z1^3
+ * h = u2 - u1
+ * r = s2 - s1
+ * x3 = r^2 - h^3 - 2 * u1 * h^2
+ * y3 = r * (u1 * h^2 - x3) - s1 * h^3
+ * z3 = h * z1 * z2
+ */
+ uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt;
+ uint32_t ret;
+
+ /*
+ * Compute u1 = x1*z2^2 (in t1) and s1 = y1*z2^3 (in t3).
+ */
+ f256_montysquare(t3, P2->z);
+ f256_montymul(t1, P1->x, t3);
+ f256_montymul(t4, P2->z, t3);
+ f256_montymul(t3, P1->y, t4);
+
+ /*
+ * Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
+ */
+ f256_montysquare(t4, P1->z);
+ f256_montymul(t2, P2->x, t4);
+ f256_montymul(t5, P1->z, t4);
+ f256_montymul(t4, P2->y, t5);
+
+ /*
+ * Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
+ * We need to test whether r is zero, so we will do some extra
+ * reduce.
+ */
+ f256_sub(t2, t2, t1);
+ f256_sub(t4, t4, t3);
+ f256_final_reduce(t4);
+ tt = t4[0] | t4[1] | t4[2] | t4[3];
+ ret = (uint32_t)(tt | (tt >> 32));
+ ret = (ret | -ret) >> 31;
+
+ /*
+ * Compute u1*h^2 (in t6) and h^3 (in t5);
+ */
+ f256_montysquare(t7, t2);
+ f256_montymul(t6, t1, t7);
+ f256_montymul(t5, t7, t2);
+
+ /*
+ * Compute x3 = r^2 - h^3 - 2*u1*h^2.
+ */
+ f256_montysquare(P1->x, t4);
+ f256_sub(P1->x, P1->x, t5);
+ f256_sub(P1->x, P1->x, t6);
+ f256_sub(P1->x, P1->x, t6);
+
+ /*
+ * Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
+ */
+ f256_sub(t6, t6, P1->x);
+ f256_montymul(P1->y, t4, t6);
+ f256_montymul(t1, t5, t3);
+ f256_sub(P1->y, P1->y, t1);
+
+ /*
+ * Compute z3 = h*z1*z2.
+ */
+ f256_montymul(t1, P1->z, P2->z);
+ f256_montymul(P1->z, t1, t2);
+
+ return ret;
+}
+
+/*
+ * Point addition (mixed coordinates): P1 is replaced with P1+P2.
+ * This is a specialised function for the case when P2 is a non-zero point
+ * in affine coordinates.
+ *
+ * This function computes the wrong result in the following cases:
+ *
+ * - If P1 == 0
+ * - If P1 == P2
+ *
+ * In both cases, P1 is set to the point at infinity.
+ *
+ * Returned value is 0 if one of the following occurs:
+ *
+ * - P1 and P2 have the same Y (affine) coordinate.
+ * - The Y coordinate of P2 is 0 and P1 is the point at infinity.
+ *
+ * The second case cannot actually happen with valid points, since a point
+ * with Y == 0 is a point of order 2, and there is no point of order 2 on
+ * curve P-256.
+ *
+ * Therefore, assuming that P1 != 0 on input, then the caller
+ * can apply the following:
+ *
+ * - If the result is not the point at infinity, then it is correct.
+ * - Otherwise, if the returned value is 1, then this is a case of
+ * P1+P2 == 0, so the result is indeed the point at infinity.
+ * - Otherwise, P1 == P2, so a "double" operation should have been
+ * performed.
+ *
+ * Again, a value of 0 may be returned in some cases where the addition
+ * result is correct.
+ */
+static uint32_t
+p256_add_mixed(p256_jacobian *P1, const p256_affine *P2)
+{
+ /*
+ * Addtions formulas are:
+ *
+ * u1 = x1
+ * u2 = x2 * z1^2
+ * s1 = y1
+ * s2 = y2 * z1^3
+ * h = u2 - u1
+ * r = s2 - s1
+ * x3 = r^2 - h^3 - 2 * u1 * h^2
+ * y3 = r * (u1 * h^2 - x3) - s1 * h^3
+ * z3 = h * z1
+ */
+ uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt;
+ uint32_t ret;
+
+ /*
+ * Compute u1 = x1 (in t1) and s1 = y1 (in t3).
+ */
+ memcpy(t1, P1->x, sizeof t1);
+ memcpy(t3, P1->y, sizeof t3);
+
+ /*
+ * Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
+ */
+ f256_montysquare(t4, P1->z);
+ f256_montymul(t2, P2->x, t4);
+ f256_montymul(t5, P1->z, t4);
+ f256_montymul(t4, P2->y, t5);
+
+ /*
+ * Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
+ * We need to test whether r is zero, so we will do some extra
+ * reduce.
+ */
+ f256_sub(t2, t2, t1);
+ f256_sub(t4, t4, t3);
+ f256_final_reduce(t4);
+ tt = t4[0] | t4[1] | t4[2] | t4[3];
+ ret = (uint32_t)(tt | (tt >> 32));
+ ret = (ret | -ret) >> 31;
+
+ /*
+ * Compute u1*h^2 (in t6) and h^3 (in t5);
+ */
+ f256_montysquare(t7, t2);
+ f256_montymul(t6, t1, t7);
+ f256_montymul(t5, t7, t2);
+
+ /*
+ * Compute x3 = r^2 - h^3 - 2*u1*h^2.
+ */
+ f256_montysquare(P1->x, t4);
+ f256_sub(P1->x, P1->x, t5);
+ f256_sub(P1->x, P1->x, t6);
+ f256_sub(P1->x, P1->x, t6);
+
+ /*
+ * Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
+ */
+ f256_sub(t6, t6, P1->x);
+ f256_montymul(P1->y, t4, t6);
+ f256_montymul(t1, t5, t3);
+ f256_sub(P1->y, P1->y, t1);
+
+ /*
+ * Compute z3 = h*z1*z2.
+ */
+ f256_montymul(P1->z, P1->z, t2);
+
+ return ret;
+}
+
+#if 0
+/* unused */
+/*
+ * Point addition (mixed coordinates, complete): P1 is replaced with P1+P2.
+ * This is a specialised function for the case when P2 is a non-zero point
+ * in affine coordinates.
+ *
+ * This function returns the correct result in all cases.
+ */
+static uint32_t
+p256_add_complete_mixed(p256_jacobian *P1, const p256_affine *P2)
+{
+ /*
+ * Addtions formulas, in the general case, are:
+ *
+ * u1 = x1
+ * u2 = x2 * z1^2
+ * s1 = y1
+ * s2 = y2 * z1^3
+ * h = u2 - u1
+ * r = s2 - s1
+ * x3 = r^2 - h^3 - 2 * u1 * h^2
+ * y3 = r * (u1 * h^2 - x3) - s1 * h^3
+ * z3 = h * z1
+ *
+ * These formulas mishandle the two following cases:
+ *
+ * - If P1 is the point-at-infinity (z1 = 0), then z3 is
+ * incorrectly set to 0.
+ *
+ * - If P1 = P2, then u1 = u2 and s1 = s2, and x3, y3 and z3
+ * are all set to 0.
+ *
+ * However, if P1 + P2 = 0, then u1 = u2 but s1 != s2, and then
+ * we correctly get z3 = 0 (the point-at-infinity).
+ *
+ * To fix the case P1 = 0, we perform at the end a copy of P2
+ * over P1, conditional to z1 = 0.
+ *
+ * For P1 = P2: in that case, both h and r are set to 0, and
+ * we get x3, y3 and z3 equal to 0. We can test for that
+ * occurrence to make a mask which will be all-one if P1 = P2,
+ * or all-zero otherwise; then we can compute the double of P2
+ * and add it, combined with the mask, to (x3,y3,z3).
+ *
+ * Using the doubling formulas in p256_double() on (x2,y2),
+ * simplifying since P2 is affine (i.e. z2 = 1, implicitly),
+ * we get:
+ * s = 4*x2*y2^2
+ * m = 3*(x2 + 1)*(x2 - 1)
+ * x' = m^2 - 2*s
+ * y' = m*(s - x') - 8*y2^4
+ * z' = 2*y2
+ * which requires only 6 multiplications. Added to the 11
+ * multiplications of the normal mixed addition in Jacobian
+ * coordinates, we get a cost of 17 multiplications in total.
+ */
+ uint64_t t1[4], t2[4], t3[4], t4[4], t5[4], t6[4], t7[4], tt, zz;
+ int i;
+
+ /*
+ * Set zz to -1 if P1 is the point at infinity, 0 otherwise.
+ */
+ zz = P1->z[0] | P1->z[1] | P1->z[2] | P1->z[3];
+ zz = ((zz | -zz) >> 63) - (uint64_t)1;
+
+ /*
+ * Compute u1 = x1 (in t1) and s1 = y1 (in t3).
+ */
+ memcpy(t1, P1->x, sizeof t1);
+ memcpy(t3, P1->y, sizeof t3);
+
+ /*
+ * Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
+ */
+ f256_montysquare(t4, P1->z);
+ f256_montymul(t2, P2->x, t4);
+ f256_montymul(t5, P1->z, t4);
+ f256_montymul(t4, P2->y, t5);
+
+ /*
+ * Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
+ * reduce.
+ */
+ f256_sub(t2, t2, t1);
+ f256_sub(t4, t4, t3);
+
+ /*
+ * If both h = 0 and r = 0, then P1 = P2, and we want to set
+ * the mask tt to -1; otherwise, the mask will be 0.
+ */
+ f256_final_reduce(t2);
+ f256_final_reduce(t4);
+ tt = t2[0] | t2[1] | t2[2] | t2[3] | t4[0] | t4[1] | t4[2] | t4[3];
+ tt = ((tt | -tt) >> 63) - (uint64_t)1;
+
+ /*
+ * Compute u1*h^2 (in t6) and h^3 (in t5);
+ */
+ f256_montysquare(t7, t2);
+ f256_montymul(t6, t1, t7);
+ f256_montymul(t5, t7, t2);
+
+ /*
+ * Compute x3 = r^2 - h^3 - 2*u1*h^2.
+ */
+ f256_montysquare(P1->x, t4);
+ f256_sub(P1->x, P1->x, t5);
+ f256_sub(P1->x, P1->x, t6);
+ f256_sub(P1->x, P1->x, t6);
+
+ /*
+ * Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
+ */
+ f256_sub(t6, t6, P1->x);
+ f256_montymul(P1->y, t4, t6);
+ f256_montymul(t1, t5, t3);
+ f256_sub(P1->y, P1->y, t1);
+
+ /*
+ * Compute z3 = h*z1.
+ */
+ f256_montymul(P1->z, P1->z, t2);
+
+ /*
+ * The "double" result, in case P1 = P2.
+ */
+
+ /*
+ * Compute z' = 2*y2 (in t1).
+ */
+ f256_add(t1, P2->y, P2->y);
+
+ /*
+ * Compute 2*(y2^2) (in t2) and s = 4*x2*(y2^2) (in t3).
+ */
+ f256_montysquare(t2, P2->y);
+ f256_add(t2, t2, t2);
+ f256_add(t3, t2, t2);
+ f256_montymul(t3, P2->x, t3);
+
+ /*
+ * Compute m = 3*(x2^2 - 1) (in t4).
+ */
+ f256_montysquare(t4, P2->x);
+ f256_sub(t4, t4, F256_R);
+ f256_add(t5, t4, t4);
+ f256_add(t4, t4, t5);
+
+ /*
+ * Compute x' = m^2 - 2*s (in t5).
+ */
+ f256_montysquare(t5, t4);
+ f256_sub(t5, t3);
+ f256_sub(t5, t3);
+
+ /*
+ * Compute y' = m*(s - x') - 8*y2^4 (in t6).
+ */
+ f256_sub(t6, t3, t5);
+ f256_montymul(t6, t6, t4);
+ f256_montysquare(t7, t2);
+ f256_sub(t6, t6, t7);
+ f256_sub(t6, t6, t7);
+
+ /*
+ * We now have the alternate (doubling) coordinates in (t5,t6,t1).
+ * We combine them with (x3,y3,z3).
+ */
+ for (i = 0; i < 4; i ++) {
+ P1->x[i] |= tt & t5[i];
+ P1->y[i] |= tt & t6[i];
+ P1->z[i] |= tt & t1[i];
+ }
+
+ /*
+ * If P1 = 0, then we get z3 = 0 (which is invalid); if z1 is 0,
+ * then we want to replace the result with a copy of P2. The
+ * test on z1 was done at the start, in the zz mask.
+ */
+ for (i = 0; i < 4; i ++) {
+ P1->x[i] ^= zz & (P1->x[i] ^ P2->x[i]);
+ P1->y[i] ^= zz & (P1->y[i] ^ P2->y[i]);
+ P1->z[i] ^= zz & (P1->z[i] ^ F256_R[i]);
+ }
+}
+#endif
+
+/*
+ * Inner function for computing a point multiplication. A window is
+ * provided, with points 1*P to 15*P in affine coordinates.
+ *
+ * Assumptions:
+ * - All provided points are valid points on the curve.
+ * - Multiplier is non-zero, and smaller than the curve order.
+ * - Everything is in Montgomery representation.
+ */
+static void
+point_mul_inner(p256_jacobian *R, const p256_affine *W,
+ const unsigned char *k, size_t klen)
+{
+ p256_jacobian Q;
+ uint32_t qz;
+
+ memset(&Q, 0, sizeof Q);
+ qz = 1;
+ while (klen -- > 0) {
+ int i;
+ unsigned bk;
+
+ bk = *k ++;
+ for (i = 0; i < 2; i ++) {
+ uint32_t bits;
+ uint32_t bnz;
+ p256_affine T;
+ p256_jacobian U;
+ uint32_t n;
+ int j;
+ uint64_t m;
+
+ p256_double(&Q);
+ p256_double(&Q);
+ p256_double(&Q);
+ p256_double(&Q);
+ bits = (bk >> 4) & 0x0F;
+ bnz = NEQ(bits, 0);
+
+ /*
+ * Lookup point in window. If the bits are 0,
+ * we get something invalid, which is not a
+ * problem because we will use it only if the
+ * bits are non-zero.
+ */
+ memset(&T, 0, sizeof T);
+ for (n = 0; n < 15; n ++) {
+ m = -(uint64_t)EQ(bits, n + 1);
+ T.x[0] |= m & W[n].x[0];
+ T.x[1] |= m & W[n].x[1];
+ T.x[2] |= m & W[n].x[2];
+ T.x[3] |= m & W[n].x[3];
+ T.y[0] |= m & W[n].y[0];
+ T.y[1] |= m & W[n].y[1];
+ T.y[2] |= m & W[n].y[2];
+ T.y[3] |= m & W[n].y[3];
+ }
+
+ U = Q;
+ p256_add_mixed(&U, &T);
+
+ /*
+ * If qz is still 1, then Q was all-zeros, and this
+ * is conserved through p256_double().
+ */
+ m = -(uint64_t)(bnz & qz);
+ for (j = 0; j < 4; j ++) {
+ Q.x[j] |= m & T.x[j];
+ Q.y[j] |= m & T.y[j];
+ Q.z[j] |= m & F256_R[j];
+ }
+ CCOPY(bnz & ~qz, &Q, &U, sizeof Q);
+ qz &= ~bnz;
+ bk <<= 4;
+ }
+ }
+ *R = Q;
+}
+
+/*
+ * Convert a window from Jacobian to affine coordinates. A single
+ * field inversion is used. This function works for windows up to
+ * 32 elements.
+ *
+ * The destination array (aff[]) and the source array (jac[]) may
+ * overlap, provided that the start of aff[] is not after the start of
+ * jac[]. Even if the arrays do _not_ overlap, the source array is
+ * modified.
+ */
+static void
+window_to_affine(p256_affine *aff, p256_jacobian *jac, int num)
+{
+ /*
+ * Convert the window points to affine coordinates. We use the
+ * following trick to mutualize the inversion computation: if
+ * we have z1, z2, z3, and z4, and want to inverse all of them,
+ * we compute u = 1/(z1*z2*z3*z4), and then we have:
+ * 1/z1 = u*z2*z3*z4
+ * 1/z2 = u*z1*z3*z4
+ * 1/z3 = u*z1*z2*z4
+ * 1/z4 = u*z1*z2*z3
+ *
+ * The partial products are computed recursively:
+ *
+ * - on input (z_1,z_2), return (z_2,z_1) and z_1*z_2
+ * - on input (z_1,z_2,... z_n):
+ * recurse on (z_1,z_2,... z_(n/2)) -> r1 and m1
+ * recurse on (z_(n/2+1),z_(n/2+2)... z_n) -> r2 and m2
+ * multiply elements of r1 by m2 -> s1
+ * multiply elements of r2 by m1 -> s2
+ * return r1||r2 and m1*m2
+ *
+ * In the example below, we suppose that we have 14 elements.
+ * Let z1, z2,... zE be the 14 values to invert (index noted in
+ * hexadecimal, starting at 1).
+ *
+ * - Depth 1:
+ * swap(z1, z2); z12 = z1*z2
+ * swap(z3, z4); z34 = z3*z4
+ * swap(z5, z6); z56 = z5*z6
+ * swap(z7, z8); z78 = z7*z8
+ * swap(z9, zA); z9A = z9*zA
+ * swap(zB, zC); zBC = zB*zC
+ * swap(zD, zE); zDE = zD*zE
+ *
+ * - Depth 2:
+ * z1 <- z1*z34, z2 <- z2*z34, z3 <- z3*z12, z4 <- z4*z12
+ * z1234 = z12*z34
+ * z5 <- z5*z78, z6 <- z6*z78, z7 <- z7*z56, z8 <- z8*z56
+ * z5678 = z56*z78
+ * z9 <- z9*zBC, zA <- zA*zBC, zB <- zB*z9A, zC <- zC*z9A
+ * z9ABC = z9A*zBC
+ *
+ * - Depth 3:
+ * z1 <- z1*z5678, z2 <- z2*z5678, z3 <- z3*z5678, z4 <- z4*z5678
+ * z5 <- z5*z1234, z6 <- z6*z1234, z7 <- z7*z1234, z8 <- z8*z1234
+ * z12345678 = z1234*z5678
+ * z9 <- z9*zDE, zA <- zA*zDE, zB <- zB*zDE, zC <- zC*zDE
+ * zD <- zD*z9ABC, zE*z9ABC
+ * z9ABCDE = z9ABC*zDE
+ *
+ * - Depth 4:
+ * multiply z1..z8 by z9ABCDE
+ * multiply z9..zE by z12345678
+ * final z = z12345678*z9ABCDE
+ */
+
+ uint64_t z[16][4];
+ int i, k, s;
+#define zt (z[15])
+#define zu (z[14])
+#define zv (z[13])
+
+ /*
+ * First recursion step (pairwise swapping and multiplication).
+ * If there is an odd number of elements, then we "invent" an
+ * extra one with coordinate Z = 1 (in Montgomery representation).
+ */
+ for (i = 0; (i + 1) < num; i += 2) {
+ memcpy(zt, jac[i].z, sizeof zt);
+ memcpy(jac[i].z, jac[i + 1].z, sizeof zt);
+ memcpy(jac[i + 1].z, zt, sizeof zt);
+ f256_montymul(z[i >> 1], jac[i].z, jac[i + 1].z);
+ }
+ if ((num & 1) != 0) {
+ memcpy(z[num >> 1], jac[num - 1].z, sizeof zt);
+ memcpy(jac[num - 1].z, F256_R, sizeof F256_R);
+ }
+
+ /*
+ * Perform further recursion steps. At the entry of each step,
+ * the process has been done for groups of 's' points. The
+ * integer k is the log2 of s.
+ */
+ for (k = 1, s = 2; s < num; k ++, s <<= 1) {
+ int n;
+
+ for (i = 0; i < num; i ++) {
+ f256_montymul(jac[i].z, jac[i].z, z[(i >> k) ^ 1]);
+ }
+ n = (num + s - 1) >> k;
+ for (i = 0; i < (n >> 1); i ++) {
+ f256_montymul(z[i], z[i << 1], z[(i << 1) + 1]);
+ }
+ if ((n & 1) != 0) {
+ memmove(z[n >> 1], z[n], sizeof zt);
+ }
+ }
+
+ /*
+ * Invert the final result, and convert all points.
+ */
+ f256_invert(zt, z[0]);
+ for (i = 0; i < num; i ++) {
+ f256_montymul(zv, jac[i].z, zt);
+ f256_montysquare(zu, zv);
+ f256_montymul(zv, zv, zu);
+ f256_montymul(aff[i].x, jac[i].x, zu);
+ f256_montymul(aff[i].y, jac[i].y, zv);
+ }
+}
+
+/*
+ * Multiply the provided point by an integer.
+ * Assumptions:
+ * - Source point is a valid curve point.
+ * - Source point is not the point-at-infinity.
+ * - Integer is not 0, and is lower than the curve order.
+ * If these conditions are not met, then the result is indeterminate
+ * (but the process is still constant-time).
+ */
+static void
+p256_mul(p256_jacobian *P, const unsigned char *k, size_t klen)
+{
+ union {
+ p256_affine aff[15];
+ p256_jacobian jac[15];
+ } window;
+ int i;
+
+ /*
+ * Compute window, in Jacobian coordinates.
+ */
+ window.jac[0] = *P;
+ for (i = 2; i < 16; i ++) {
+ window.jac[i - 1] = window.jac[(i >> 1) - 1];
+ if ((i & 1) == 0) {
+ p256_double(&window.jac[i - 1]);
+ } else {
+ p256_add(&window.jac[i - 1], &window.jac[i >> 1]);
+ }
+ }
+
+ /*
+ * Convert the window points to affine coordinates. Point
+ * window[0] is the source point, already in affine coordinates.
+ */
+ window_to_affine(window.aff, window.jac, 15);
+
+ /*
+ * Perform point multiplication.
+ */
+ point_mul_inner(P, window.aff, k, klen);
+}
+
+/*
+ * Precomputed window for the conventional generator: P256_Gwin[n]
+ * contains (n+1)*G (affine coordinates, in Montgomery representation).
+ */
+static const p256_affine P256_Gwin[] = {
+ {
+ { 0x79E730D418A9143C, 0x75BA95FC5FEDB601,
+ 0x79FB732B77622510, 0x18905F76A53755C6 },
+ { 0xDDF25357CE95560A, 0x8B4AB8E4BA19E45C,
+ 0xD2E88688DD21F325, 0x8571FF1825885D85 }
+ },
+ {
+ { 0x850046D410DDD64D, 0xAA6AE3C1A433827D,
+ 0x732205038D1490D9, 0xF6BB32E43DCF3A3B },
+ { 0x2F3648D361BEE1A5, 0x152CD7CBEB236FF8,
+ 0x19A8FB0E92042DBE, 0x78C577510A5B8A3B }
+ },
+ {
+ { 0xFFAC3F904EEBC127, 0xB027F84A087D81FB,
+ 0x66AD77DD87CBBC98, 0x26936A3FB6FF747E },
+ { 0xB04C5C1FC983A7EB, 0x583E47AD0861FE1A,
+ 0x788208311A2EE98E, 0xD5F06A29E587CC07 }
+ },
+ {
+ { 0x74B0B50D46918DCC, 0x4650A6EDC623C173,
+ 0x0CDAACACE8100AF2, 0x577362F541B0176B },
+ { 0x2D96F24CE4CBABA6, 0x17628471FAD6F447,
+ 0x6B6C36DEE5DDD22E, 0x84B14C394C5AB863 }
+ },
+ {
+ { 0xBE1B8AAEC45C61F5, 0x90EC649A94B9537D,
+ 0x941CB5AAD076C20C, 0xC9079605890523C8 },
+ { 0xEB309B4AE7BA4F10, 0x73C568EFE5EB882B,
+ 0x3540A9877E7A1F68, 0x73A076BB2DD1E916 }
+ },
+ {
+ { 0x403947373E77664A, 0x55AE744F346CEE3E,
+ 0xD50A961A5B17A3AD, 0x13074B5954213673 },
+ { 0x93D36220D377E44B, 0x299C2B53ADFF14B5,
+ 0xF424D44CEF639F11, 0xA4C9916D4A07F75F }
+ },
+ {
+ { 0x0746354EA0173B4F, 0x2BD20213D23C00F7,
+ 0xF43EAAB50C23BB08, 0x13BA5119C3123E03 },
+ { 0x2847D0303F5B9D4D, 0x6742F2F25DA67BDD,
+ 0xEF933BDC77C94195, 0xEAEDD9156E240867 }
+ },
+ {
+ { 0x27F14CD19499A78F, 0x462AB5C56F9B3455,
+ 0x8F90F02AF02CFC6B, 0xB763891EB265230D },
+ { 0xF59DA3A9532D4977, 0x21E3327DCF9EBA15,
+ 0x123C7B84BE60BBF0, 0x56EC12F27706DF76 }
+ },
+ {
+ { 0x75C96E8F264E20E8, 0xABE6BFED59A7A841,
+ 0x2CC09C0444C8EB00, 0xE05B3080F0C4E16B },
+ { 0x1EB7777AA45F3314, 0x56AF7BEDCE5D45E3,
+ 0x2B6E019A88B12F1A, 0x086659CDFD835F9B }
+ },
+ {
+ { 0x2C18DBD19DC21EC8, 0x98F9868A0FCF8139,
+ 0x737D2CD648250B49, 0xCC61C94724B3428F },
+ { 0x0C2B407880DD9E76, 0xC43A8991383FBE08,
+ 0x5F7D2D65779BE5D2, 0x78719A54EB3B4AB5 }
+ },
+ {
+ { 0xEA7D260A6245E404, 0x9DE407956E7FDFE0,
+ 0x1FF3A4158DAC1AB5, 0x3E7090F1649C9073 },
+ { 0x1A7685612B944E88, 0x250F939EE57F61C8,
+ 0x0C0DAA891EAD643D, 0x68930023E125B88E }
+ },
+ {
+ { 0x04B71AA7D2697768, 0xABDEDEF5CA345A33,
+ 0x2409D29DEE37385E, 0x4EE1DF77CB83E156 },
+ { 0x0CAC12D91CBB5B43, 0x170ED2F6CA895637,
+ 0x28228CFA8ADE6D66, 0x7FF57C9553238ACA }
+ },
+ {
+ { 0xCCC425634B2ED709, 0x0E356769856FD30D,
+ 0xBCBCD43F559E9811, 0x738477AC5395B759 },
+ { 0x35752B90C00EE17F, 0x68748390742ED2E3,
+ 0x7CD06422BD1F5BC1, 0xFBC08769C9E7B797 }
+ },
+ {
+ { 0xA242A35BB0CF664A, 0x126E48F77F9707E3,
+ 0x1717BF54C6832660, 0xFAAE7332FD12C72E },
+ { 0x27B52DB7995D586B, 0xBE29569E832237C2,
+ 0xE8E4193E2A65E7DB, 0x152706DC2EAA1BBB }
+ },
+ {
+ { 0x72BCD8B7BC60055B, 0x03CC23EE56E27E4B,
+ 0xEE337424E4819370, 0xE2AA0E430AD3DA09 },
+ { 0x40B8524F6383C45D, 0xD766355442A41B25,
+ 0x64EFA6DE778A4797, 0x2042170A7079ADF4 }
+ }
+};
+
+/*
+ * Multiply the conventional generator of the curve by the provided
+ * integer. Return is written in *P.
+ *
+ * Assumptions:
+ * - Integer is not 0, and is lower than the curve order.
+ * If this conditions is not met, then the result is indeterminate
+ * (but the process is still constant-time).
+ */
+static void
+p256_mulgen(p256_jacobian *P, const unsigned char *k, size_t klen)
+{
+ point_mul_inner(P, P256_Gwin, k, klen);
+}
+
+/*
+ * Return 1 if all of the following hold:
+ * - klen <= 32
+ * - k != 0
+ * - k is lower than the curve order
+ * Otherwise, return 0.
+ *
+ * Constant-time behaviour: only klen may be observable.
+ */
+static uint32_t
+check_scalar(const unsigned char *k, size_t klen)
+{
+ uint32_t z;
+ int32_t c;
+ size_t u;
+
+ if (klen > 32) {
+ return 0;
+ }
+ z = 0;
+ for (u = 0; u < klen; u ++) {
+ z |= k[u];
+ }
+ if (klen == 32) {
+ c = 0;
+ for (u = 0; u < klen; u ++) {
+ c |= -(int32_t)EQ0(c) & CMP(k[u], P256_N[u]);
+ }
+ } else {
+ c = -1;
+ }
+ return NEQ(z, 0) & LT0(c);
+}
+
+static uint32_t
+api_mul(unsigned char *G, size_t Glen,
+ const unsigned char *k, size_t klen, int curve)
+{
+ uint32_t r;
+ p256_jacobian P;
+
+ (void)curve;
+ if (Glen != 65) {
+ return 0;
+ }
+ r = check_scalar(k, klen);
+ r &= point_decode(&P, G);
+ p256_mul(&P, k, klen);
+ r &= point_encode(G, &P);
+ return r;
+}
+
+static size_t
+api_mulgen(unsigned char *R,
+ const unsigned char *k, size_t klen, int curve)
+{
+ p256_jacobian P;
+
+ (void)curve;
+ p256_mulgen(&P, k, klen);
+ point_encode(R, &P);
+ return 65;
+}
+
+static uint32_t
+api_muladd(unsigned char *A, const unsigned char *B, size_t len,
+ const unsigned char *x, size_t xlen,
+ const unsigned char *y, size_t ylen, int curve)
+{
+ /*
+ * We might want to use Shamir's trick here: make a composite
+ * window of u*P+v*Q points, to merge the two doubling-ladders
+ * into one. This, however, has some complications:
+ *
+ * - During the computation, we may hit the point-at-infinity.
+ * Thus, we would need p256_add_complete_mixed() (complete
+ * formulas for point addition), with a higher cost (17 muls
+ * instead of 11).
+ *
+ * - A 4-bit window would be too large, since it would involve
+ * 16*16-1 = 255 points. For the same window size as in the
+ * p256_mul() case, we would need to reduce the window size
+ * to 2 bits, and thus perform twice as many non-doubling
+ * point additions.
+ *
+ * - The window may itself contain the point-at-infinity, and
+ * thus cannot be in all generality be made of affine points.
+ * Instead, we would need to make it a window of points in
+ * Jacobian coordinates. Even p256_add_complete_mixed() would
+ * be inappropriate.
+ *
+ * For these reasons, the code below performs two separate
+ * point multiplications, then computes the final point addition
+ * (which is both a "normal" addition, and a doubling, to handle
+ * all cases).
+ */
+
+ p256_jacobian P, Q;
+ uint32_t r, t, s;
+ uint64_t z;
+
+ (void)curve;
+ if (len != 65) {
+ return 0;
+ }
+ r = point_decode(&P, A);
+ p256_mul(&P, x, xlen);
+ if (B == NULL) {
+ p256_mulgen(&Q, y, ylen);
+ } else {
+ r &= point_decode(&Q, B);
+ p256_mul(&Q, y, ylen);
+ }
+
+ /*
+ * The final addition may fail in case both points are equal.
+ */
+ t = p256_add(&P, &Q);
+ f256_final_reduce(P.z);
+ z = P.z[0] | P.z[1] | P.z[2] | P.z[3];
+ s = EQ((uint32_t)(z | (z >> 32)), 0);
+ p256_double(&Q);
+
+ /*
+ * If s is 1 then either P+Q = 0 (t = 1) or P = Q (t = 0). So we
+ * have the following:
+ *
+ * s = 0, t = 0 return P (normal addition)
+ * s = 0, t = 1 return P (normal addition)
+ * s = 1, t = 0 return Q (a 'double' case)
+ * s = 1, t = 1 report an error (P+Q = 0)
+ */
+ CCOPY(s & ~t, &P, &Q, sizeof Q);
+ point_encode(A, &P);
+ r &= ~(s & t);
+ return r;
+}
+
+/* see bearssl_ec.h */
+const br_ec_impl br_ec_p256_m64 = {
+ (uint32_t)0x00800000,
+ &api_generator,
+ &api_order,
+ &api_xoff,
+ &api_mul,
+ &api_mulgen,
+ &api_muladd
+};
+
+/* see bearssl_ec.h */
+const br_ec_impl *
+br_ec_p256_m64_get(void)
+{
+ return &br_ec_p256_m64;
+}
+
+#else
+
+/* see bearssl_ec.h */
+const br_ec_impl *
+br_ec_p256_m64_get(void)
+{
+ return 0;
+}
+
+#endif
projects / BearSSL / blobdiff