Add an optimized 32-bit implementation of the NIST P-256 elliptic curve.

mozilla/security/nss/lib/freebl/ecl/ecp_256_32.c

+/* This Source Code Form is subject to the terms of the Mozilla Public
+ * License, v. 2.0. If a copy of the MPL was not distributed with this
+ * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
+
+/* A 32-bit implementation of the NIST P-256 elliptic curve. */

[wtc, 2013/01/25 02:32:49]

Lines 1-5 used to be one comment block. I separated it into
a license header and a comment.

I also removed <stdint.h>, <stdio.h>, and <stdlib.h> and kept
only <string.h> because the only standard C library functions
I saw in this file are memcpy, memcmp, memset.

+
+#include <string.h>
+
+#include "prtypes.h"
+#include "mpi.h"
+#include "mpi-priv.h"
+#include "ecp.h"
+
+typedef PRUint8 u8;
+typedef PRUint32 u32;
+typedef PRInt32 s32;
+typedef PRUint64 u64;
+
+/* Our field elements are represented as nine, unsigned 32-bit words. Freebl's
+ * MPI library calls them digits, but here they are called limbs, which is
+ * GMP's terminology.
+ *
+ * The value of an felem (field element) is:
+ *   x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
+ *
+ * That is, each limb is alternately 29 or 28-bits wide in little-endian
+ * order.
+ *
+ * This means that an felem hits 2**257, rather than 2**256 as we would like. A
+ * 28, 29, ... pattern would cause us to hit 2**256, but that causes problems
+ * when multiplying as terms end up one bit short of a limb which would require
+ * much bit-shifting to correct.
+ *
+ * Finally, the values stored in an felem are in Montgomery form. So the value
+ * |y| is stored as (y*R) mod p, where p is the P-256 prime and R is 2**257.
+ */
+typedef u32 limb;
+#define NLIMBS 9
+typedef limb felem[NLIMBS];
+
+static const limb kBottom28Bits = 0xfffffff;
+static const limb kBottom29Bits = 0x1fffffff;
+
+/* kOne is the number 1 as an felem. It's 2**257 mod p split up into 29 and
+ * 28-bit words.
+ */
+static const felem kOne = {
+    2, 0, 0, 0xffff800,
+    0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff,
+    0
+};
+static const felem kZero = {0};
+static const felem kP = {
+    0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff,
+    0, 0, 0x200000, 0xf000000,
+    0xfffffff
+};
+static const felem k2P = {
+    0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff,
+    0, 0, 0x400000, 0xe000000,
+    0x1fffffff
+};
+
+/* kPrecomputed contains precomputed values to aid the calculation of scalar
+ * multiples of the base point, G. It's actually two, equal length, tables
+ * concatenated.
+ *
+ * The first table contains (x,y) felem pairs for 16 multiples of the base
+ * point, G.
+ *
+ *   Index  |  Index (binary) | Value
+ *       0  |           0000  | 0G (all zeros, omitted)
+ *       1  |           0001  | G
+ *       2  |           0010  | 2**64G
+ *       3  |           0011  | 2**64G + G
+ *       4  |           0100  | 2**128G
+ *       5  |           0101  | 2**128G + G
+ *       6  |           0110  | 2**128G + 2**64G
+ *       7  |           0111  | 2**128G + 2**64G + G
+ *       8  |           1000  | 2**192G
+ *       9  |           1001  | 2**192G + G
+ *      10  |           1010  | 2**192G + 2**64G
+ *      11  |           1011  | 2**192G + 2**64G + G
+ *      12  |           1100  | 2**192G + 2**128G
+ *      13  |           1101  | 2**192G + 2**128G + G
+ *      14  |           1110  | 2**192G + 2**128G + 2**64G
+ *      15  |           1111  | 2**192G + 2**128G + 2**64G + G
+ *
+ * The second table follows the same style, but the terms are 2**32G,
+ * 2**96G, 2**160G, 2**224G.
+ *
+ * This is ~2KB of data.
+ */
+static const limb kPrecomputed[NLIMBS * 2 * 15 * 2] = {
+    0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee,
+    0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3,
+    0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c,
+    0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22,
+    0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050,
+    0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b,
+    0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa,
+    0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2,
+    0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609,
+    0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581,
+    0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca,
+    0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33,
+    0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6,
+    0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd,
+    0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0,
+    0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881,
+    0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a,
+    0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26,
+    0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b,
+    0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023,
+    0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133,
+    0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa,
+    0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29,
+    0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc,
+    0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8,
+    0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59,
+    0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39,
+    0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689,
+    0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa,
+    0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3,
+    0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1,
+    0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f,
+    0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72,
+    0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d,
+    0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b,
+    0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a,
+    0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a,
+    0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f,
+    0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb,
+    0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc,
+    0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9,
+    0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce,
+    0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2,
+    0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca,
+    0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229,
+    0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57,
+    0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c,
+    0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa,
+    0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651,
+    0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec,
+    0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7,
+    0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c,
+    0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927,
+    0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298,
+    0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8,
+    0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2,
+    0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d,
+    0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4,
+    0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8,
+    0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
+};
+
+/* Field element operations:
+ */
+
+/* NON_ZERO_TO_ALL_ONES returns:
+ *   0xffffffff for 0 < x <= 2**31
+ *   0 for x == 0 or x > 2**31.
+ *
+ * This macro assumes that right-shifting a signed number shifts in the MSB on
+ * the left. This is not ensured by the C standard, but is true on the CPUs
+ * that we're targetting with this code (x86 and ARM).
+ */
+#define NON_ZERO_TO_ALL_ONES(x) (~((u32) (((s32) ((x)-1)) >> 31)))

[Ryan Sleevi, 2013/01/25 03:50:36]

Just because this is slipping into more and more code, does it make since to
place this standard-but-undefined behaviour somewhere just as a gatekeeper for
'things might go wrong here'? Are we relying on the fact that a unittest should
fail, were this to not be true?

Should their be a bitsmear alternative 'just in case'?

eg:
x &= 0x80000000; x |= x >> 16;
x |= x >> 8; x |= x >> 4;
x |= x >> 2; x |= x >> 1;

[agl, 2013/01/25 16:04:03]

I'd love to put it in nspr or somewhere similar. Preferably with an #if
defined(__x86_64__) || defined(__whatever_arm_uses__) with the bitsmeared
version as an alternative.

However, in the interests of keeping the patches limited so far I've been
copying it as it's only a one-liner and somewhat obscure.

[wtc, 2013/01/31 16:59:02]

After I worked out the detail (including plugging in Ryan's
constantTimeCondition code from the NSS RSA-OAEP patch),
I found that the result is simply

u32 NonZeroToAllOnes(u32 x) {
  x = (x - 1) >> 31;
  return x - 1;
}

or in macro form:

#define NON_ZERO_TO_ALL_ONES(x) ((((x)-1) >> 31) - 1)

This should be equally fast. I hope I didn't make a mistake.

+
+/* felem_reduce_carry adds a multiple of p in order to cancel |carry|,
+ * which is a term at 2**257.
+ *
+ * On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
+ * On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
+ */
+static void felem_reduce_carry(felem inout, limb carry)
+{
+    const u32 carry_mask = NON_ZERO_TO_ALL_ONES(carry);
+
+    inout[0] += carry << 1;
+    inout[3] += 0x10000000 & carry_mask;
+    /* carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
+     * previous line therefore this doesn't underflow.
+     */
+    inout[3] -= carry << 11;
+    inout[4] += (0x20000000 - 1) & carry_mask;
+    inout[5] += (0x10000000 - 1) & carry_mask;
+    inout[6] += (0x20000000 - 1) & carry_mask;
+    inout[6] -= carry << 22;
+    /* This may underflow if carry is non-zero but, if so, we'll fix it in the
+     * next line.
+     */
+    inout[7] -= 1 & carry_mask;
+    inout[7] += carry << 25;
+}
+
+/* felem_sum sets out = in+in2.
+ *
+ * On entry, in[i]+in2[i] must not overflow a 32-bit word.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
+ */
+static void felem_sum(felem out, const felem in, const felem in2)
+{
+    limb carry = 0;
+    unsigned int i;
+    for (i = 0;; i++) {
+        out[i] = in[i] + in2[i];
+        out[i] += carry;
+        carry = out[i] >> 29;
+        out[i] &= kBottom29Bits;
+
+        i++;
+        if (i == NLIMBS)
+            break;
+
+        out[i] = in[i] + in2[i];
+        out[i] += carry;
+        carry = out[i] >> 28;
+        out[i] &= kBottom28Bits;
+    }
+
+    felem_reduce_carry(out, carry);
+}
+
+#define two31m3 (((limb)1) << 31) - (((limb)1) << 3)
+#define two30m2 (((limb)1) << 30) - (((limb)1) << 2)
+#define two30p13m2 (((limb)1) << 30) + (((limb)1) << 13) - (((limb)1) << 2)
+#define two31m2 (((limb)1) << 31) - (((limb)1) << 2)
+#define two31p24m2 (((limb)1) << 31) + (((limb)1) << 24) - (((limb)1) << 2)
+#define two30m27m2 (((limb)1) << 30) - (((limb)1) << 27) - (((limb)1) << 2)
+
+/* zero31 is 0 mod p.
+ */
+static const felem zero31 = {
+    two31m3, two30m2, two31m2, two30p13m2,
+    two31m2, two30m2, two31p24m2, two30m27m2,
+    two31m2
+};
+
+/* felem_diff sets out = in-in2.
+ *
+ * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+ *           in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_diff(felem out, const felem in, const felem in2)
+{
+    limb carry = 0;
+    unsigned int i;
+
+    for (i = 0;; i++) {
+        out[i] = in[i] - in2[i];
+        out[i] += zero31[i];
+        out[i] += carry;
+        carry = out[i] >> 29;
+        out[i] &= kBottom29Bits;
+
+        i++;
+        if (i == NLIMBS)
+            break;
+
+        out[i] = in[i] - in2[i];
+        out[i] += zero31[i];
+        out[i] += carry;
+        carry = out[i] >> 28;
+        out[i] &= kBottom28Bits;
+    }
+
+    felem_reduce_carry(out, carry);
+}
+
+/* felem_reduce_degree sets out = tmp/R mod p where tmp contains 64-bit words
+ * with the same 29,28,... bit positions as an felem.
+ *
+ * The values in felems are in Montgomery form: x*R mod p where R = 2**257.
+ * Since we just multiplied two Montgomery values together, the result is
+ * x*y*R*R mod p. We wish to divide by R in order for the result also to be
+ * in Montgomery form.
+ *
+ * On entry: tmp[i] < 2**64
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
+ */
+static void felem_reduce_degree(felem out, u64 tmp[17])
+{
+    /* The following table may be helpful when reading this code:
+     *
+     * Limb number:   0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
+     * Width (bits):  29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
+     * Start bit:     0 | 29| 57| 86|114|143|171|200|228|257|285
+     *   (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
+     */
+    limb tmp2[18], carry, x, xMask;
+    unsigned int i;
+
+    /* tmp contains 64-bit words with the same 29,28,29-bit positions as an
+     * felem. So the top of an element of tmp might overlap with another
+     * element two positions down. The following loop eliminates this
+     * overlap.
+     */
+    tmp2[0] = tmp[0] & kBottom29Bits;
+
+    /* In the following we use "(limb) tmp[x]" and "(limb) (tmp[x]>>32)" to try
+     * and hint to the compiler that it can do a single-word shift by selecting
+     * the right register rather than doing a double-word shift and truncating
+     * afterwards.
+     */
+    tmp2[1] = ((limb) tmp[0]) >> 29;
+    tmp2[1] |= (((limb) (tmp[0] >> 32)) << 3) & kBottom28Bits;
+    tmp2[1] += ((limb) tmp[1]) & kBottom28Bits;
+    carry = tmp2[1] >> 28;
+    tmp2[1] &= kBottom28Bits;
+
+    for (i = 2; i < 17; i++) {
+        tmp2[i] = ((limb) (tmp[i - 2] >> 32)) >> 25;
+        tmp2[i] += ((limb) (tmp[i - 1])) >> 28;
+        tmp2[i] += (((limb) (tmp[i - 1] >> 32)) << 4) & kBottom29Bits;
+        tmp2[i] += ((limb) tmp[i]) & kBottom29Bits;
+        tmp2[i] += carry;
+        carry = tmp2[i] >> 29;
+        tmp2[i] &= kBottom29Bits;
+
+        i++;
+        if (i == 17)
+            break;
+        tmp2[i] = ((limb) (tmp[i - 2] >> 32)) >> 25;
+        tmp2[i] += ((limb) (tmp[i - 1])) >> 29;
+        tmp2[i] += (((limb) (tmp[i - 1] >> 32)) << 3) & kBottom28Bits;
+        tmp2[i] += ((limb) tmp[i]) & kBottom28Bits;
+        tmp2[i] += carry;
+        carry = tmp2[i] >> 28;
+        tmp2[i] &= kBottom28Bits;
+    }
+
+    tmp2[17] = ((limb) (tmp[15] >> 32)) >> 25;
+    tmp2[17] += ((limb) (tmp[16])) >> 29;
+    tmp2[17] += (((limb) (tmp[16] >> 32)) << 3);
+    tmp2[17] += carry;
+
+    /* Montgomery elimination of terms:
+     *
+     * Since R is 2**257, we can divide by R with a bitwise shift if we can
+     * ensure that the right-most 257 bits are all zero. We can make that true
+     * by adding multiplies of p without affecting the value.
+     *
+     * So we eliminate limbs from right to left. Since the bottom 29 bits of p
+     * are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
+     * We can do that for 8 further limbs and then right shift to eliminate the
+     * extra factor of R.
+     */
+    for (i = 0;; i += 2) {
+        tmp2[i + 1] += tmp2[i] >> 29;
+        x = tmp2[i] & kBottom29Bits;
+        xMask = NON_ZERO_TO_ALL_ONES(x);
+        tmp2[i] = 0;
+
+        /* The bounds calculations for this loop are tricky. Each iteration of
+         * the loop eliminates two words by adding values to words to their
+         * right.
+         *
+         * The following table contains the amounts added to each word (as an
+         * offset from the value of i at the top of the loop). The amounts are
+         * accounted for from the first and second half of the loop separately
+         * and are written as, for example, 28 to mean a value <2**28.
+         *
+         * Word:                   3   4   5   6   7   8   9   10
+         * Added in top half:     28  11      29  21  29  28
+         *                                        28  29
+         *                                            29
+         * Added in bottom half:      29  10      28  21  28   28
+         *                                            29
+         *
+         * The value that is currently offset 7 will be offset 5 for the next
+         * iteration and then offset 3 for the iteration after that. Therefore
+         * the total value added will be the values added at 7, 5 and 3.
+         *
+         * The following table accumulates these values. The sums at the bottom
+         * are written as, for example, 29+28, to mean a value < 2**29+2**28.
+         *
+         * Word:                   3   4   5   6   7   8   9  10  11  12  13
+         *                        28  11  10  29  21  29  28  28  28  28  28
+         *                            29  28  11  28  29  28  29  28  29  28
+         *                                    29  28  21  21  29  21  29  21
+         *                                        10  29  28  21  28  21  28
+         *                                        28  29  28  29  28  29  28
+         *                                            11  10  29  10  29  10
+         *                                            29  28  11  28  11
+         *                                                    29      29
+         *                        --------------------------------------------
+         *                                                30+ 31+ 30+ 31+ 30+
+         *                                                28+ 29+ 28+ 29+ 21+
+         *                                                21+ 28+ 21+ 28+ 10
+         *                                                10  21+ 10  21+
+         *                                                    11      11
+         *
+         * So the greatest amount is added to tmp2[10] and tmp2[12]. If
+         * tmp2[10/12] has an initial value of <2**29, then the maximum value
+         * will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
+         * as required.
+         */
+        tmp2[i + 3] += (x << 10) & kBottom28Bits;
+        tmp2[i + 4] += (x >> 18);
+
+        tmp2[i + 6] += (x << 21) & kBottom29Bits;
+        tmp2[i + 7] += x >> 8;
+
+        /* At position 200, which is the starting bit position for word 7, we
+         * have a factor of 0xf000000 = 2**28 - 2**24.
+         */
+        tmp2[i + 7] += 0x10000000 & xMask;
+        /* Word 7 is 28 bits wide, so the 2**28 term exactly hits word 8. */
+        tmp2[i + 8] += (x - 1) & xMask;
+        tmp2[i + 7] -= (x << 24) & kBottom28Bits;
+        tmp2[i + 8] -= x >> 4;
+
+        tmp2[i + 8] += 0x20000000 & xMask;
+        tmp2[i + 8] -= x;
+        tmp2[i + 8] += (x << 28) & kBottom29Bits;
+        tmp2[i + 9] += ((x >> 1) - 1) & xMask;
+
+        if (i+1 == NLIMBS)
+            break;
+        tmp2[i + 2] += tmp2[i + 1] >> 28;
+        x = tmp2[i + 1] & kBottom28Bits;
+        xMask = NON_ZERO_TO_ALL_ONES(x);
+        tmp2[i + 1] = 0;
+
+        tmp2[i + 4] += (x << 11) & kBottom29Bits;
+        tmp2[i + 5] += (x >> 18);
+
+        tmp2[i + 7] += (x << 21) & kBottom28Bits;
+        tmp2[i + 8] += x >> 7;
+
+        /* At position 199, which is the starting bit of the 8th word when
+         * dealing with a context starting on an odd word, we have a factor of
+         * 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
+         * word from i+1 is i+8.
+         */
+        tmp2[i + 8] += 0x20000000 & xMask;
+        tmp2[i + 9] += (x - 1) & xMask;
+        tmp2[i + 8] -= (x << 25) & kBottom29Bits;
+        tmp2[i + 9] -= x >> 4;
+
+        tmp2[i + 9] += 0x10000000 & xMask;
+        tmp2[i + 9] -= x;
+        tmp2[i + 10] += (x - 1) & xMask;
+    }
+
+    /* We merge the right shift with a carry chain. The words above 2**257 have
+     * widths of 28,29,... which we need to correct when copying them down.
+     */
+    carry = 0;
+    for (i = 0; i < 8; i++) {
+        /* The maximum value of tmp2[i + 9] occurs on the first iteration and
+         * is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
+         * therefore safe.
+         */
+        out[i] = tmp2[i + 9];
+        out[i] += carry;
+        out[i] += (tmp2[i + 10] << 28) & kBottom29Bits;
+        carry = out[i] >> 29;
+        out[i] &= kBottom29Bits;
+
+        i++;
+        out[i] = tmp2[i + 9] >> 1;
+        out[i] += carry;
+        carry = out[i] >> 28;
+        out[i] &= kBottom28Bits;
+    }
+
+    out[8] = tmp2[17];
+    out[8] += carry;
+    carry = out[8] >> 29;
+    out[8] &= kBottom29Bits;
+
+    felem_reduce_carry(out, carry);
+}
+
+/* felem_square sets out=in*in.
+ *
+ * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_square(felem out, const felem in)
+{
+    u64 tmp[17];
+
+    tmp[0] = ((u64) in[0]) * in[0];
+    tmp[1] = ((u64) in[0]) * (in[1] << 1);
+    tmp[2] = ((u64) in[0]) * (in[2] << 1) +
+             ((u64) in[1]) * (in[1] << 1);
+    tmp[3] = ((u64) in[0]) * (in[3] << 1) +
+             ((u64) in[1]) * (in[2] << 1);
+    tmp[4] = ((u64) in[0]) * (in[4] << 1) +
+             ((u64) in[1]) * (in[3] << 2) +
+             ((u64) in[2]) * in[2];
+    tmp[5] = ((u64) in[0]) * (in[5] << 1) +
+             ((u64) in[1]) * (in[4] << 1) +
+             ((u64) in[2]) * (in[3] << 1);
+    tmp[6] = ((u64) in[0]) * (in[6] << 1) +
+             ((u64) in[1]) * (in[5] << 2) +
+             ((u64) in[2]) * (in[4] << 1) +
+             ((u64) in[3]) * (in[3] << 1);
+    tmp[7] = ((u64) in[0]) * (in[7] << 1) +
+             ((u64) in[1]) * (in[6] << 1) +
+             ((u64) in[2]) * (in[5] << 1) +
+             ((u64) in[3]) * (in[4] << 1);
+    /* tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
+     * which is < 2**64 as required.
+     */
+    tmp[8] = ((u64) in[0]) * (in[8] << 1) +
+             ((u64) in[1]) * (in[7] << 2) +
+             ((u64) in[2]) * (in[6] << 1) +
+             ((u64) in[3]) * (in[5] << 2) +
+             ((u64) in[4]) * in[4];
+    tmp[9] = ((u64) in[1]) * (in[8] << 1) +
+             ((u64) in[2]) * (in[7] << 1) +
+             ((u64) in[3]) * (in[6] << 1) +
+             ((u64) in[4]) * (in[5] << 1);
+    tmp[10] = ((u64) in[2]) * (in[8] << 1) +
+              ((u64) in[3]) * (in[7] << 2) +
+              ((u64) in[4]) * (in[6] << 1) +
+              ((u64) in[5]) * (in[5] << 1);
+    tmp[11] = ((u64) in[3]) * (in[8] << 1) +
+              ((u64) in[4]) * (in[7] << 1) +
+              ((u64) in[5]) * (in[6] << 1);
+    tmp[12] = ((u64) in[4]) * (in[8] << 1) +
+              ((u64) in[5]) * (in[7] << 2) +
+              ((u64) in[6]) * in[6];
+    tmp[13] = ((u64) in[5]) * (in[8] << 1) +
+              ((u64) in[6]) * (in[7] << 1);
+    tmp[14] = ((u64) in[6]) * (in[8] << 1) +
+              ((u64) in[7]) * (in[7] << 1);
+    tmp[15] = ((u64) in[7]) * (in[8] << 1);
+    tmp[16] = ((u64) in[8]) * in[8];
+
+    felem_reduce_degree(out, tmp);
+}
+
+/* felem_mul sets out=in*in2.
+ *
+ * On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+ *           in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_mul(felem out, const felem in, const felem in2)
+{
+    u64 tmp[17];
+
+    tmp[0] = ((u64) in[0]) * in2[0];
+    tmp[1] = ((u64) in[0]) * (in2[1] << 0) +
+             ((u64) in[1]) * (in2[0] << 0);
+    tmp[2] = ((u64) in[0]) * (in2[2] << 0) +
+             ((u64) in[1]) * (in2[1] << 1) +
+             ((u64) in[2]) * (in2[0] << 0);
+    tmp[3] = ((u64) in[0]) * (in2[3] << 0) +
+             ((u64) in[1]) * (in2[2] << 0) +
+             ((u64) in[2]) * (in2[1] << 0) +
+             ((u64) in[3]) * (in2[0] << 0);
+    tmp[4] = ((u64) in[0]) * (in2[4] << 0) +
+             ((u64) in[1]) * (in2[3] << 1) +
+             ((u64) in[2]) * (in2[2] << 0) +
+             ((u64) in[3]) * (in2[1] << 1) +
+             ((u64) in[4]) * (in2[0] << 0);
+    tmp[5] = ((u64) in[0]) * (in2[5] << 0) +
+             ((u64) in[1]) * (in2[4] << 0) +
+             ((u64) in[2]) * (in2[3] << 0) +
+             ((u64) in[3]) * (in2[2] << 0) +
+             ((u64) in[4]) * (in2[1] << 0) +
+             ((u64) in[5]) * (in2[0] << 0);
+    tmp[6] = ((u64) in[0]) * (in2[6] << 0) +
+             ((u64) in[1]) * (in2[5] << 1) +
+             ((u64) in[2]) * (in2[4] << 0) +
+             ((u64) in[3]) * (in2[3] << 1) +
+             ((u64) in[4]) * (in2[2] << 0) +
+             ((u64) in[5]) * (in2[1] << 1) +
+             ((u64) in[6]) * (in2[0] << 0);
+    tmp[7] = ((u64) in[0]) * (in2[7] << 0) +
+             ((u64) in[1]) * (in2[6] << 0) +
+             ((u64) in[2]) * (in2[5] << 0) +
+             ((u64) in[3]) * (in2[4] << 0) +
+             ((u64) in[4]) * (in2[3] << 0) +
+             ((u64) in[5]) * (in2[2] << 0) +
+             ((u64) in[6]) * (in2[1] << 0) +
+             ((u64) in[7]) * (in2[0] << 0);
+    /* tmp[8] has the greatest value but doesn't overflow. See logic in
+     * felem_square.
+     */
+    tmp[8] = ((u64) in[0]) * (in2[8] << 0) +
+             ((u64) in[1]) * (in2[7] << 1) +
+             ((u64) in[2]) * (in2[6] << 0) +
+             ((u64) in[3]) * (in2[5] << 1) +
+             ((u64) in[4]) * (in2[4] << 0) +
+             ((u64) in[5]) * (in2[3] << 1) +
+             ((u64) in[6]) * (in2[2] << 0) +
+             ((u64) in[7]) * (in2[1] << 1) +
+             ((u64) in[8]) * (in2[0] << 0);
+    tmp[9] = ((u64) in[1]) * (in2[8] << 0) +
+             ((u64) in[2]) * (in2[7] << 0) +
+             ((u64) in[3]) * (in2[6] << 0) +
+             ((u64) in[4]) * (in2[5] << 0) +
+             ((u64) in[5]) * (in2[4] << 0) +
+             ((u64) in[6]) * (in2[3] << 0) +
+             ((u64) in[7]) * (in2[2] << 0) +
+             ((u64) in[8]) * (in2[1] << 0);
+    tmp[10] = ((u64) in[2]) * (in2[8] << 0) +
+              ((u64) in[3]) * (in2[7] << 1) +
+              ((u64) in[4]) * (in2[6] << 0) +
+              ((u64) in[5]) * (in2[5] << 1) +
+              ((u64) in[6]) * (in2[4] << 0) +
+              ((u64) in[7]) * (in2[3] << 1) +
+              ((u64) in[8]) * (in2[2] << 0);
+    tmp[11] = ((u64) in[3]) * (in2[8] << 0) +
+              ((u64) in[4]) * (in2[7] << 0) +
+              ((u64) in[5]) * (in2[6] << 0) +
+              ((u64) in[6]) * (in2[5] << 0) +
+              ((u64) in[7]) * (in2[4] << 0) +
+              ((u64) in[8]) * (in2[3] << 0);
+    tmp[12] = ((u64) in[4]) * (in2[8] << 0) +
+              ((u64) in[5]) * (in2[7] << 1) +
+              ((u64) in[6]) * (in2[6] << 0) +
+              ((u64) in[7]) * (in2[5] << 1) +
+              ((u64) in[8]) * (in2[4] << 0);
+    tmp[13] = ((u64) in[5]) * (in2[8] << 0) +
+              ((u64) in[6]) * (in2[7] << 0) +
+              ((u64) in[7]) * (in2[6] << 0) +
+              ((u64) in[8]) * (in2[5] << 0);
+    tmp[14] = ((u64) in[6]) * (in2[8] << 0) +
+              ((u64) in[7]) * (in2[7] << 1) +
+              ((u64) in[8]) * (in2[6] << 0);
+    tmp[15] = ((u64) in[7]) * (in2[8] << 0) +
+              ((u64) in[8]) * (in2[7] << 0);
+    tmp[16] = ((u64) in[8]) * (in2[8] << 0);
+
+    felem_reduce_degree(out, tmp);
+}
+
+static void felem_assign(felem out, const felem in)
+{
+    memcpy(out, in, sizeof(felem));
+}
+
+/* felem_inv calculates |out| = |in|^{-1}
+ *
+ * Based on Fermat's Little Theorem:
+ *   a^p = a (mod p)
+ *   a^{p-1} = 1 (mod p)
+ *   a^{p-2} = a^{-1} (mod p)
+ */
+static void felem_inv(felem out, const felem in)
+{
+    felem ftmp, ftmp2;
+    /* each e_I will hold |in|^{2^I - 1} */
+    felem e2, e4, e8, e16, e32, e64;
+    unsigned int i;
+
+    felem_square(ftmp, in);             /* 2^1 */
+    felem_mul(ftmp, in, ftmp);          /* 2^2 - 2^0 */
+    felem_assign(e2, ftmp);
+    felem_square(ftmp, ftmp);           /* 2^3 - 2^1 */
+    felem_square(ftmp, ftmp);           /* 2^4 - 2^2 */
+    felem_mul(ftmp, ftmp, e2);          /* 2^4 - 2^0 */
+    felem_assign(e4, ftmp);
+    felem_square(ftmp, ftmp);           /* 2^5 - 2^1 */
+    felem_square(ftmp, ftmp);           /* 2^6 - 2^2 */
+    felem_square(ftmp, ftmp);           /* 2^7 - 2^3 */
+    felem_square(ftmp, ftmp);           /* 2^8 - 2^4 */
+    felem_mul(ftmp, ftmp, e4);          /* 2^8 - 2^0 */
+    felem_assign(e8, ftmp);
+    for (i = 0; i < 8; i++) {
+        felem_square(ftmp, ftmp);
+    }                                   /* 2^16 - 2^8 */
+    felem_mul(ftmp, ftmp, e8);          /* 2^16 - 2^0 */
+    felem_assign(e16, ftmp);
+    for (i = 0; i < 16; i++) {
+        felem_square(ftmp, ftmp);
+    }                                   /* 2^32 - 2^16 */
+    felem_mul(ftmp, ftmp, e16);         /* 2^32 - 2^0 */
+    felem_assign(e32, ftmp);
+    for (i = 0; i < 32; i++) {
+        felem_square(ftmp, ftmp);
+    }                                   /* 2^64 - 2^32 */
+    felem_assign(e64, ftmp);
+    felem_mul(ftmp, ftmp, in);          /* 2^64 - 2^32 + 2^0 */
+    for (i = 0; i < 192; i++) {
+        felem_square(ftmp, ftmp);
+    }                                   /* 2^256 - 2^224 + 2^192 */
+
+    felem_mul(ftmp2, e64, e32);         /* 2^64 - 2^0 */
+    for (i = 0; i < 16; i++) {
+        felem_square(ftmp2, ftmp2);
+    }                                   /* 2^80 - 2^16 */
+    felem_mul(ftmp2, ftmp2, e16);       /* 2^80 - 2^0 */
+    for (i = 0; i < 8; i++) {
+        felem_square(ftmp2, ftmp2);
+    }                                   /* 2^88 - 2^8 */
+    felem_mul(ftmp2, ftmp2, e8);        /* 2^88 - 2^0 */
+    for (i = 0; i < 4; i++) {
+        felem_square(ftmp2, ftmp2);
+    }                                   /* 2^92 - 2^4 */
+    felem_mul(ftmp2, ftmp2, e4);        /* 2^92 - 2^0 */
+    felem_square(ftmp2, ftmp2);         /* 2^93 - 2^1 */
+    felem_square(ftmp2, ftmp2);         /* 2^94 - 2^2 */
+    felem_mul(ftmp2, ftmp2, e2);        /* 2^94 - 2^0 */
+    felem_square(ftmp2, ftmp2);         /* 2^95 - 2^1 */
+    felem_square(ftmp2, ftmp2);         /* 2^96 - 2^2 */
+    felem_mul(ftmp2, ftmp2, in);        /* 2^96 - 3 */
+
+    felem_mul(out, ftmp2, ftmp);        /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
+}
+
+/* felem_scalar_3 sets out=3*out.
+ *
+ * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_scalar_3(felem out)
+{
+    limb carry = 0;
+    unsigned int i;
+
+    for (i = 0;; i++) {
+        out[i] *= 3;
+        out[i] += carry;
+        carry = out[i] >> 29;
+        out[i] &= kBottom29Bits;
+
+        i++;
+        if (i == NLIMBS)
+            break;
+
+        out[i] *= 3;
+        out[i] += carry;
+        carry = out[i] >> 28;
+        out[i] &= kBottom28Bits;
+    }
+
+    felem_reduce_carry(out, carry);
+}
+
+/* felem_scalar_4 sets out=4*out.
+ *
+ * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_scalar_4(felem out)
+{
+    limb carry = 0, next_carry;
+    unsigned int i;
+
+    for (i = 0;; i++) {
+        next_carry = out[i] >> 27;
+        out[i] <<= 2;
+        out[i] &= kBottom29Bits;
+        out[i] += carry;
+        carry = next_carry + (out[i] >> 29);
+        out[i] &= kBottom29Bits;
+
+        i++;
+        if (i == NLIMBS)
+            break;
+        next_carry = out[i] >> 26;
+        out[i] <<= 2;
+        out[i] &= kBottom28Bits;
+        out[i] += carry;
+        carry = next_carry + (out[i] >> 28);
+        out[i] &= kBottom28Bits;
+    }
+
+    felem_reduce_carry(out, carry);
+}
+
+/* felem_scalar_8 sets out=8*out.
+ *
+ * On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ * On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+ */
+static void felem_scalar_8(felem out)
+{
+    limb carry = 0, next_carry;
+    unsigned int i;
+
+    for (i = 0;; i++) {
+        next_carry = out[i] >> 26;
+        out[i] <<= 3;
+        out[i] &= kBottom29Bits;
+        out[i] += carry;
+        carry = next_carry + (out[i] >> 29);
+        out[i] &= kBottom29Bits;
+
+        i++;
+        if (i == NLIMBS)
+            break;
+        next_carry = out[i] >> 25;
+        out[i] <<= 3;
+        out[i] &= kBottom28Bits;
+        out[i] += carry;
+        carry = next_carry + (out[i] >> 28);
+        out[i] &= kBottom28Bits;
+    }
+
+    felem_reduce_carry(out, carry);
+}
+
+/* felem_is_zero_vartime returns 1 iff |in| == 0. It takes a variable amount of
+ * time depending on the value of |in|.
+ */
+static char felem_is_zero_vartime(const felem in)
+{
+    limb carry;
+    int i;
+    limb tmp[NLIMBS];
+    felem_assign(tmp, in);
+
+    /* First, reduce tmp to a minimal form.
+     */
+    do {
+        carry = 0;
+        for (i = 0;; i++) {
+            tmp[i] += carry;
+            carry = tmp[i] >> 29;
+            tmp[i] &= kBottom29Bits;
+
+            i++;
+            if (i == NLIMBS)
+                break;
+
+            tmp[i] += carry;
+            carry = tmp[i] >> 28;
+            tmp[i] &= kBottom28Bits;
+        }
+
+        felem_reduce_carry(tmp, carry);
+    } while (carry);
+
+    /* tmp < 2**257, so the only possible zero values are 0, p and 2p.
+     */
+    return memcmp(tmp, kZero, sizeof(tmp)) == 0 ||
+           memcmp(tmp, kP, sizeof(tmp)) == 0 ||
+           memcmp(tmp, k2P, sizeof(tmp)) == 0;
+}
+
+/* Group operations:
+ *
+ * Elements of the elliptic curve group are represented in Jacobian
+ * coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in
+ * Jacobian form.
+ */
+
+/* point_double sets {x_out,y_out,z_out} = 2*{x,y,z}.
+ *
+ * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
+ */
+static void point_double(felem x_out, felem y_out, felem z_out,
+                         const felem x, const felem y, const felem z)
+{
+    felem delta, gamma, alpha, beta, tmp, tmp2;
+
+    felem_square(delta, z);
+    felem_square(gamma, y);
+    felem_mul(beta, x, gamma);
+
+    felem_sum(tmp, x, delta);
+    felem_diff(tmp2, x, delta);
+    felem_mul(alpha, tmp, tmp2);
+    felem_scalar_3(alpha);
+
+    felem_sum(tmp, y, z);
+    felem_square(tmp, tmp);
+    felem_diff(tmp, tmp, gamma);
+    felem_diff(z_out, tmp, delta);
+
+    felem_scalar_4(beta);
+    felem_square(x_out, alpha);
+    felem_diff(x_out, x_out, beta);
+    felem_diff(x_out, x_out, beta);
+
+    felem_diff(tmp, beta, x_out);
+    felem_mul(tmp, alpha, tmp);
+    felem_square(tmp2, gamma);
+    felem_scalar_8(tmp2);
+    felem_diff(y_out, tmp, tmp2);
+}
+
+/* point_add_mixed sets {x_out,y_out,z_out} = {x1,y1,z1} + {x2,y2,1}.
+ * (i.e. the second point is affine.)
+ *
+ * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+ *
+ * Note that this function does not handle P+P, infinity+P nor P+infinity
+ * correctly.
+ */
+static void point_add_mixed(felem x_out, felem y_out, felem z_out,
+                            const felem x1, const felem y1, const felem z1,
+                            const felem x2, const felem y2)
+{
+    felem z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp;
+
+    felem_square(z1z1, z1);
+    felem_sum(tmp, z1, z1);
+
+    felem_mul(u2, x2, z1z1);
+    felem_mul(z1z1z1, z1, z1z1);
+    felem_mul(s2, y2, z1z1z1);
+    felem_diff(h, u2, x1);
+    felem_sum(i, h, h);
+    felem_square(i, i);
+    felem_mul(j, h, i);
+    felem_diff(r, s2, y1);
+    felem_sum(r, r, r);
+    felem_mul(v, x1, i);
+
+    felem_mul(z_out, tmp, h);
+    felem_square(rr, r);
+    felem_diff(x_out, rr, j);
+    felem_diff(x_out, x_out, v);
+    felem_diff(x_out, x_out, v);
+
+    felem_diff(tmp, v, x_out);
+    felem_mul(y_out, tmp, r);
+    felem_mul(tmp, y1, j);
+    felem_diff(y_out, y_out, tmp);
+    felem_diff(y_out, y_out, tmp);
+}
+
+/* point_add sets {x_out,y_out,z_out} = {x1,y1,z1} + {x2,y2,z2}.
+ *
+ * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+ *
+ * Note that this function does not handle P+P, infinity+P nor P+infinity
+ * correctly.
+ */
+static void point_add(felem x_out, felem y_out, felem z_out,
+                      const felem x1, const felem y1, const felem z1,
+                      const felem x2, const felem y2, const felem z2)
+{
+    felem z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp;
+
+    felem_square(z1z1, z1);
+    felem_square(z2z2, z2);
+    felem_mul(u1, x1, z2z2);
+
+    felem_sum(tmp, z1, z2);
+    felem_square(tmp, tmp);
+    felem_diff(tmp, tmp, z1z1);
+    felem_diff(tmp, tmp, z2z2);
+
+    felem_mul(z2z2z2, z2, z2z2);
+    felem_mul(s1, y1, z2z2z2);
+
+    felem_mul(u2, x2, z1z1);
+    felem_mul(z1z1z1, z1, z1z1);
+    felem_mul(s2, y2, z1z1z1);
+    felem_diff(h, u2, u1);
+    felem_sum(i, h, h);
+    felem_square(i, i);
+    felem_mul(j, h, i);
+    felem_diff(r, s2, s1);
+    felem_sum(r, r, r);
+    felem_mul(v, u1, i);
+
+    felem_mul(z_out, tmp, h);
+    felem_square(rr, r);
+    felem_diff(x_out, rr, j);
+    felem_diff(x_out, x_out, v);
+    felem_diff(x_out, x_out, v);
+
+    felem_diff(tmp, v, x_out);
+    felem_mul(y_out, tmp, r);
+    felem_mul(tmp, s1, j);
+    felem_diff(y_out, y_out, tmp);
+    felem_diff(y_out, y_out, tmp);
+}
+
+/* point_add_or_double_vartime sets {x_out,y_out,z_out} = {x1,y1,z1} +
+ *                                                        {x2,y2,z2}.
+ *
+ * See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+ *
+ * This function handles the case where {x1,y1,z1}={x2,y2,z2}.
+ */
+static void point_add_or_double_vartime(
+    felem x_out, felem y_out, felem z_out,
+    const felem x1, const felem y1, const felem z1,
+    const felem x2, const felem y2, const felem z2)
+{
+    felem z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp;
+    char x_equal, y_equal;
+
+    felem_square(z1z1, z1);
+    felem_square(z2z2, z2);
+    felem_mul(u1, x1, z2z2);
+
+    felem_sum(tmp, z1, z2);
+    felem_square(tmp, tmp);
+    felem_diff(tmp, tmp, z1z1);
+    felem_diff(tmp, tmp, z2z2);
+
+    felem_mul(z2z2z2, z2, z2z2);
+    felem_mul(s1, y1, z2z2z2);
+
+    felem_mul(u2, x2, z1z1);
+    felem_mul(z1z1z1, z1, z1z1);
+    felem_mul(s2, y2, z1z1z1);
+    felem_diff(h, u2, u1);
+    x_equal = felem_is_zero_vartime(h);
+    felem_sum(i, h, h);
+    felem_square(i, i);
+    felem_mul(j, h, i);
+    felem_diff(r, s2, s1);
+    y_equal = felem_is_zero_vartime(r);
+    if (x_equal && y_equal) {
+        point_double(x_out, y_out, z_out, x1, y1, z1);
+        return;
+    }
+    felem_sum(r, r, r);
+    felem_mul(v, u1, i);
+
+    felem_mul(z_out, tmp, h);
+    felem_square(rr, r);
+    felem_diff(x_out, rr, j);
+    felem_diff(x_out, x_out, v);
+    felem_diff(x_out, x_out, v);
+
+    felem_diff(tmp, v, x_out);
+    felem_mul(y_out, tmp, r);
+    felem_mul(tmp, s1, j);
+    felem_diff(y_out, y_out, tmp);
+    felem_diff(y_out, y_out, tmp);
+}
+
+/* copy_conditional sets out=in if mask = 0xffffffff in constant time.
+ *
+ * On entry: mask is either 0 or 0xffffffff.
+ */
+static void copy_conditional(felem out, const felem in, limb mask)
+{
+    int i;
+
+    for (i = 0; i < NLIMBS; i++) {
+        const limb tmp = mask & (in[i] ^ out[i]);
+        out[i] ^= tmp;
+    }
+}
+
+/* select_affine_point sets {out_x,out_y} to the index'th entry of table.
+ * On entry: index < 16, table[0] must be zero.
+ */
+static void select_affine_point(felem out_x, felem out_y,
+                                const limb *table, limb index)
+{
+    limb i, j;
+
+    memset(out_x, 0, sizeof(felem));
+    memset(out_y, 0, sizeof(felem));
+
+    for (i = 1; i < 16; i++) {
+        limb mask = i ^ index;
+        mask |= mask >> 2;
+        mask |= mask >> 1;
+        mask &= 1;
+        mask--;
+        for (j = 0; j < NLIMBS; j++, table++) {
+            out_x[j] |= *table & mask;
+        }
+        for (j = 0; j < NLIMBS; j++, table++) {
+            out_y[j] |= *table & mask;
+        }
+    }
+}
+
+/* select_jacobian_point sets {out_x,out_y,out_z} to the index'th entry of
+ * table.  On entry: index < 16, table[0] must be zero.
+ */
+static void select_jacobian_point(felem out_x, felem out_y, felem out_z,
+                                  const limb *table, limb index)
+{
+    limb i, j;
+
+    memset(out_x, 0, sizeof(felem));
+    memset(out_y, 0, sizeof(felem));
+    memset(out_z, 0, sizeof(felem));
+
+    /* The implicit value at index 0 is all zero. We don't need to perform that
+     * iteration of the loop because we already set out_* to zero.
+     */
+    table += 3*NLIMBS;
+
+    for (i = 1; i < 16; i++) {
+        limb mask = i ^ index;
+        mask |= mask >> 2;
+        mask |= mask >> 1;
+        mask &= 1;
+        mask--;
+        for (j = 0; j < NLIMBS; j++, table++) {
+            out_x[j] |= *table & mask;
+        }
+        for (j = 0; j < NLIMBS; j++, table++) {
+            out_y[j] |= *table & mask;
+        }
+        for (j = 0; j < NLIMBS; j++, table++) {
+            out_z[j] |= *table & mask;
+        }
+    }
+}
+
+/* get_bit returns the bit'th bit of scalar. */
+static char get_bit(const u8 scalar[32], int bit)
+{
+    return ((scalar[bit >> 3]) >> (bit & 7)) & 1;
+}
+
+/* scalar_base_mult sets {nx,ny,nz} = scalar*G where scalar is a little-endian
+ * number. Note that the value of scalar must be less than the order of the
+ * group.
+ */
+static void scalar_base_mult(felem nx, felem ny, felem nz, const u8 scalar[32])
+{
+    int i, j;
+    limb n_is_infinity_mask = -1, p_is_noninfinite_mask, mask;
+    u32 table_offset;
+
+    felem px, py;
+    felem tx, ty, tz;
+
+    memset(nx, 0, sizeof(felem));
+    memset(ny, 0, sizeof(felem));
+    memset(nz, 0, sizeof(felem));
+
+    /* The loop adds bits at positions 0, 64, 128 and 192, followed by
+     * positions 32,96,160 and 224 and does this 32 times.
+     */
+    for (i = 0; i < 32; i++) {
+        if (i) {
+            point_double(nx, ny, nz, nx, ny, nz);
+        }
+        for (j = 0; j <= 32; j += 32) {
+            char bit0 = get_bit(scalar, 31 - i + j);
+            char bit1 = get_bit(scalar, 95 - i + j);
+            char bit2 = get_bit(scalar, 159 - i + j);
+            char bit3 = get_bit(scalar, 223 - i + j);
+            limb index = bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3);
+
+            table_offset = ((((s32)j) << (32-6)) >> 31) & (30*NLIMBS);
+            select_affine_point(px, py, kPrecomputed + table_offset, index);
+
+            /* Since scalar is less than the order of the group, we know that
+             * {nx,ny,nz} != {px,py,1}, unless both are zero, which we handle
+             * below.
+             */
+            point_add_mixed(tx, ty, tz, nx, ny, nz, px, py);
+            /* The result of point_add_mixed is incorrect if {nx,ny,nz} is zero
+             * (a.k.a.  the point at infinity). We handle that situation by
+             * copying the point from the table.
+             */
+            copy_conditional(nx, px, n_is_infinity_mask);
+            copy_conditional(ny, py, n_is_infinity_mask);
+            copy_conditional(nz, kOne, n_is_infinity_mask);
+
+            /* Equally, the result is also wrong if the point from the table is
+             * zero, which happens when the index is zero. We handle that by
+             * only copying from {tx,ty,tz} to {nx,ny,nz} if index != 0.
+             */
+            p_is_noninfinite_mask = NON_ZERO_TO_ALL_ONES(index);
+            mask = p_is_noninfinite_mask & ~n_is_infinity_mask;
+            copy_conditional(nx, tx, mask);
+            copy_conditional(ny, ty, mask);
+            copy_conditional(nz, tz, mask);
+            /* If p was not zero, then n is now non-zero. */
+            n_is_infinity_mask &= ~p_is_noninfinite_mask;
+        }
+    }
+}
+
+/* point_to_affine converts a Jacobian point to an affine point. If the input
+ * is the point at infinity then it returns (0, 0) in constant time.
+ */
+static void point_to_affine(felem x_out, felem y_out,
+                            const felem nx, const felem ny, const felem nz) {
+    felem z_inv, z_inv_sq;
+    felem_inv(z_inv, nz);
+    felem_square(z_inv_sq, z_inv);
+    felem_mul(x_out, nx, z_inv_sq);
+    felem_mul(z_inv, z_inv, z_inv_sq);
+    felem_mul(y_out, ny, z_inv);
+}
+
+/* scalar_mult sets {nx,ny,nz} = scalar*{x,y}. */
+static void scalar_mult(felem nx, felem ny, felem nz,
+                        const felem x, const felem y, const u8 scalar[32])
+{
+    int i;
+    felem px, py, pz, tx, ty, tz;
+    felem precomp[16][3];
+    limb n_is_infinity_mask, index, p_is_noninfinite_mask, mask;
+
+    /* We precompute 0,1,2,... times {x,y}. */
+    memset(precomp, 0, sizeof(felem) * 3);
+    memcpy(&precomp[1][0], x, sizeof(felem));
+    memcpy(&precomp[1][1], y, sizeof(felem));
+    memcpy(&precomp[1][2], kOne, sizeof(felem));
+
+    for (i = 2; i < 16; i += 2) {
+        point_double(precomp[i][0], precomp[i][1], precomp[i][2],
+                     precomp[i / 2][0], precomp[i / 2][1], precomp[i / 2][2]);
+
+        point_add_mixed(precomp[i + 1][0], precomp[i + 1][1], precomp[i + 1][2],
+                        precomp[i][0], precomp[i][1], precomp[i][2], x, y);
+    }
+
+    memset(nx, 0, sizeof(felem));
+    memset(ny, 0, sizeof(felem));
+    memset(nz, 0, sizeof(felem));
+    n_is_infinity_mask = -1;
+
+    /* We add in a window of four bits each iteration and do this 64 times. */
+    for (i = 0; i < 64; i++) {
+        if (i) {
+            point_double(nx, ny, nz, nx, ny, nz);
+            point_double(nx, ny, nz, nx, ny, nz);
+            point_double(nx, ny, nz, nx, ny, nz);
+            point_double(nx, ny, nz, nx, ny, nz);
+        }
+
+        index = scalar[31 - i / 2];
+        if ((i & 1) == 1) {
+            index &= 15;
+        } else {
+            index >>= 4;
+        }
+
+        /* See the comments in scalar_base_mult about handling infinities. */
+        select_jacobian_point(px, py, pz, (limb *) precomp, index);
+        point_add(tx, ty, tz, nx, ny, nz, px, py, pz);
+        copy_conditional(nx, px, n_is_infinity_mask);
+        copy_conditional(ny, py, n_is_infinity_mask);
+        copy_conditional(nz, pz, n_is_infinity_mask);
+
+        p_is_noninfinite_mask = ((s32) ~ (index - 1)) >> 31;
+        mask = p_is_noninfinite_mask & ~n_is_infinity_mask;
+        copy_conditional(nx, tx, mask);
+        copy_conditional(ny, ty, mask);
+        copy_conditional(nz, tz, mask);
+        n_is_infinity_mask &= ~p_is_noninfinite_mask;
+    }
+}
+
+/* Interface with Freebl: */
+
+#ifdef IS_BIG_ENDIAN

[wtc, 2013/01/25 02:32:49]

I changed #if to #ifdef here.

+#error "This code needs a little-endian processor"
+#endif
+
+static const u32 kRInvDigits[8] = {
+    0x80000000, 1, 0xffffffff, 0,
+    0x80000001, 0xfffffffe, 1, 0x7fffffff
+};
+#define MP_DIGITS_IN_256_BITS (32/sizeof(mp_digit))
+static const mp_int kRInv = {
+    MP_ZPOS,
+    MP_DIGITS_IN_256_BITS,
+    MP_DIGITS_IN_256_BITS,
+    /* Because we are running on a little-endian processor, this cast works for
+     * both 32 and 64-bit processors.
+     */
+    (mp_digit*) kRInvDigits
+};
+
+static const limb kTwo28 = 0x10000000;
+static const limb kTwo29 = 0x20000000;
+
+/* to_montgomery sets out = R*in. */
+static mp_err to_montgomery(felem out, const mp_int *in, const ECGroup *group)
+{
+    /* There are no MPI functions for bitshift operations and we wish to shift
+     * in 257 bits left so we move the digits 256-bits left and then multiply
+     * by two.
+     */
+    mp_int in_shifted;
+    int i;
+    mp_err res;
+
+    mp_init(&in_shifted);
+    s_mp_pad(&in_shifted, MP_USED(in) + MP_DIGITS_IN_256_BITS);
+    memcpy(&MP_DIGIT(&in_shifted, MP_DIGITS_IN_256_BITS),
+           MP_DIGITS(in),
+           MP_USED(in)*sizeof(mp_digit));
+    mp_mul_2(&in_shifted, &in_shifted);
+    MP_CHECKOK(group->meth->field_mod(&in_shifted, &in_shifted, group->meth));
+
+    for (i = 0;; i++) {
+        out[i] = MP_DIGIT(&in_shifted, 0) & kBottom29Bits;
+        mp_div_d(&in_shifted, kTwo29, &in_shifted, NULL);
+
+        i++;
+        if (i == NLIMBS)
+            break;
+        out[i] = MP_DIGIT(&in_shifted, 0) & kBottom28Bits;
+        mp_div_d(&in_shifted, kTwo28, &in_shifted, NULL);
+    }
+
+CLEANUP:
+    mp_clear(&in_shifted);
+    return res;
+}
+
+/* from_montgomery sets out=in/R. */
+static mp_err from_montgomery(mp_int *out, const felem in,
+                              const ECGroup *group)
+{
+    mp_int result, tmp;
+    mp_err res;
+    int i;
+
+    mp_init(&result);
+    mp_init(&tmp);
+
+    MP_CHECKOK(mp_add_d(&tmp, in[NLIMBS-1], &result));
+    for (i = NLIMBS-2; i >= 0; i--) {
+        if ((i & 1) == 0) {
+            MP_CHECKOK(mp_mul_d(&result, kTwo29, &tmp));
+        } else {
+            MP_CHECKOK(mp_mul_d(&result, kTwo28, &tmp));
+        }
+        MP_CHECKOK(mp_add_d(&tmp, in[i], &result));
+    }
+
+    MP_CHECKOK(mp_mul(&result, &kRInv, out));
+    MP_CHECKOK(group->meth->field_mod(out, out, group->meth));
+
+CLEANUP:
+    mp_clear(&result);
+    mp_clear(&tmp);
+    return res;
+}
+
+/* scalar_from_mp_int sets out_scalar=n, where n < the group order. */
+static void scalar_from_mp_int(u8 out_scalar[32], const mp_int *n)
+{
+    /* We require that |n| is less than the order of the group and therefore it
+     * will fit into |scalar|. However, these is a timing side-channel here that
+     * we cannot avoid: if |n| is sufficiently small it may be one or more words
+     * too short and we'll copy less data.
+     */
+    memset(out_scalar, 0, 32);
+    memcpy(out_scalar, MP_DIGITS(n), MP_USED(n) * sizeof(mp_digit));
+}
+
+/* ec_GFp_nistp256_base_point_mul sets {out_x,out_y} = nG, where n is < the
+ * order of the group.
+ */
+mp_err ec_GFp_nistp256_base_point_mul(const mp_int *n,
+                                      mp_int *out_x, mp_int *out_y,
+                                      const ECGroup *group)
+{
+    u8 scalar[32];
+    felem x, y, z, x_affine, y_affine;
+    mp_err res;
+
+    /* FIXME(agl): test that n < order. */
+
+    scalar_from_mp_int(scalar, n);
+    scalar_base_mult(x, y, z, scalar);
+    point_to_affine(x_affine, y_affine, x, y, z);
+    MP_CHECKOK(from_montgomery(out_x, x_affine, group));
+    MP_CHECKOK(from_montgomery(out_y, y_affine, group));
+
+CLEANUP:
+    return res;
+}
+
+/* ec_GFp_nistp256_point_mul sets {out_x,out_y} = n*{in_x,in_y}, where n is <
+ * the order of the group.
+ */
+mp_err ec_GFp_nistp256_point_mul(const mp_int *n,
+                                 const mp_int *in_x, const mp_int *in_y,
+                                 mp_int *out_x, mp_int *out_y,
+                                 const ECGroup *group)
+{
+    u8 scalar[32];
+    felem x, y, z, x_affine, y_affine, px, py;
+    mp_err res;
+
+    scalar_from_mp_int(scalar, n);
+
+    MP_CHECKOK(to_montgomery(px, in_x, group));
+    MP_CHECKOK(to_montgomery(py, in_y, group));
+
+    scalar_mult(x, y, z, px, py, scalar);
+    point_to_affine(x_affine, y_affine, x, y, z);
+    MP_CHECKOK(from_montgomery(out_x, x_affine, group));
+    MP_CHECKOK(from_montgomery(out_y, y_affine, group));
+
+CLEANUP:
+    return res;
+}
+
+/* ec_GFp_nistp256_point_mul_vartime sets {out_x,out_y} = n1*G +
+ * n2*{in_x,in_y}, where n1 and n2 are < the order of the group.
+ *
+ * As indicated by the name, this function operates in variable time. This
+ * is safe because it's used for signature validation which doesn't deal
+ * with secrets.
+ */
+mp_err ec_GFp_nistp256_points_mul_vartime(
+        const mp_int *n1, const mp_int *n2,
+        const mp_int *in_x, const mp_int *in_y,
+        mp_int *out_x, mp_int *out_y,
+        const ECGroup *group)
+{
+    u8 scalar1[32], scalar2[32];
+    felem x1, y1, z1, x2, y2, z2, x_affine, y_affine, px, py;
+    mp_err res = MP_OKAY;
+
+    /* If n2 == NULL, this is just a base-point multiplication. */
+    if (n2 == NULL) {
+        return ec_GFp_nistp256_base_point_mul(n1, out_x, out_y, group);
+    }
+
+    /* If n1 == nULL, this is just an arbitary-point multiplication. */
+    if (n1 == NULL) {
+        return ec_GFp_nistp256_point_mul(n2, in_x, in_y, out_x, out_y, group);
+    }
+
+    /* If both scalars are zero, then the result is the point at infinity. */
+    if (mp_cmp_z(n1) == 0 && mp_cmp_z(n2) == 0) {
+        mp_zero(out_x);
+        mp_zero(out_y);
+        return res;
+    }
+
+    scalar_from_mp_int(scalar1, n1);
+    scalar_from_mp_int(scalar2, n2);
+
+    MP_CHECKOK(to_montgomery(px, in_x, group));
+    MP_CHECKOK(to_montgomery(py, in_y, group));
+    scalar_base_mult(x1, y1, z1, scalar1);
+    scalar_mult(x2, y2, z2, px, py, scalar2);
+
+    if (mp_cmp_z(n2) == 0) {
+        /* If n2 == 0, then {x2,y2,z2} is zero and the result is just
+         * {x1,y1,z1}. */
+    } else if (mp_cmp_z(n1) == 0) {
+        /* If n1 == 0, then {x1,y1,z1} is zero and the result is just
+         * {x2,y2,z2}. */
+        memcpy(x1, x2, sizeof(x2));
+        memcpy(y1, y2, sizeof(y2));
+        memcpy(z1, z2, sizeof(z2));
+    } else {
+        /* This function handles the case where {x1,y1,z1} == {x2,y2,z2}. */
+        point_add_or_double_vartime(x1, y1, z1, x1, y1, z1, x2, y2, z2);
+    }
+
+    point_to_affine(x_affine, y_affine, x1, y1, z1);
+    MP_CHECKOK(from_montgomery(out_x, x_affine, group));
+    MP_CHECKOK(from_montgomery(out_y, y_affine, group));
+
+CLEANUP:
+    return res;
+}