libuecc/src/ec25519_gf.c

/*
  Copyright (c) 2012, Matthias Schiffer <mschiffer@universe-factory.net>
  Partly based on public domain code by Matthew Dempsky and D. J. Bernstein.
  All rights reserved.

  Redistribution and use in source and binary forms, with or without
  modification, are permitted provided that the following conditions are met:

    1. Redistributions of source code must retain the above copyright notice,
       this list of conditions and the following disclaimer.
    2. Redistributions in binary form must reproduce the above copyright notice,
       this list of conditions and the following disclaimer in the documentation
       and/or other materials provided with the distribution.

  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
  AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
  DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
  FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
  SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
  CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
  OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
  OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/

/** \file
  Simple finite field operations on the prime field \f$ F_q \f$ for
  \f$ q = 2^{252} + 27742317777372353535851937790883648493 \f$, which
  is the order of the base point used for ec25519
*/

#include <libuecc/ecc.h>


/** Checks if the highest bit of an unsigned integer is set */
#define IS_NEGATIVE(n) ((int)((((unsigned)n) >> (8*sizeof(n)-1))&1))

/** Performs an arithmetic right shift */
#define ASR(n,s) (((n) >> s)|(IS_NEGATIVE(n)*((unsigned)-1) << (8*sizeof(n)-s)))


/**
 * The order of the prime field
 *
 * The order is \f$ 2^{252} + 27742317777372353535851937790883648493 \f$.
 */
const ecc_int256_t ecc_25519_gf_order = {{
	0xed, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58,
	0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10
}};

/** An internal alias for \ref ecc_25519_gf_order */
static const unsigned char *q = ecc_25519_gf_order.p;

/**
 * Copies the content of r into out if b == 0, the contents of s if b == 1
 */
static void select(unsigned char out[32], const unsigned char r[32], const unsigned char s[32], unsigned int b) {
	unsigned int j;
	unsigned int t;
	unsigned int bminus1;

	bminus1 = b - 1;
	for (j = 0;j < 32;++j) {
		t = bminus1 & (r[j] ^ s[j]);
		out[j] = s[j] ^ t;
	}
}

/** Checks if an integer is equal to zero (after reduction) */
int ecc_25519_gf_is_zero(const ecc_int256_t *in) {
	int i;
	ecc_int256_t r;
	unsigned int bits = 0;

	ecc_25519_gf_reduce(&r, in);

	for (i = 0; i < 32; i++)
		bits |= r.p[i];

	return (((bits-1)>>8) & 1);
}

/**
 * Adds two integers as Galois field elements
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_add(ecc_int256_t *out, const ecc_int256_t *in1, const ecc_int256_t *in2) {
	unsigned int j;
	unsigned int u;
	int nq = 1 - (in1->p[31]>>4) - (in2->p[31]>>4);

	u = 0;
	for (j = 0; j < 32; ++j) {
		u += in1->p[j] + in2->p[j] + nq*q[j];

		out->p[j] = u;
		u = ASR(u, 8);
	}
}

/**
 * Subtracts two integers as Galois field elements
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_sub(ecc_int256_t *out, const ecc_int256_t *in1, const ecc_int256_t *in2) {
	unsigned int j;
	unsigned int u;
	int nq = 8 - (in1->p[31]>>4) + (in2->p[31]>>4);

	u = 0;
	for (j = 0; j < 32; ++j) {
		u += in1->p[j] - in2->p[j] + nq*q[j];

		out->p[j] = u;
		u = ASR(u, 8);
	}
}

/** Reduces an integer to a unique representation in the range \f$ [0,q-1] \f$ */
static void reduce(unsigned char a[32]) {
	unsigned int j;
	unsigned int nq = a[31] >> 4;
	unsigned int u1, u2;
	unsigned char out1[32], out2[32];

	u1 = u2 = 0;
	for (j = 0; j < 31; ++j) {
		u1 += a[j] - nq*q[j];
		u2 += a[j] - (nq-1)*q[j];

		out1[j] = u1; out2[j] = u2;
		u1 = ASR(u1, 8);
		u2 = ASR(u2, 8);
	}
	u1 += a[31] - nq*q[31];
	u2 += a[31] - (nq-1)*q[31];
	out1[31] = u1; out2[31] = u2;

	select(a, out1, out2, IS_NEGATIVE(u1));
}

/**
 * Reduces an integer to a unique representation in the range \f$ [0,q-1] \f$
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_reduce(ecc_int256_t *out, const ecc_int256_t *in) {
	int i;

	for (i = 0; i < 32; i++)
		out->p[i] = in->p[i];

	reduce(out->p);
}

/** Montgomery modular multiplication algorithm */
static void montgomery(unsigned char out[32], const unsigned char a[32], const unsigned char b[32]) {
	unsigned int i, j;
	unsigned int nq;
	unsigned int u;

	for (i = 0; i < 32; i++)
		out[i] = 0;

	for (i = 0; i < 32; i++) {
		u = out[0] + a[i]*b[0];
		nq = (u*27) & 255;
		u += nq*q[0];

		for (j = 1; j < 32; ++j) {
			u += (out[j] + a[i]*b[j] + nq*q[j]) << 8;
			u >>= 8;
			out[j-1] = u;
		}

		out[31] = u >> 8;
	}
}

/**
 * Multiplies two integers as Galois field elements
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_mult(ecc_int256_t *out, const ecc_int256_t *in1, const ecc_int256_t *in2) {
	/* 2^512 mod q */
	static const unsigned char C[32] = {
		0x01, 0x0f, 0x9c, 0x44, 0xe3, 0x11, 0x06, 0xa4,
		0x47, 0x93, 0x85, 0x68, 0xa7, 0x1b, 0x0e, 0xd0,
		0x65, 0xbe, 0xf5, 0x17, 0xd2, 0x73, 0xec, 0xce,
		0x3d, 0x9a, 0x30, 0x7c, 0x1b, 0x41, 0x99, 0x03
	};

	unsigned char B[32];
	unsigned char R[32];
	unsigned int i;

	for (i = 0; i < 32; i++)
		B[i] = in2->p[i];

	reduce(B);

	montgomery(R, in1->p, B);
	montgomery(out->p, R, C);
}

/**
 * Computes the reciprocal of a Galois field element
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_recip(ecc_int256_t *out, const ecc_int256_t *in) {
	static const unsigned char C[32] = {
		0x01
	};

	unsigned char A[32], B[32];
	unsigned char R1[32], R2[32];
	int use_r2 = 0;
	unsigned int i, j;

	for (i = 0; i < 32; i++) {
		R1[i] = (i == 0);
		A[i] = in->p[i];
	}

	reduce(A);

	for (i = 0; i < 32; i++) {
		unsigned char c;

		if (i == 0)
			c = 0xeb; /* q[0] - 2 */
		else
			c = q[i];

		for (j = 0; j < 8; j+=2) {
			if (c & (1 << j)) {
				if (use_r2)
					montgomery(R1, R2, A);
				else
					montgomery(R2, R1, A);

				use_r2 = !use_r2;
			}

			montgomery(B, A, A);

			if (c & (2 << j)) {
				if (use_r2)
					montgomery(R1, R2, B);
				else
					montgomery(R2, R1, B);

				use_r2 = !use_r2;
			}

			montgomery(A, B, B);
		}
	}

	montgomery(out->p, R2, C);
}

/**
 * Ensures some properties of a Galois field element to make it fit for use as a secret key
 *
 * The same pointers may be used for input and output.
 */
void ecc_25519_gf_sanitize_secret(ecc_int256_t *out, const ecc_int256_t *in) {
	int i;

	for (i = 0; i < 32; i++)
		out->p[i] = in->p[i];

	out->p[0] &= 0xf8;
	out->p[31] &= 0x7f;
	out->p[31] |= 0x40;
}