usign/ed25519.c

/* Edwards curve operations
 * Daniel Beer <dlbeer@gmail.com>, 9 Jan 2014
 *
 * This file is in the public domain.
 */

#include "ed25519.h"

/* Base point is (numbers wrapped):
 *
 *     x = 151122213495354007725011514095885315114
 *         54012693041857206046113283949847762202
 *     y = 463168356949264781694283940034751631413
 *         07993866256225615783033603165251855960
 *
 * y is derived by transforming the original Montgomery base (u=9). x
 * is the corresponding positive coordinate for the new curve equation.
 * t is x*y.
 */
const struct ed25519_pt ed25519_base = {
	.x = {
		0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9,
		0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69,
		0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0,
		0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21
	},
	.y = {
		0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
		0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
		0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
		0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66
	},
	.t = {
		0xa3, 0xdd, 0xb7, 0xa5, 0xb3, 0x8a, 0xde, 0x6d,
		0xf5, 0x52, 0x51, 0x77, 0x80, 0x9f, 0xf0, 0x20,
		0x7d, 0xe3, 0xab, 0x64, 0x8e, 0x4e, 0xea, 0x66,
		0x65, 0x76, 0x8b, 0xd7, 0x0f, 0x5f, 0x87, 0x67
	},
	.z = {1, 0}
};

static const struct ed25519_pt ed25519_neutral = {
	.x = {0},
	.y = {1, 0},
	.t = {0},
	.z = {1, 0}
};

/* Conversion to and from projective coordinates */
void ed25519_project(struct ed25519_pt *p,
		     const uint8_t *x, const uint8_t *y)
{
	f25519_copy(p->x, x);
	f25519_copy(p->y, y);
	f25519_load(p->z, 1);
	f25519_mul__distinct(p->t, x, y);
}

void ed25519_unproject(uint8_t *x, uint8_t *y,
		       const struct ed25519_pt *p)
{
	uint8_t z1[F25519_SIZE];

	f25519_inv__distinct(z1, p->z);
	f25519_mul__distinct(x, p->x, z1);
	f25519_mul__distinct(y, p->y, z1);

	f25519_normalize(x);
	f25519_normalize(y);
}

/* Compress/uncompress points. We compress points by storing the x
 * coordinate and the parity of the y coordinate.
 *
 * Rearranging the curve equation, we obtain explicit formulae for the
 * coordinates:
 *
 *     x = sqrt((y^2-1) / (1+dy^2))
 *     y = sqrt((x^2+1) / (1-dx^2))
 *
 * Where d = (-121665/121666), or:
 *
 *     d = 370957059346694393431380835087545651895
 *         42113879843219016388785533085940283555
 */

static const uint8_t ed25519_d[F25519_SIZE] = {
	0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75,
	0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00,
	0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c,
	0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52
};

void ed25519_pack(uint8_t *c, const uint8_t *x, const uint8_t *y)
{
	uint8_t tmp[F25519_SIZE];
	uint8_t parity;

	f25519_copy(tmp, x);
	f25519_normalize(tmp);
	parity = (tmp[0] & 1) << 7;

	f25519_copy(c, y);
	f25519_normalize(c);
	c[31] |= parity;
}

uint8_t ed25519_try_unpack(uint8_t *x, uint8_t *y, const uint8_t *comp)
{
	const int parity = comp[31] >> 7;
	uint8_t a[F25519_SIZE];
	uint8_t b[F25519_SIZE];
	uint8_t c[F25519_SIZE];

	/* Unpack y */
	f25519_copy(y, comp);
	y[31] &= 127;

	/* Compute c = y^2 */
	f25519_mul__distinct(c, y, y);

	/* Compute b = (1+dy^2)^-1 */
	f25519_mul__distinct(b, c, ed25519_d);
	f25519_add(a, b, f25519_one);
	f25519_inv__distinct(b, a);

	/* Compute a = y^2-1 */
	f25519_sub(a, c, f25519_one);

	/* Compute c = a*b = (y^2-1)/(1-dy^2) */
	f25519_mul__distinct(c, a, b);

	/* Compute a, b = +/-sqrt(c), if c is square */
	f25519_sqrt(a, c);
	f25519_neg(b, a);

	/* Select one of them, based on the compressed parity bit */
	f25519_select(x, a, b, (a[0] ^ parity) & 1);

	/* Verify that x^2 = c */
	f25519_mul__distinct(a, x, x);
	f25519_normalize(a);
	f25519_normalize(c);

	return f25519_eq(a, c);
}

/* k = 2d */
static const uint8_t ed25519_k[F25519_SIZE] = {
	0x59, 0xf1, 0xb2, 0x26, 0x94, 0x9b, 0xd6, 0xeb,
	0x56, 0xb1, 0x83, 0x82, 0x9a, 0x14, 0xe0, 0x00,
	0x30, 0xd1, 0xf3, 0xee, 0xf2, 0x80, 0x8e, 0x19,
	0xe7, 0xfc, 0xdf, 0x56, 0xdc, 0xd9, 0x06, 0x24
};

void ed25519_add(struct ed25519_pt *r,
		 const struct ed25519_pt *p1, const struct ed25519_pt *p2)
{
	/* Explicit formulas database: add-2008-hwcd-3
	 *
	 * source 2008 Hisil--Wong--Carter--Dawson,
	 *     http://eprint.iacr.org/2008/522, Section 3.1
	 * appliesto extended-1
	 * parameter k
	 * assume k = 2 d
	 * compute A = (Y1-X1)(Y2-X2)
	 * compute B = (Y1+X1)(Y2+X2)
	 * compute C = T1 k T2
	 * compute D = Z1 2 Z2
	 * compute E = B - A
	 * compute F = D - C
	 * compute G = D + C
	 * compute H = B + A
	 * compute X3 = E F
	 * compute Y3 = G H
	 * compute T3 = E H
	 * compute Z3 = F G
	 */
	uint8_t a[F25519_SIZE];
	uint8_t b[F25519_SIZE];
	uint8_t c[F25519_SIZE];
	uint8_t d[F25519_SIZE];
	uint8_t e[F25519_SIZE];
	uint8_t f[F25519_SIZE];
	uint8_t g[F25519_SIZE];
	uint8_t h[F25519_SIZE];

	/* A = (Y1-X1)(Y2-X2) */
	f25519_sub(c, p1->y, p1->x);
	f25519_sub(d, p2->y, p2->x);
	f25519_mul__distinct(a, c, d);

	/* B = (Y1+X1)(Y2+X2) */
	f25519_add(c, p1->y, p1->x);
	f25519_add(d, p2->y, p2->x);
	f25519_mul__distinct(b, c, d);

	/* C = T1 k T2 */
	f25519_mul__distinct(d, p1->t, p2->t);
	f25519_mul__distinct(c, d, ed25519_k);

	/* D = Z1 2 Z2 */
	f25519_mul__distinct(d, p1->z, p2->z);
	f25519_add(d, d, d);

	/* E = B - A */
	f25519_sub(e, b, a);

	/* F = D - C */
	f25519_sub(f, d, c);

	/* G = D + C */
	f25519_add(g, d, c);

	/* H = B + A */
	f25519_add(h, b, a);

	/* X3 = E F */
	f25519_mul__distinct(r->x, e, f);

	/* Y3 = G H */
	f25519_mul__distinct(r->y, g, h);

	/* T3 = E H */
	f25519_mul__distinct(r->t, e, h);

	/* Z3 = F G */
	f25519_mul__distinct(r->z, f, g);
}

static void ed25519_double(struct ed25519_pt *r, const struct ed25519_pt *p)
{
	/* Explicit formulas database: dbl-2008-hwcd
	 *
	 * source 2008 Hisil--Wong--Carter--Dawson,
	 *     http://eprint.iacr.org/2008/522, Section 3.3
	 * compute A = X1^2
	 * compute B = Y1^2
	 * compute C = 2 Z1^2
	 * compute D = a A
	 * compute E = (X1+Y1)^2-A-B
	 * compute G = D + B
	 * compute F = G - C
	 * compute H = D - B
	 * compute X3 = E F
	 * compute Y3 = G H
	 * compute T3 = E H
	 * compute Z3 = F G
	 */
	uint8_t a[F25519_SIZE];
	uint8_t b[F25519_SIZE];
	uint8_t c[F25519_SIZE];
	uint8_t e[F25519_SIZE];
	uint8_t f[F25519_SIZE];
	uint8_t g[F25519_SIZE];
	uint8_t h[F25519_SIZE];

	/* A = X1^2 */
	f25519_mul__distinct(a, p->x, p->x);

	/* B = Y1^2 */
	f25519_mul__distinct(b, p->y, p->y);

	/* C = 2 Z1^2 */
	f25519_mul__distinct(c, p->z, p->z);
	f25519_add(c, c, c);

	/* D = a A (alter sign) */
	/* E = (X1+Y1)^2-A-B */
	f25519_add(f, p->x, p->y);
	f25519_mul__distinct(e, f, f);
	f25519_sub(e, e, a);
	f25519_sub(e, e, b);

	/* G = D + B */
	f25519_sub(g, b, a);

	/* F = G - C */
	f25519_sub(f, g, c);

	/* H = D - B */
	f25519_neg(h, b);
	f25519_sub(h, h, a);

	/* X3 = E F */
	f25519_mul__distinct(r->x, e, f);

	/* Y3 = G H */
	f25519_mul__distinct(r->y, g, h);

	/* T3 = E H */
	f25519_mul__distinct(r->t, e, h);

	/* Z3 = F G */
	f25519_mul__distinct(r->z, f, g);
}

void ed25519_smult(struct ed25519_pt *r_out, const struct ed25519_pt *p,
		   const uint8_t *e)
{
	struct ed25519_pt r;
	int i;

	ed25519_copy(&r, &ed25519_neutral);

	for (i = 255; i >= 0; i--) {
		const uint8_t bit = (e[i >> 3] >> (i & 7)) & 1;
		struct ed25519_pt s;

		ed25519_double(&r, &r);
		ed25519_add(&s, &r, p);

		f25519_select(r.x, r.x, s.x, bit);
		f25519_select(r.y, r.y, s.y, bit);
		f25519_select(r.z, r.z, s.z, bit);
		f25519_select(r.t, r.t, s.t, bit);
	}

	ed25519_copy(r_out, &r);
}