ec_c25519_m62.c

/*
 * Copyright (c) 2018 Thomas Pornin <[email protected]>
 *
 * 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 GEN[] = {
	0x09, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00
};

static const unsigned char ORDER[] = {
	0x7F, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
	0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
	0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
	0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF
};

static const unsigned char *
api_generator(int curve, size_t *len)
{
	(void)curve;
	*len = 32;
	return GEN;
}

static const unsigned char *
api_order(int curve, size_t *len)
{
	(void)curve;
	*len = 32;
	return ORDER;
}

static size_t
api_xoff(int curve, size_t *len)
{
	(void)curve;
	*len = 32;
	return 0;
}

/*
 * A field element is encoded as five 64-bit integers, in basis 2^51.
 * Limbs may be occasionally larger than 2^51, to save on carry
 * propagation costs.
 */

#define MASK51   (((uint64_t)1 << 51) - (uint64_t)1)

/*
 * Swap two field elements, conditionally on a flag.
 */
static inline void
f255_cswap(uint64_t *a, uint64_t *b, uint32_t ctl)
{
	uint64_t m, w;

	m = -(uint64_t)ctl;
	w = m & (a[0] ^ b[0]); a[0] ^= w; b[0] ^= w;
	w = m & (a[1] ^ b[1]); a[1] ^= w; b[1] ^= w;
	w = m & (a[2] ^ b[2]); a[2] ^= w; b[2] ^= w;
	w = m & (a[3] ^ b[3]); a[3] ^= w; b[3] ^= w;
	w = m & (a[4] ^ b[4]); a[4] ^= w; b[4] ^= w;
}

/*
 * Addition with no carry propagation. Limbs double in size.
 */
static inline void
f255_add(uint64_t *d, const uint64_t *a, const uint64_t *b)
{
	d[0] = a[0] + b[0];
	d[1] = a[1] + b[1];
	d[2] = a[2] + b[2];
	d[3] = a[3] + b[3];
	d[4] = a[4] + b[4];
}

/*
 * Subtraction.
 * On input, limbs must fit on 60 bits each. On output, result is
 * partially reduced, with max value 2^255+19456; moreover, all
 * limbs will fit on 51 bits, except the low limb, which may have
 * value up to 2^51+19455.
 */
static inline void
f255_sub(uint64_t *d, const uint64_t *a, const uint64_t *b)
{
	uint64_t cc, w;

	/*
	 * We compute d = (2^255-19)*1024 + a - b. Since the limbs
	 * fit on 60 bits, the maximum value of operands are slightly
	 * more than 2^264, but much less than 2^265-19456. This
	 * ensures that the result is positive.
	 */

	/*
	 * Initial carry is 19456, since we add 2^265-19456. Each
	 * individual subtraction may yield a carry up to 513.
	 */
	w = a[0] - b[0] - 19456;
	d[0] = w & MASK51;
	cc = -(w >> 51) & 0x3FF;
	w = a[1] - b[1] - cc;
	d[1] = w & MASK51;
	cc = -(w >> 51) & 0x3FF;
	w = a[2] - b[2] - cc;
	d[2] = w & MASK51;
	cc = -(w >> 51) & 0x3FF;
	w = a[3] - b[3] - cc;
	d[3] = w & MASK51;
	cc = -(w >> 51) & 0x3FF;
	d[4] = ((uint64_t)1 << 61) + a[4] - b[4] - cc;

	/*
	 * Partial reduction. The intermediate result may be up to
	 * slightly above 2^265, but less than 2^265+2^255. When we
	 * truncate to 255 bits, the upper bits will be at most 1024.
	 */
	d[0] += 19 * (d[4] >> 51);
	d[4] &= MASK51;
}

/*
 * UMUL51(hi, lo, x, y) computes:
 *
 *   hi = floor((x * y) / (2^51))
 *   lo = x * y mod 2^51
 *
 * Note that lo < 2^51, but "hi" may be larger, if the input operands are
 * larger.
 */
#if BR_INT128

#define UMUL51(hi, lo, x, y)   do { \
		unsigned __int128 umul_tmp; \
		umul_tmp = (unsigned __int128)(x) * (unsigned __int128)(y); \
		(hi) = (uint64_t)(umul_tmp >> 51); \
		(lo) = (uint64_t)umul_tmp & MASK51; \
	} while (0)

#elif BR_UMUL128

#define UMUL51(hi, lo, x, y)   do { \
		uint64_t umul_hi, umul_lo; \
		umul_lo = _umul128((x), (y), &umul_hi); \
		(hi) = (umul_hi << 13) | (umul_lo >> 51); \
		(lo) = umul_lo & MASK51; \
	} while (0)

#endif

/*
 * Multiplication.
 * On input, limbs must fit on 54 bits each.
 * On output, limb 0 is at most 2^51 + 155647, and other limbs fit
 * on 51 bits each.
 */
static inline void
f255_mul(uint64_t *d, uint64_t *a, uint64_t *b)
{
	uint64_t t[10], hi, lo, w, cc;

	/*
	 * Perform cross products, accumulating values without carry
	 * propagation.
	 *
	 * Since input limbs fit on 54 bits each, each individual
	 * UMUL51 will produce a "hi" of less than 2^57. The maximum
	 * sum will be at most 5*(2^57-1) + 4*(2^51-1) (for t[5]),
	 * i.e. less than 324*2^51.
	 */

	UMUL51(t[1], t[0], a[0], b[0]);

	UMUL51(t[2], lo, a[1], b[0]); t[1] += lo;
	UMUL51(hi, lo, a[0], b[1]); t[1] += lo; t[2] += hi;

	UMUL51(t[3], lo, a[2], b[0]); t[2] += lo;
	UMUL51(hi, lo, a[1], b[1]); t[2] += lo; t[3] += hi;
	UMUL51(hi, lo, a[0], b[2]); t[2] += lo; t[3] += hi;

	UMUL51(t[4], lo, a[3], b[0]); t[3] += lo;
	UMUL51(hi, lo, a[2], b[1]); t[3] += lo; t[4] += hi;
	UMUL51(hi, lo, a[1], b[2]); t[3] += lo; t[4] += hi;
	UMUL51(hi, lo, a[0], b[3]); t[3] += lo; t[4] += hi;

	UMUL51(t[5], lo, a[4], b[0]); t[4] += lo;
	UMUL51(hi, lo, a[3], b[1]); t[4] += lo; t[5] += hi;
	UMUL51(hi, lo, a[2], b[2]); t[4] += lo; t[5] += hi;
	UMUL51(hi, lo, a[1], b[3]); t[4] += lo; t[5] += hi;
	UMUL51(hi, lo, a[0], b[4]); t[4] += lo; t[5] += hi;

	UMUL51(t[6], lo, a[4], b[1]); t[5] += lo;
	UMUL51(hi, lo, a[3], b[2]); t[5] += lo; t[6] += hi;
	UMUL51(hi, lo, a[2], b[3]); t[5] += lo; t[6] += hi;
	UMUL51(hi, lo, a[1], b[4]); t[5] += lo; t[6] += hi;

	UMUL51(t[7], lo, a[4], b[2]); t[6] += lo;
	UMUL51(hi, lo, a[3], b[3]); t[6] += lo; t[7] += hi;
	UMUL51(hi, lo, a[2], b[4]); t[6] += lo; t[7] += hi;

	UMUL51(t[8], lo, a[4], b[3]); t[7] += lo;
	UMUL51(hi, lo, a[3], b[4]); t[7] += lo; t[8] += hi;

	UMUL51(t[9], lo, a[4], b[4]); t[8] += lo;

	/*
	 * The upper words t[5]..t[9] are folded back into the lower
	 * words, using the rule that 2^255 = 19 in the field.
	 *
	 * Since each t[i] is less than 324*2^51, the additions below
	 * will yield less than 6480*2^51 in each limb; this fits in
	 * 64 bits (6480*2^51 < 8192*2^51 = 2^64), hence there is
	 * no overflow.
	 */
	t[0] += 19 * t[5];
	t[1] += 19 * t[6];
	t[2] += 19 * t[7];
	t[3] += 19 * t[8];
	t[4] += 19 * t[9];

	/*
	 * Propagate carries.
	 */
	w = t[0];
	d[0] = w & MASK51;
	cc = w >> 51;
	w = t[1] + cc;
	d[1] = w & MASK51;
	cc = w >> 51;
	w = t[2] + cc;
	d[2] = w & MASK51;
	cc = w >> 51;
	w = t[3] + cc;
	d[3] = w & MASK51;
	cc = w >> 51;
	w = t[4] + cc;
	d[4] = w & MASK51;
	cc = w >> 51;

	/*
	 * Since the limbs were 64-bit values, the top carry is at
	 * most 8192 (in practice, that cannot be reached). We simply
	 * performed a partial reduction.
	 */
	d[0] += 19 * cc;
}

/*
 * Multiplication by A24 = 121665.
 * Input must have limbs of 60 bits at most.
 */
static inline void
f255_mul_a24(uint64_t *d, const uint64_t *a)
{
	uint64_t t[5], cc, w;

	/*
	 * 121665 = 15 * 8111. We first multiply by 15, with carry
	 * propagation and partial reduction.
	 */
	w = a[0] * 15;
	t[0] = w & MASK51;
	cc = w >> 51;
	w = a[1] * 15 + cc;
	t[1] = w & MASK51;
	cc = w >> 51;
	w = a[2] * 15 + cc;
	t[2] = w & MASK51;
	cc = w >> 51;
	w = a[3] * 15 + cc;
	t[3] = w & MASK51;
	cc = w >> 51;
	w = a[4] * 15 + cc;
	t[4] = w & MASK51;
	t[0] += 19 * (w >> 51);

	/*
	 * Then multiplication by 8111. At that point, we known that
	 * t[0] is less than 2^51 + 19*8192, and other limbs are less
	 * than 2^51; thus, there will be no overflow.
	 */
	w = t[0] * 8111;
	d[0] = w & MASK51;
	cc = w >> 51;
	w = t[1] * 8111 + cc;
	d[1] = w & MASK51;
	cc = w >> 51;
	w = t[2] * 8111 + cc;
	d[2] = w & MASK51;
	cc = w >> 51;
	w = t[3] * 8111 + cc;
	d[3] = w & MASK51;
	cc = w >> 51;
	w = t[4] * 8111 + cc;
	d[4] = w & MASK51;
	d[0] += 19 * (w >> 51);
}

/*
 * Finalize reduction.
 * On input, limbs must fit on 51 bits, except possibly the low limb,
 * which may be slightly above 2^51.
 */
static inline void
f255_final_reduce(uint64_t *a)
{
	uint64_t t[5], cc, w;

	/*
	 * We add 19. If the result (in t[]) is below 2^255, then a[]
	 * is already less than 2^255-19, thus already reduced.
	 * Otherwise, we subtract 2^255 from t[], in which case we
	 * have t = a - (2^255-19), and that's our result.
	 */
	w = a[0] + 19;
	t[0] = w & MASK51;
	cc = w >> 51;
	w = a[1] + cc;
	t[1] = w & MASK51;
	cc = w >> 51;
	w = a[2] + cc;
	t[2] = w & MASK51;
	cc = w >> 51;
	w = a[3] + cc;
	t[3] = w & MASK51;
	cc = w >> 51;
	w = a[4] + cc;
	t[4] = w & MASK51;
	cc = w >> 51;

	/*
	 * The bit 255 of t is in cc. If that bit is 0, when a[] must
	 * be unchanged; otherwise, it must be replaced with t[].
	 */
	cc = -cc;
	a[0] ^= cc & (a[0] ^ t[0]);
	a[1] ^= cc & (a[1] ^ t[1]);
	a[2] ^= cc & (a[2] ^ t[2]);
	a[3] ^= cc & (a[3] ^ t[3]);
	a[4] ^= cc & (a[4] ^ t[4]);
}

static uint32_t
api_mul(unsigned char *G, size_t Glen,
	const unsigned char *kb, size_t kblen, int curve)
{
	unsigned char k[32];
	uint64_t x1[5], x2[5], z2[5], x3[5], z3[5];
	uint32_t swap;
	int i;

	(void)curve;

	/*
	 * Points are encoded over exactly 32 bytes. Multipliers must fit
	 * in 32 bytes as well.
	 */
	if (Glen != 32 || kblen > 32) {
		return 0;
	}

	/*
	 * RFC 7748 mandates that the high bit of the last point byte must
	 * be ignored/cleared; the "& MASK51" in the initialization for
	 * x1[4] clears that bit.
	 */
	x1[0] = br_dec64le(&G[0]) & MASK51;
	x1[1] = (br_dec64le(&G[6]) >> 3) & MASK51;
	x1[2] = (br_dec64le(&G[12]) >> 6) & MASK51;
	x1[3] = (br_dec64le(&G[19]) >> 1) & MASK51;
	x1[4] = (br_dec64le(&G[24]) >> 12) & MASK51;

	/*
	 * We can use memset() to clear values, because exact-width types
	 * like uint64_t are guaranteed to have no padding bits or
	 * trap representations.
	 */
	memset(x2, 0, sizeof x2);
	x2[0] = 1;
	memset(z2, 0, sizeof z2);
	memcpy(x3, x1, sizeof x1);
	memcpy(z3, x2, sizeof x2);

	/*
	 * The multiplier is provided in big-endian notation, and
	 * possibly shorter than 32 bytes.
	 */
	memset(k, 0, (sizeof k) - kblen);
	memcpy(k + (sizeof k) - kblen, kb, kblen);
	k[31] &= 0xF8;
	k[0] &= 0x7F;
	k[0] |= 0x40;

	swap = 0;

	for (i = 254; i >= 0; i --) {
		uint64_t a[5], aa[5], b[5], bb[5], e[5];
		uint64_t c[5], d[5], da[5], cb[5];
		uint32_t kt;

		kt = (k[31 - (i >> 3)] >> (i & 7)) & 1;
		swap ^= kt;
		f255_cswap(x2, x3, swap);
		f255_cswap(z2, z3, swap);
		swap = kt;

		/*
		 * At that point, limbs of x_2 and z_2 are assumed to fit
		 * on at most 52 bits each.
		 *
		 * Each f255_add() adds one bit to the maximum range of
		 * the values, but f255_sub() and f255_mul() bring back
		 * the limbs into 52 bits. All f255_add() outputs are
		 * used only as inputs for f255_mul(), which ensures
		 * that limbs remain in the proper range.
		 */

		/* A = x_2 + z_2   -- limbs fit on 53 bits each */
		f255_add(a, x2, z2);

		/* AA = A^2 */
		f255_mul(aa, a, a);

		/* B = x_2 - z_2 */
		f255_sub(b, x2, z2);

		/* BB = B^2 */
		f255_mul(bb, b, b);

		/* E = AA - BB */
		f255_sub(e, aa, bb);

		/* C = x_3 + z_3   -- limbs fit on 53 bits each */
		f255_add(c, x3, z3);

		/* D = x_3 - z_3 */
		f255_sub(d, x3, z3);

		/* DA = D * A */
		f255_mul(da, d, a);

		/* CB = C * B */
		f255_mul(cb, c, b);

		/* x_3 = (DA + CB)^2 */
		f255_add(x3, da, cb);
		f255_mul(x3, x3, x3);

		/* z_3 = x_1 * (DA - CB)^2 */
		f255_sub(z3, da, cb);
		f255_mul(z3, z3, z3);
		f255_mul(z3, x1, z3);

		/* x_2 = AA * BB */
		f255_mul(x2, aa, bb);

		/* z_2 = E * (AA + a24 * E) */
		f255_mul_a24(z2, e);
		f255_add(z2, aa, z2);
		f255_mul(z2, e, z2);
	}

	f255_cswap(x2, x3, swap);
	f255_cswap(z2, z3, swap);

	/*
	 * Compute 1/z2 = z2^(p-2). Since p = 2^255-19, we can mutualize
	 * most non-squarings. We use x1 and x3, now useless, as temporaries.
	 */
	memcpy(x1, z2, sizeof z2);
	for (i = 0; i < 15; i ++) {
		f255_mul(x1, x1, x1);
		f255_mul(x1, x1, z2);
	}
	memcpy(x3, x1, sizeof x1);
	for (i = 0; i < 14; i ++) {
		int j;

		for (j = 0; j < 16; j ++) {
			f255_mul(x3, x3, x3);
		}
		f255_mul(x3, x3, x1);
	}
	for (i = 14; i >= 0; i --) {
		f255_mul(x3, x3, x3);
		if ((0xFFEB >> i) & 1) {
			f255_mul(x3, z2, x3);
		}
	}

	/*
	 * Compute x2/z2. We have 1/z2 in x3.
	 */
	f255_mul(x2, x2, x3);
	f255_final_reduce(x2);

	/*
	 * Encode the final x2 value in little-endian. We first assemble
	 * the limbs into 64-bit values.
	 */
	x2[0] |= x2[1] << 51;
	x2[1] = (x2[1] >> 13) | (x2[2] << 38);
	x2[2] = (x2[2] >> 26) | (x2[3] << 25);
	x2[3] = (x2[3] >> 39) | (x2[4] << 12);
	br_enc64le(G, x2[0]);
	br_enc64le(G + 8, x2[1]);
	br_enc64le(G + 16, x2[2]);
	br_enc64le(G + 24, x2[3]);
	return 1;
}

static size_t
api_mulgen(unsigned char *R,
	const unsigned char *x, size_t xlen, int curve)
{
	const unsigned char *G;
	size_t Glen;

	G = api_generator(curve, &Glen);
	memcpy(R, G, Glen);
	api_mul(R, Glen, x, xlen, curve);
	return Glen;
}

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 don't implement this method, since it is used for ECDSA
	 * only, and there is no ECDSA over Curve25519 (which instead
	 * uses EdDSA).
	 */
	(void)A;
	(void)B;
	(void)len;
	(void)x;
	(void)xlen;
	(void)y;
	(void)ylen;
	(void)curve;
	return 0;
}

/* see bearssl_ec.h */
const br_ec_impl br_ec_c25519_m62 = {
	(uint32_t)0x20000000,
	&api_generator,
	&api_order,
	&api_xoff,
	&api_mul,
	&api_mulgen,
	&api_muladd
};

/* see bearssl_ec.h */
const br_ec_impl *
br_ec_c25519_m62_get(void)
{
	return &br_ec_c25519_m62;
}

#else

/* see bearssl_ec.h */
const br_ec_impl *
br_ec_c25519_m62_get(void)
{
	return 0;
}

#endif