Improve optional 64-bit NIST-P224 implementation, and add NIST-P256 and

NIST-P521. (Now -DEC_NISTP_64_GCC_128 enables all three of these;
-DEC_NISTP224_64_GCC_128 no longer works.)

Submitted by: Google Inc.

1 files changed, 2158 insertions, 0 deletions

diff --git a/crypto/ec/ecp_nistp256.c b/crypto/ec/ecp_nistp256.c
new file mode 100644
index 0000000000..a1cef699a0
--- /dev/null
+++ b/crypto/ec/ecp_nistp256.c
@@ -0,0 +1,2158 @@
+/* crypto/ec/ecp_nistp256.c */
+/*
+ * Written by Adam Langley (Google) for the OpenSSL project
+ */
+/* Copyright 2011 Google Inc.
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ *
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ *     http://www.apache.org/licenses/LICENSE-2.0
+ *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
+ */
+
+/*
+ * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
+ *
+ * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
+ * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
+ * work which got its smarts from Daniel J. Bernstein's work on the same.
+ */
+
+#ifdef EC_NISTP_64_GCC_128
+
+#include <stdint.h>
+#include <string.h>
+#include <openssl/err.h>
+#include "ec_lcl.h"
+
+#if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
+  /* even with gcc, the typedef won't work for 32-bit platforms */
+  typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
+  typedef __int128_t int128_t;
+#else
+  #error "Need GCC 3.1 or later to define type uint128_t"
+#endif
+
+typedef uint8_t u8;
+typedef uint32_t u32;
+typedef uint64_t u64;
+typedef int64_t s64;
+
+/* The underlying field.
+ *
+ * P256 operates over GF(2^256-2^224+2^192+2^96-1). We can serialise an element
+ * of this field into 32 bytes. We call this an felem_bytearray. */
+
+typedef u8 felem_bytearray[32];
+
+/* These are the parameters of P256, taken from FIPS 186-3, page 86. These
+ * values are big-endian. */
+static const felem_bytearray nistp256_curve_params[5] = {
+	{0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* p */
+	 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+	 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
+	 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
+	{0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* a = -3 */
+	 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
+	 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
+	 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc},      /* b */
+	{0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
+	 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
+	 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
+	 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
+	{0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47,       /* x */
+	 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,       /* y */
+	 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
+	 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
+	 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
+};
+
+/* The representation of field elements.
+ * ------------------------------------
+ *
+ * We represent field elements with either four 128-bit values, eight 128-bit
+ * values, or four 64-bit values. The field element represented is:
+ *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192  (mod p)
+ * or:
+ *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512  (mod p)
+ *
+ * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
+ * apart, but are 128-bits wide, the most significant bits of each limb overlap
+ * with the least significant bits of the next.
+ *
+ * A field element with four limbs is an 'felem'. One with eight limbs is a
+ * 'longfelem'
+ *
+ * A field element with four, 64-bit values is called a 'smallfelem'. Small
+ * values are used as intermediate values before multiplication.
+ */
+
+#define NLIMBS 4
+
+typedef uint128_t limb;
+typedef limb felem[NLIMBS];
+typedef limb longfelem[NLIMBS * 2];
+typedef u64 smallfelem[NLIMBS];
+
+/* This is the value of the prime as four 64-bit words, little-endian. */
+static const u64 kPrime[4] = { 0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul };
+static const limb bottom32bits = 0xffffffff;
+static const u64 bottom63bits = 0x7ffffffffffffffful;
+
+/* bin32_to_felem takes a little-endian byte array and converts it into felem
+ * form. This assumes that the CPU is little-endian. */
+static void bin32_to_felem(felem out, const u8 in[32])
+	{
+	out[0] = *((u64*) &in[0]);
+	out[1] = *((u64*) &in[8]);
+	out[2] = *((u64*) &in[16]);
+	out[3] = *((u64*) &in[24]);
+	}
+
+/* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian,
+ * 32 byte array. This assumes that the CPU is little-endian. */
+static void smallfelem_to_bin32(u8 out[32], const smallfelem in)
+	{
+	*((u64*) &out[0]) = in[0];
+	*((u64*) &out[8]) = in[1];
+	*((u64*) &out[16]) = in[2];
+	*((u64*) &out[24]) = in[3];
+	}
+
+/* To preserve endianness when using BN_bn2bin and BN_bin2bn */
+static void flip_endian(u8 *out, const u8 *in, unsigned len)
+	{
+	unsigned i;
+	for (i = 0; i < len; ++i)
+		out[i] = in[len-1-i];
+	}
+
+/* BN_to_felem converts an OpenSSL BIGNUM into an felem */
+static int BN_to_felem(felem out, const BIGNUM *bn)
+	{
+	felem_bytearray b_in;
+	felem_bytearray b_out;
+	unsigned num_bytes;
+
+	/* BN_bn2bin eats leading zeroes */
+	memset(b_out, 0, sizeof b_out);
+	num_bytes = BN_num_bytes(bn);
+	if (num_bytes > sizeof b_out)
+		{
+		ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
+		return 0;
+		}
+	if (BN_is_negative(bn))
+		{
+		ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
+		return 0;
+		}
+	num_bytes = BN_bn2bin(bn, b_in);
+	flip_endian(b_out, b_in, num_bytes);
+	bin32_to_felem(out, b_out);
+	return 1;
+	}
+
+/* felem_to_BN converts an felem into an OpenSSL BIGNUM */
+static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in)
+	{
+	felem_bytearray b_in, b_out;
+	smallfelem_to_bin32(b_in, in);
+	flip_endian(b_out, b_in, sizeof b_out);
+	return BN_bin2bn(b_out, sizeof b_out, out);
+	}
+
+
+/* Field operations
+ * ---------------- */
+
+static void smallfelem_one(smallfelem out)
+	{
+	out[0] = 1;
+	out[1] = 0;
+	out[2] = 0;
+	out[3] = 0;
+	}
+
+static void smallfelem_assign(smallfelem out, const smallfelem in)
+	{
+	out[0] = in[0];
+	out[1] = in[1];
+	out[2] = in[2];
+	out[3] = in[3];
+	}
+
+static void felem_assign(felem out, const felem in)
+	{
+	out[0] = in[0];
+	out[1] = in[1];
+	out[2] = in[2];
+	out[3] = in[3];
+	}
+
+/* felem_sum sets out = out + in. */
+static void felem_sum(felem out, const felem in)
+	{
+	out[0] += in[0];
+	out[1] += in[1];
+	out[2] += in[2];
+	out[3] += in[3];
+	}
+
+/* felem_small_sum sets out = out + in. */
+static void felem_small_sum(felem out, const smallfelem in)
+	{
+	out[0] += in[0];
+	out[1] += in[1];
+	out[2] += in[2];
+	out[3] += in[3];
+	}
+
+/* felem_scalar sets out = out * scalar */
+static void felem_scalar(felem out, const u64 scalar)
+	{
+	out[0] *= scalar;
+	out[1] *= scalar;
+	out[2] *= scalar;
+	out[3] *= scalar;
+	}
+
+/* longfelem_scalar sets out = out * scalar */
+static void longfelem_scalar(longfelem out, const u64 scalar)
+	{
+	out[0] *= scalar;
+	out[1] *= scalar;
+	out[2] *= scalar;
+	out[3] *= scalar;
+	out[4] *= scalar;
+	out[5] *= scalar;
+	out[6] *= scalar;
+	out[7] *= scalar;
+	}
+
+#define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
+#define two105 (((limb)1) << 105)
+#define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
+
+/* zero105 is 0 mod p */
+static const felem zero105 = { two105m41m9, two105, two105m41p9, two105m41p9 };
+
+/* smallfelem_neg sets |out| to |-small|
+ * On exit:
+ *   out[i] < out[i] + 2^105
+ */
+static void smallfelem_neg(felem out, const smallfelem small)
+	{
+	/* In order to prevent underflow, we subtract from 0 mod p. */
+	out[0] = zero105[0] - small[0];
+	out[1] = zero105[1] - small[1];
+	out[2] = zero105[2] - small[2];
+	out[3] = zero105[3] - small[3];
+	}
+
+/* felem_diff subtracts |in| from |out|
+ * On entry:
+ *   in[i] < 2^104
+ * On exit:
+ *   out[i] < out[i] + 2^105
+ */
+static void felem_diff(felem out, const felem in)
+	{
+	/* In order to prevent underflow, we add 0 mod p before subtracting. */
+	out[0] += zero105[0];
+	out[1] += zero105[1];
+	out[2] += zero105[2];
+	out[3] += zero105[3];
+
+	out[0] -= in[0];
+	out[1] -= in[1];
+	out[2] -= in[2];
+	out[3] -= in[3];
+	}
+
+#define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
+#define two107 (((limb)1) << 107)
+#define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
+
+/* zero107 is 0 mod p */
+static const felem zero107 = { two107m43m11, two107, two107m43p11, two107m43p11 };
+
+/* An alternative felem_diff for larger inputs |in|
+ * felem_diff_zero107 subtracts |in| from |out|
+ * On entry:
+ *   in[i] < 2^106
+ * On exit:
+ *   out[i] < out[i] + 2^107
+ */
+static void felem_diff_zero107(felem out, const felem in)
+	{
+	/* In order to prevent underflow, we add 0 mod p before subtracting. */
+	out[0] += zero107[0];
+	out[1] += zero107[1];
+	out[2] += zero107[2];
+	out[3] += zero107[3];
+
+	out[0] -= in[0];
+	out[1] -= in[1];
+	out[2] -= in[2];
+	out[3] -= in[3];
+	}
+
+/* longfelem_diff subtracts |in| from |out|
+ * On entry:
+ *   in[i] < 7*2^67
+ * On exit:
+ *   out[i] < out[i] + 2^70 + 2^40
+ */
+static void longfelem_diff(longfelem out, const longfelem in)
+	{
+	static const limb two70m8p6 = (((limb)1) << 70) - (((limb)1) << 8) + (((limb)1) << 6);
+	static const limb two70p40 = (((limb)1) << 70) + (((limb)1) << 40);
+	static const limb two70 = (((limb)1) << 70);
+	static const limb two70m40m38p6 = (((limb)1) << 70) - (((limb)1) << 40) - (((limb)1) << 38) + (((limb)1) << 6);
+	static const limb two70m6 = (((limb)1) << 70) - (((limb)1) << 6);
+
+	/* add 0 mod p to avoid underflow */
+	out[0] += two70m8p6;
+	out[1] += two70p40;
+	out[2] += two70;
+	out[3] += two70m40m38p6;
+	out[4] += two70m6;
+	out[5] += two70m6;
+	out[6] += two70m6;
+	out[7] += two70m6;
+
+	/* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
+	out[0] -= in[0];
+	out[1] -= in[1];
+	out[2] -= in[2];
+	out[3] -= in[3];
+	out[4] -= in[4];
+	out[5] -= in[5];
+	out[6] -= in[6];
+	out[7] -= in[7];
+	}
+
+#define two64m0 (((limb)1) << 64) - 1
+#define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
+#define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
+#define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
+
+/* zero110 is 0 mod p */
+static const felem zero110 = { two64m0, two110p32m0, two64m46, two64m32 };
+
+/* felem_shrink converts an felem into a smallfelem. The result isn't quite
+ * minimal as the value may be greater than p.
+ *
+ * On entry:
+ *   in[i] < 2^109
+ * On exit:
+ *   out[i] < 2^64
+ */
+static void felem_shrink(smallfelem out, const felem in)
+	{
+	felem tmp;
+	/* Carry 2->3 */
+	tmp[3] = zero110[3] + in[3] + ((u64) (in[2] >> 64));
+	/* tmp[3] < 2^110 */
+
+	tmp[2] = zero110[2] + (u64) in[2];
+	tmp[0] = zero110[0] + in[0];
+	tmp[1] = zero110[1] + in[1];
+	/* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
+
+	/* We perform two partial reductions where we eliminate the
+	 * high-word of tmp[3]. We don't update the other words till the end.
+	 */
+	u64 a = tmp[3] >> 64; /* a < 2^46 */
+	tmp[3] = (u64) tmp[3];
+	tmp[3] -= a;
+	tmp[3] += ((limb)a) << 32;
+	/* tmp[3] < 2^79 */
+
+	u64 b = a;
+	a = tmp[3] >> 64; /* a < 2^15 */
+	b += a; /* b < 2^46 + 2^15 < 2^47 */
+	tmp[3] = (u64) tmp[3];
+	tmp[3] -= a;
+	tmp[3] += ((limb)a) << 32;
+	/* tmp[3] < 2^64 + 2^47 */
+
+	/* This adjusts the other two words to complete the two partial
+	 * reductions. */
+	tmp[0] += b;
+	tmp[1] -= (((limb)b) << 32);
+
+	/* In order to make space in tmp[3] for the carry from 2 -> 3, we
+	 * conditionally subtract kPrime if tmp[3] is large enough. */
+	static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */
+	s64 high = tmp[3] >> 64;
+	/* As tmp[3] < 2^65, high is either 1 or 0 */
+	high <<= 63;
+	high >>= 63;
+	/* high is:
+	 *   all ones   if the high word of tmp[3] is 1
+	 *   all zeros  if the high word of tmp[3] if 0 */
+	s64 low = tmp[3];
+	u64 mask = low >> 63;
+	/* mask is:
+	 *   all ones   if the MSB of low is 1
+	 *   all zeros  if the MSB of low if 0 */
+	low &= bottom63bits;
+	low -= kPrime3Test;
+	/* if low was greater than kPrime3Test then the MSB is zero */
+	low = ~low;
+	low >>= 63;
+	/* low is:
+	 *   all ones   if low was > kPrime3Test
+	 *   all zeros  if low was <= kPrime3Test */
+	mask = (mask & low) | high;
+	tmp[0] -= mask & kPrime[0];
+	tmp[1] -= mask & kPrime[1];
+	/* kPrime[2] is zero, so omitted */
+	tmp[3] -= mask & kPrime[3];
+	/* tmp[3] < 2**64 - 2**32 + 1 */
+
+	tmp[1] += ((u64) (tmp[0] >> 64)); tmp[0] = (u64) tmp[0];
+	tmp[2] += ((u64) (tmp[1] >> 64)); tmp[1] = (u64) tmp[1];
+	tmp[3] += ((u64) (tmp[2] >> 64)); tmp[2] = (u64) tmp[2];
+	/* tmp[i] < 2^64 */
+
+	out[0] = tmp[0];
+	out[1] = tmp[1];
+	out[2] = tmp[2];
+	out[3] = tmp[3];
+	}
+
+/* smallfelem_expand converts a smallfelem to an felem */
+static void smallfelem_expand(felem out, const smallfelem in)
+	{
+	out[0] = in[0];
+	out[1] = in[1];
+	out[2] = in[2];
+	out[3] = in[3];
+	}
+
+/* smallfelem_square sets |out| = |small|^2
+ * On entry:
+ *   small[i] < 2^64
+ * On exit:
+ *   out[i] < 7 * 2^64 < 2^67
+ */
+static void smallfelem_square(longfelem out, const smallfelem small)
+	{
+	limb a;
+	u64 high, low;
+
+	a = ((uint128_t) small[0]) * small[0];
+	low = a;
+	high = a >> 64;
+	out[0] = low;
+	out[1] = high;
+
+	a = ((uint128_t) small[0]) * small[1];
+	low = a;
+	high = a >> 64;
+	out[1] += low;
+	out[1] += low;
+	out[2] = high;
+
+	a = ((uint128_t) small[0]) * small[2];
+	low = a;
+	high = a >> 64;
+	out[2] += low;
+	out[2] *= 2;
+	out[3] = high;
+
+	a = ((uint128_t) small[0]) * small[3];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[4] = high;
+
+	a = ((uint128_t) small[1]) * small[2];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[3] *= 2;
+	out[4] += high;
+
+	a = ((uint128_t) small[1]) * small[1];
+	low = a;
+	high = a >> 64;
+	out[2] += low;
+	out[3] += high;
+
+	a = ((uint128_t) small[1]) * small[3];
+	low = a;
+	high = a >> 64;
+	out[4] += low;
+	out[4] *= 2;
+	out[5] = high;
+
+	a = ((uint128_t) small[2]) * small[3];
+	low = a;
+	high = a >> 64;
+	out[5] += low;
+	out[5] *= 2;
+	out[6] = high;
+	out[6] += high;
+
+	a = ((uint128_t) small[2]) * small[2];
+	low = a;
+	high = a >> 64;
+	out[4] += low;
+	out[5] += high;
+
+	a = ((uint128_t) small[3]) * small[3];
+	low = a;
+	high = a >> 64;
+	out[6] += low;
+	out[7] = high;
+	}
+
+/* felem_square sets |out| = |in|^2
+ * On entry:
+ *   in[i] < 2^109
+ * On exit:
+ *   out[i] < 7 * 2^64 < 2^67
+ */
+static void felem_square(longfelem out, const felem in)
+	{
+	u64 small[4];
+	felem_shrink(small, in);
+	smallfelem_square(out, small);
+	}
+
+/* smallfelem_mul sets |out| = |small1| * |small2|
+ * On entry:
+ *   small1[i] < 2^64
+ *   small2[i] < 2^64
+ * On exit:
+ *   out[i] < 7 * 2^64 < 2^67
+ */
+static void smallfelem_mul(longfelem out, const smallfelem small1, const smallfelem small2)
+	{
+	limb a;
+	u64 high, low;
+
+	a = ((uint128_t) small1[0]) * small2[0];
+	low = a;
+	high = a >> 64;
+	out[0] = low;
+	out[1] = high;
+
+
+	a = ((uint128_t) small1[0]) * small2[1];
+	low = a;
+	high = a >> 64;
+	out[1] += low;
+	out[2] = high;
+
+	a = ((uint128_t) small1[1]) * small2[0];
+	low = a;
+	high = a >> 64;
+	out[1] += low;
+	out[2] += high;
+
+
+	a = ((uint128_t) small1[0]) * small2[2];
+	low = a;
+	high = a >> 64;
+	out[2] += low;
+	out[3] = high;
+
+	a = ((uint128_t) small1[1]) * small2[1];
+	low = a;
+	high = a >> 64;
+	out[2] += low;
+	out[3] += high;
+
+	a = ((uint128_t) small1[2]) * small2[0];
+	low = a;
+	high = a >> 64;
+	out[2] += low;
+	out[3] += high;
+
+
+	a = ((uint128_t) small1[0]) * small2[3];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[4] = high;
+
+	a = ((uint128_t) small1[1]) * small2[2];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[4] += high;
+
+	a = ((uint128_t) small1[2]) * small2[1];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[4] += high;
+
+	a = ((uint128_t) small1[3]) * small2[0];
+	low = a;
+	high = a >> 64;
+	out[3] += low;
+	out[4] += high;
+
+
+	a = ((uint128_t) small1[1]) * small2[3];
+	low = a;
+	high = a >> 64;
+	out[4] += low;
+	out[5] = high;
+
+	a = ((uint128_t) small1[2]) * small2[2];
+	low = a;
+	high = a >> 64;
+	out[4] += low;
+	out[5] += high;
+
+	a = ((uint128_t) small1[3]) * small2[1];
+	low = a;
+	high = a >> 64;
+	out[4] += low;
+	out[5] += high;
+
+
+	a = ((uint128_t) small1[2]) * small2[3];
+	low = a;
+	high = a >> 64;
+	out[5] += low;
+	out[6] = high;
+
+	a = ((uint128_t) small1[3]) * small2[2];
+	low = a;
+	high = a >> 64;
+	out[5] += low;
+	out[6] += high;
+
+
+	a = ((uint128_t) small1[3]) * small2[3];
+	low = a;
+	high = a >> 64;
+	out[6] += low;
+	out[7] = high;
+	}
+
+/* felem_mul sets |out| = |in1| * |in2|
+ * On entry:
+ *   in1[i] < 2^109
+ *   in2[i] < 2^109
+ * On exit:
+ *   out[i] < 7 * 2^64 < 2^67
+ */
+static void felem_mul(longfelem out, const felem in1, const felem in2)
+	{
+	smallfelem small1, small2;
+	felem_shrink(small1, in1);
+	felem_shrink(small2, in2);
+	smallfelem_mul(out, small1, small2);
+	}
+
+/* felem_small_mul sets |out| = |small1| * |in2|
+ * On entry:
+ *   small1[i] < 2^64
+ *   in2[i] < 2^109
+ * On exit:
+ *   out[i] < 7 * 2^64 < 2^67
+ */
+static void felem_small_mul(longfelem out, const smallfelem small1, const felem in2)
+	{
+	smallfelem small2;
+	felem_shrink(small2, in2);
+	smallfelem_mul(out, small1, small2);
+	}
+
+#define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
+#define two100 (((limb)1) << 100)
+#define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
+/* zero100 is 0 mod p */
+static const felem zero100 = { two100m36m4, two100, two100m36p4, two100m36p4 };
+
+/* Internal function for the different flavours of felem_reduce.
+ * felem_reduce_ reduces the higher coefficients in[4]-in[7].
+ * On entry:
+ *   out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7] 
+ *   out[1] >= in[7] + 2^32*in[4]
+ *   out[2] >= in[5] + 2^32*in[5]
+ *   out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
+ * On exit:
+ *   out[0] <= out[0] + in[4] + 2^32*in[5]
+ *   out[1] <= out[1] + in[5] + 2^33*in[6]
+ *   out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
+ *   out[3] <= out[3] + 2^32*in[4] + 3*in[7]
+ */
+static void felem_reduce_(felem out, const longfelem in)
+	{
+	int128_t c;
+	/* combine common terms from below */
+	c = in[4] + (in[5] << 32);
+	out[0] += c;
+	out[3] -= c;
+
+	c = in[5] - in[7];
+	out[1] += c;
+	out[2] -= c;
+
+	/* the remaining terms */
+	/* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
+	out[1] -= (in[4] << 32);
+	out[3] += (in[4] << 32);
+
+	/* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
+	out[2] -= (in[5] << 32);
+
+	/* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
+	out[0] -= in[6];
+	out[0] -= (in[6] << 32);
+	out[1] += (in[6] << 33);
+	out[2] += (in[6] * 2);
+	out[3] -= (in[6] << 32);
+
+	/* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
+	out[0] -= in[7];
+	out[0] -= (in[7] << 32);
+	out[2] += (in[7] << 33);
+	out[3] += (in[7] * 3);
+	}
+
+/* felem_reduce converts a longfelem into an felem.
+ * To be called directly after felem_square or felem_mul.
+ * On entry:
+ *   in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
+ *   in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
+ * On exit:
+ *   out[i] < 2^101
+ */
+static void felem_reduce(felem out, const longfelem in)
+	{
+	out[0] = zero100[0] + in[0];
+	out[1] = zero100[1] + in[1];
+	out[2] = zero100[2] + in[2];
+	out[3] = zero100[3] + in[3];
+
+	felem_reduce_(out, in);
+
+	/* out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
+	 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
+	 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
+	 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
+	 *
+	 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
+	 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
+	 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
+	 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101
+	 */
+	}
+
+/* felem_reduce_zero105 converts a larger longfelem into an felem.
+ * On entry:
+ *   in[0] < 2^71
+ * On exit:
+ *   out[i] < 2^106
+ */
+static void felem_reduce_zero105(felem out, const longfelem in)
+	{
+	out[0] = zero105[0] + in[0];
+	out[1] = zero105[1] + in[1];
+	out[2] = zero105[2] + in[2];
+	out[3] = zero105[3] + in[3];
+
+	felem_reduce_(out, in);
+
+	/* out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
+	 * out[1] > 2^105 - 2^71 - 2^103 > 0
+	 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
+	 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
+	 *
+	 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
+	 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
+	 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
+	 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
+	 */
+	}
+
+/* subtract_u64 sets *result = *result - v and *carry to one if the subtraction
+ * underflowed. */
+static void subtract_u64(u64* result, u64* carry, u64 v)
+	{
+	uint128_t r = *result;
+	r -= v;
+	*carry = (r >> 64) & 1;
+	*result = (u64) r;
+	}
+
+/* felem_contract converts |in| to its unique, minimal representation.
+ * On entry:
+ *   in[i] < 2^109
+ */
+static void felem_contract(smallfelem out, const felem in)
+	{
+	unsigned i;
+	u64 all_equal_so_far = 0, result = 0, carry;
+
+	felem_shrink(out, in);
+	/* small is minimal except that the value might be > p */
+
+	all_equal_so_far--;
+	/* We are doing a constant time test if out >= kPrime. We need to
+	 * compare each u64, from most-significant to least significant. For
+	 * each one, if all words so far have been equal (m is all ones) then a
+	 * non-equal result is the answer. Otherwise we continue. */
+	for (i = 3; i < 4; i--) {
+		uint128_t a = ((uint128_t) kPrime[i]) - out[i];
+		/* if out[i] > kPrime[i] then a will underflow and the high
+		 * 64-bits will all be set. */
+		result |= all_equal_so_far & ((u64) (a >> 64));
+
+		/* if kPrime[i] == out[i] then |equal| will be all zeros and
+		 * the decrement will make it all ones. */
+		u64 equal = kPrime[i] ^ out[i];
+		equal--;
+		equal &= equal << 32;
+		equal &= equal << 16;
+		equal &= equal << 8;
+		equal &= equal << 4;
+		equal &= equal << 2;
+		equal &= equal << 1;
+		equal = ((s64) equal) >> 63;
+
+		all_equal_so_far &= equal;
+	}
+
+	/* if all_equal_so_far is still all ones then the two values are equal
+	 * and so out >= kPrime is true. */
+	result |= all_equal_so_far;
+
+	/* if out >= kPrime then we subtract kPrime. */
+	subtract_u64(&out[0], &carry, result & kPrime[0]);
+	subtract_u64(&out[1], &carry, carry);
+	subtract_u64(&out[2], &carry, carry);
+	subtract_u64(&out[3], &carry, carry);
+
+	subtract_u64(&out[1], &carry, result & kPrime[1]);
+	subtract_u64(&out[2], &carry, carry);
+	subtract_u64(&out[3], &carry, carry);
+
+	subtract_u64(&out[2], &carry, result & kPrime[2]);
+	subtract_u64(&out[3], &carry, carry);
+
+	subtract_u64(&out[3], &carry, result & kPrime[3]);
+	}
+
+static void smallfelem_square_contract(smallfelem out, const smallfelem in)
+	{
+	longfelem longtmp;
+	felem tmp;
+
+	smallfelem_square(longtmp, in);
+	felem_reduce(tmp, longtmp);
+	felem_contract(out, tmp);
+	}
+
+static void smallfelem_mul_contract(smallfelem out, const smallfelem in1, const smallfelem in2)
+	{
+	longfelem longtmp;
+	felem tmp;
+
+	smallfelem_mul(longtmp, in1, in2);
+	felem_reduce(tmp, longtmp);
+	felem_contract(out, tmp);
+	}
+
+/* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
+ * otherwise.
+ * On entry:
+ *   small[i] < 2^64
+ */
+static limb smallfelem_is_zero(const smallfelem small)
+	{
+	limb result;
+
+	u64 is_zero = small[0] | small[1] | small[2] | small[3];
+	is_zero--;
+	is_zero &= is_zero << 32;
+	is_zero &= is_zero << 16;
+	is_zero &= is_zero << 8;
+	is_zero &= is_zero << 4;
+	is_zero &= is_zero << 2;
+	is_zero &= is_zero << 1;
+	is_zero = ((s64) is_zero) >> 63;
+
+	u64 is_p = (small[0] ^ kPrime[0]) |
+		   (small[1] ^ kPrime[1]) |
+		   (small[2] ^ kPrime[2]) |
+		   (small[3] ^ kPrime[3]);
+	is_p--;
+	is_p &= is_p << 32;
+	is_p &= is_p << 16;
+	is_p &= is_p << 8;
+	is_p &= is_p << 4;
+	is_p &= is_p << 2;
+	is_p &= is_p << 1;
+	is_p = ((s64) is_p) >> 63;
+
+	is_zero |= is_p;
+
+	result = is_zero;
+	result |= ((limb) is_zero) << 64;
+	return result;
+	}
+
+static int smallfelem_is_zero_int(const smallfelem small)
+	{
+	return (int) (smallfelem_is_zero(small) & ((limb)1));
+	}
+
+/* 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;
+	longfelem tmp;
+	unsigned i;
+
+	felem_square(tmp, in); felem_reduce(ftmp, tmp);			/* 2^1 */
+	felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);		/* 2^2 - 2^0 */
+	felem_assign(e2, ftmp);
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^3 - 2^1 */
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^4 - 2^2 */
+	felem_mul(tmp, ftmp, e2); felem_reduce(ftmp, tmp);		/* 2^4 - 2^0 */
+	felem_assign(e4, ftmp);
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^5 - 2^1 */
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^6 - 2^2 */
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^7 - 2^3 */
+	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);		/* 2^8 - 2^4 */
+	felem_mul(tmp, ftmp, e4); felem_reduce(ftmp, tmp);		/* 2^8 - 2^0 */
+	felem_assign(e8, ftmp);
+	for (i = 0; i < 8; i++) {
+		felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
+	}								/* 2^16 - 2^8 */
+	felem_mul(tmp, ftmp, e8); felem_reduce(ftmp, tmp);		/* 2^16 - 2^0 */
+	felem_assign(e16, ftmp);
+	for (i = 0; i < 16; i++) {
+		felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
+	}								/* 2^32 - 2^16 */
+	felem_mul(tmp, ftmp, e16); felem_reduce(ftmp, tmp);		/* 2^32 - 2^0 */
+	felem_assign(e32, ftmp);
+	for (i = 0; i < 32; i++) {
+		felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
+	}								/* 2^64 - 2^32 */
+	felem_assign(e64, ftmp);
+	felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp);		/* 2^64 - 2^32 + 2^0 */
+	for (i = 0; i < 192; i++) {
+		felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
+	}								/* 2^256 - 2^224 + 2^192 */
+
+	felem_mul(tmp, e64, e32); felem_reduce(ftmp2, tmp);		/* 2^64 - 2^0 */
+	for (i = 0; i < 16; i++) {
+		felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
+	}								/* 2^80 - 2^16 */
+	felem_mul(tmp, ftmp2, e16); felem_reduce(ftmp2, tmp);		/* 2^80 - 2^0 */
+	for (i = 0; i < 8; i++) {
+		felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
+	}								/* 2^88 - 2^8 */
+	felem_mul(tmp, ftmp2, e8); felem_reduce(ftmp2, tmp);		/* 2^88 - 2^0 */
+	for (i = 0; i < 4; i++) {
+		felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
+	}								/* 2^92 - 2^4 */
+	felem_mul(tmp, ftmp2, e4); felem_reduce(ftmp2, tmp);		/* 2^92 - 2^0 */
+	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);		/* 2^93 - 2^1 */
+	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);		/* 2^94 - 2^2 */
+	felem_mul(tmp, ftmp2, e2); felem_reduce(ftmp2, tmp);		/* 2^94 - 2^0 */
+	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);		/* 2^95 - 2^1 */
+	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);		/* 2^96 - 2^2 */
+	felem_mul(tmp, ftmp2, in); felem_reduce(ftmp2, tmp);		/* 2^96 - 3 */
+
+	felem_mul(tmp, ftmp2, ftmp); felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
+	}
+
+static void smallfelem_inv_contract(smallfelem out, const smallfelem in)
+	{
+	felem tmp;
+
+	smallfelem_expand(tmp, in);
+	felem_inv(tmp, tmp);
+	felem_contract(out, tmp);
+	}
+
+/* Group operations
+ * ----------------
+ *
+ * Building on top of the field operations we have the operations on the
+ * elliptic curve group itself. Points on the curve are represented in Jacobian
+ * coordinates */
+
+/* point_double calculates 2*(x_in, y_in, z_in)
+ *
+ * The method is taken from:
+ *   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
+ *
+ * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
+ * while x_out == y_in is not (maybe this works, but it's not tested). */
+static void
+point_double(felem x_out, felem y_out, felem z_out,
+	     const felem x_in, const felem y_in, const felem z_in)
+	{
+	longfelem tmp, tmp2;
+	felem delta, gamma, beta, alpha, ftmp, ftmp2;
+	smallfelem small1, small2;
+
+	felem_assign(ftmp, x_in);
+	/* ftmp[i] < 2^106 */
+	felem_assign(ftmp2, x_in);
+	/* ftmp2[i] < 2^106 */
+
+	/* delta = z^2 */
+	felem_square(tmp, z_in);
+	felem_reduce(delta, tmp);
+	/* delta[i] < 2^101 */
+
+	/* gamma = y^2 */
+	felem_square(tmp, y_in);
+	felem_reduce(gamma, tmp);
+	/* gamma[i] < 2^101 */
+	felem_shrink(small1, gamma);
+
+	/* beta = x*gamma */
+	felem_small_mul(tmp, small1, x_in);
+	felem_reduce(beta, tmp);
+	/* beta[i] < 2^101 */
+
+	/* alpha = 3*(x-delta)*(x+delta) */
+	felem_diff(ftmp, delta);
+	/* ftmp[i] < 2^105 + 2^106 < 2^107 */
+	felem_sum(ftmp2, delta);
+	/* ftmp2[i] < 2^105 + 2^106 < 2^107 */
+	felem_scalar(ftmp2, 3);
+	/* ftmp2[i] < 3 * 2^107 < 2^109 */
+	felem_mul(tmp, ftmp, ftmp2);
+	felem_reduce(alpha, tmp);
+	/* alpha[i] < 2^101 */
+	felem_shrink(small2, alpha);
+
+	/* x' = alpha^2 - 8*beta */
+	smallfelem_square(tmp, small2);
+	felem_reduce(x_out, tmp);
+	felem_assign(ftmp, beta);
+	felem_scalar(ftmp, 8);
+	/* ftmp[i] < 8 * 2^101 = 2^104 */
+	felem_diff(x_out, ftmp);
+	/* x_out[i] < 2^105 + 2^101 < 2^106 */
+
+	/* z' = (y + z)^2 - gamma - delta */
+	felem_sum(delta, gamma);
+	/* delta[i] < 2^101 + 2^101 = 2^102 */
+	felem_assign(ftmp, y_in);
+	felem_sum(ftmp, z_in);
+	/* ftmp[i] < 2^106 + 2^106 = 2^107 */
+	felem_square(tmp, ftmp);
+	felem_reduce(z_out, tmp);
+	felem_diff(z_out, delta);
+	/* z_out[i] < 2^105 + 2^101 < 2^106 */
+
+	/* y' = alpha*(4*beta - x') - 8*gamma^2 */
+	felem_scalar(beta, 4);
+	/* beta[i] < 4 * 2^101 = 2^103 */
+	felem_diff_zero107(beta, x_out);
+	/* beta[i] < 2^107 + 2^103 < 2^108 */
+	felem_small_mul(tmp, small2, beta);
+	/* tmp[i] < 7 * 2^64 < 2^67 */
+	smallfelem_square(tmp2, small1);
+	/* tmp2[i] < 7 * 2^64 */
+	longfelem_scalar(tmp2, 8);
+	/* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
+	longfelem_diff(tmp, tmp2);
+	/* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
+	felem_reduce_zero105(y_out, tmp);
+	/* y_out[i] < 2^106 */
+	}
+
+/* point_double_small is the same as point_double, except that it operates on
+ * smallfelems */
+static void
+point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out,
+		   const smallfelem x_in, const smallfelem y_in, const smallfelem z_in)
+	{
+	felem felem_x_out, felem_y_out, felem_z_out;
+	felem felem_x_in, felem_y_in, felem_z_in;
+
+	smallfelem_expand(felem_x_in, x_in);
+	smallfelem_expand(felem_y_in, y_in);
+	smallfelem_expand(felem_z_in, z_in);