OpenHarmony-v4.0.1-Release/s

/* ecc.c - TinyCrypt implementation of common ECC functions */

/*
 * Copyright (c) 2014, Kenneth MacKay
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 * * Redistributions of source code must retain the above copyright notice,
 * this list of conditions and the following disclaimer.
 * * 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.
 *
 *  Copyright (C) 2017 by Intel Corporation, All Rights Reserved.
 *
 *  Redistribution and use in source and binary forms, with or without
 *  modification, are permitted provided that the following conditions are met:
 *
 *    - Redistributions of source code must retain the above copyright notice,
 *     this list of conditions and the following disclaimer.
 *
 *    - 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.
 *
 *    - Neither the name of Intel Corporation nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 *  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 OWNER 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.
 */

#include <tinycrypt/ecc.h>
#include <tinycrypt/ecc_platform_specific.h>
#include <string.h>

/* IMPORTANT: Make sure a cryptographically-secure PRNG is set and the platform
 * has access to enough entropy in order to feed the PRNG regularly. */
#if default_RNG_defined
static uECC_RNG_Function g_rng_function = &default_CSPRNG;
#else
static uECC_RNG_Function g_rng_function = 0;
#endif

void uECC_set_rng(uECC_RNG_Function rng_function)
{
    g_rng_function = rng_function;
}

uECC_RNG_Function uECC_get_rng(void)
{
    return g_rng_function;
}

int uECC_curve_private_key_size(uECC_Curve curve)
{
    return BITS_TO_BYTES(curve->num_n_bits);
}

int uECC_curve_public_key_size(uECC_Curve curve)
{
    return 2 * curve->num_bytes; // 2:byte alignment
}

void uECC_vli_clear(uECC_word_t *vli, wordcount_t num_words)
{
    wordcount_t i;

    for (i = 0; i < num_words; ++i) {
        vli[i] = 0;
    }
}

uECC_word_t uECC_vli_isZero(const uECC_word_t *vli, wordcount_t num_words)
{
    uECC_word_t bits = 0;
    wordcount_t i;

    for (i = 0; i < num_words; ++i) {
        bits |= vli[i];
    }
    return (bits == 0);
}

uECC_word_t uECC_vli_testBit(const uECC_word_t *vli, bitcount_t bit)
{
    return (vli[bit >> uECC_WORD_BITS_SHIFT] &
            ((uECC_word_t)1 << (bit & uECC_WORD_BITS_MASK)));
}

/* Counts the number of words in vli. */
static wordcount_t vli_numDigits(const uECC_word_t *vli,
                                 const wordcount_t max_words)
{
    wordcount_t i;

    /* Search from the end until we find a non-zero digit. We do it in reverse
     * because we expect that most digits will be nonzero. */
    for (i = max_words - 1; i >= 0 && vli[i] == 0; --i) {
    }
    return (i + 1);
}

bitcount_t uECC_vli_numBits(const uECC_word_t *vli,
                            const wordcount_t max_words)
{
    uECC_word_t i;
    uECC_word_t digit;
    wordcount_t num_digits = vli_numDigits(vli, max_words);
    if (num_digits == 0) {
        return 0;
    }

    digit = vli[num_digits - 1];

    for (i = 0; digit; ++i) {
        digit >>= 1;
    }

    return (((bitcount_t)(num_digits - 1) << uECC_WORD_BITS_SHIFT) + i);
}

void uECC_vli_set(uECC_word_t *dest, const uECC_word_t *src,
                  wordcount_t num_words)
{
    wordcount_t i;

    for (i = 0; i < num_words; ++i) {
        dest[i] = src[i];
    }
}

cmpresult_t uECC_vli_cmp_unsafe(const uECC_word_t *left,
                                const uECC_word_t *right,
                                wordcount_t num_words)
{
    wordcount_t i;

    for (i = num_words - 1; i >= 0; --i) {
        if (left[i] > right[i]) {
            return 1;
        } else if (left[i] < right[i]) {
            return -1;
        }
    }

    return 0;
}

uECC_word_t uECC_vli_equal(const uECC_word_t *left, const uECC_word_t *right,
                           wordcount_t num_words)
{
    uECC_word_t diff = 0;
    wordcount_t i;

    for (i = num_words - 1; i >= 0; --i) {
        diff |= (left[i] ^ right[i]);
    }

    return !(diff == 0);
}

uECC_word_t cond_set(uECC_word_t p_true, uECC_word_t p_false, unsigned int cond)
{
    return (p_true * (cond)) | (p_false * (!cond));
}

/* Computes result = left - right, returning borrow, in constant time.
 * Can modify in place. */
uECC_word_t uECC_vli_sub(uECC_word_t *result, const uECC_word_t *left,
                         const uECC_word_t *right, wordcount_t num_words)
{
    uECC_word_t borrow = 0;
    wordcount_t i;

    for (i = 0; i < num_words; ++i) {
        uECC_word_t diff = left[i] - right[i] - borrow;
        uECC_word_t val = (diff > left[i]);
        borrow = cond_set(val, borrow, (diff != left[i]));
        result[i] = diff;
    }

    return borrow;
}

/* Computes result = left + right, returning carry, in constant time.
 * Can modify in place. */
static uECC_word_t uECC_vli_add(uECC_word_t *result, const uECC_word_t *left,
                                const uECC_word_t *right, wordcount_t num_words)
{
    uECC_word_t carry = 0;
    wordcount_t i;

    for (i = 0; i < num_words; ++i) {
        uECC_word_t sum = left[i] + right[i] + carry;
        uECC_word_t val = (sum < left[i]);
        carry = cond_set(val, carry, (sum != left[i]));
        result[i] = sum;
    }

    return carry;
}

cmpresult_t uECC_vli_cmp(const uECC_word_t *left, const uECC_word_t *right,
                         wordcount_t num_words)
{
    uECC_word_t tmp[NUM_ECC_WORDS];
    uECC_word_t neg = !!uECC_vli_sub(tmp, left, right, num_words);
    uECC_word_t equal = uECC_vli_isZero(tmp, num_words);
    return (!equal - 2 * neg); // 2:byte alignment
}

/* Computes vli = vli >> 1. */
static void uECC_vli_rshift1(uECC_word_t *vli, wordcount_t num_words)
{
    uECC_word_t *end = vli;
    uECC_word_t carry = 0;
    vli += num_words;

    while (vli-- > end) {
        uECC_word_t temp = *vli;
        *vli = (temp >> 1) | carry;
        carry = temp << (uECC_WORD_BITS - 1);
    }
}

static void muladd(uECC_word_t a, uECC_word_t b, uECC_word_t *r0,
                   uECC_word_t *r1, uECC_word_t *r2)
{
    uECC_dword_t p = (uECC_dword_t)a * b;
    uECC_dword_t r01 = ((uECC_dword_t)(*r1) << uECC_WORD_BITS) | *r0;
    r01 += p;
    *r2 += (r01 < p);
    *r1 = r01 >> uECC_WORD_BITS;
    *r0 = (uECC_word_t)r01;
}

/* Computes result = left * right. Result must be 2 * num_words long. */
static void uECC_vli_mult(uECC_word_t *result, const uECC_word_t *left,
                          const uECC_word_t *right, wordcount_t num_words)
{
    uECC_word_t r0 = 0;
    uECC_word_t r1 = 0;
    uECC_word_t r2 = 0;
    wordcount_t i, k;

    /* Compute each digit of result in sequence, maintaining the carries. */
    for (k = 0; k < num_words; ++k) {
        for (i = 0; i <= k; ++i) {
            muladd(left[i], right[k - i], &r0, &r1, &r2);
        }

        result[k] = r0;
        r0 = r1;
        r1 = r2;
        r2 = 0;
    }

    for (k = num_words; k < num_words * 2 - 1; ++k) {
        for (i = (k + 1) - num_words; i < num_words; ++i) {
            muladd(left[i], right[k - i], &r0, &r1, &r2);
        }

        result[k] = r0;
        r0 = r1;
        r1 = r2;
        r2 = 0;
    }

    result[num_words * 2 - 1] = r0; // 2:byte alignment
}

void uECC_vli_modAdd(uECC_word_t *result, const uECC_word_t *left,
                     const uECC_word_t *right, const uECC_word_t *mod,
                     wordcount_t num_words)
{
    uECC_word_t carry = uECC_vli_add(result, left, right, num_words);
    if (carry || uECC_vli_cmp_unsafe(mod, result, num_words) != 1) {
        /* result > mod (result = mod + remainder), so subtract mod to get
         * remainder. */
        uECC_vli_sub(result, result, mod, num_words);
    }
}

void uECC_vli_modSub(uECC_word_t *result, const uECC_word_t *left,
                     const uECC_word_t *right, const uECC_word_t *mod,
                     wordcount_t num_words)
{
    uECC_word_t l_borrow = uECC_vli_sub(result, left, right, num_words);
    if (l_borrow) {
        /* In this case, result == -diff == (max int) - diff. Since -x % d == d - x,
         * we can get the correct result from result + mod (with overflow). */
        uECC_vli_add(result, result, mod, num_words);
    }
}

/* Computes result = product % mod, where product is 2N words long. */
/* Currently only designed to work for curve_p or curve_n. */
void uECC_vli_mmod(uECC_word_t *result, uECC_word_t *product,
                   const uECC_word_t *mod, wordcount_t num_words)
{
    uECC_word_t mod_multiple[2 * NUM_ECC_WORDS]; // 2:byte alignment
    uECC_word_t tmp[2 * NUM_ECC_WORDS]; // 2:byte alignment
    uECC_word_t *v[2] = {tmp, product}; // 2:array element
    uECC_word_t index;
    /* Shift mod so its highest set bit is at the maximum position. */
    bitcount_t shift = (num_words * 2 * uECC_WORD_BITS) - // 2:byte alignment
                       uECC_vli_numBits(mod, num_words);
    wordcount_t word_shift = shift / uECC_WORD_BITS;
    wordcount_t bit_shift = shift % uECC_WORD_BITS;
    uECC_word_t carry = 0;
    uECC_vli_clear(mod_multiple, word_shift);

    if (bit_shift > 0) {
        for (index = 0; index < (uECC_word_t)num_words; ++index) {
            mod_multiple[word_shift + index] = (mod[index] << bit_shift) | carry;
            carry = mod[index] >> (uECC_WORD_BITS - bit_shift);
        }
    } else {
        uECC_vli_set(mod_multiple + word_shift, mod, num_words);
    }

    for (index = 1; shift >= 0; --shift) {
        uECC_word_t borrow = 0;
        wordcount_t i;

        for (i = 0; i < num_words * 2; ++i) { // 2:byte alignment
            uECC_word_t diff = v[index][i] - mod_multiple[i] - borrow;

            if (diff != v[index][i]) {
                borrow = (diff > v[index][i]);
            }

            v[1 - index][i] = diff;
        }

        /* Swap the index if there was no borrow */
        index = !(index ^ borrow);
        uECC_vli_rshift1(mod_multiple, num_words);
        mod_multiple[num_words - 1] |= mod_multiple[num_words] <<
                                       (uECC_WORD_BITS - 1);
        uECC_vli_rshift1(mod_multiple + num_words, num_words);
    }

    uECC_vli_set(result, v[index], num_words);
}

void uECC_vli_modMult(uECC_word_t *result, const uECC_word_t *left,
                      const uECC_word_t *right, const uECC_word_t *mod,
                      wordcount_t num_words)
{
    uECC_word_t product[2 * NUM_ECC_WORDS]; // 2:byte alignment
    uECC_vli_mult(product, left, right, num_words);
    uECC_vli_mmod(result, product, mod, num_words);
}

void uECC_vli_modMult_fast(uECC_word_t *result, const uECC_word_t *left,
                           const uECC_word_t *right, uECC_Curve curve)
{
    uECC_word_t product[2 * NUM_ECC_WORDS]; // 2:byte alignment
    uECC_vli_mult(product, left, right, curve->num_words);
    curve->mmod_fast(result, product);
}

static void uECC_vli_modSquare_fast(uECC_word_t *result,
                                    const uECC_word_t *left,
                                    uECC_Curve curve)
{
    uECC_vli_modMult_fast(result, left, left, curve);
}

#define EVEN(vli) (!((vli)[0] & 1))

static void vli_modInv_update(uECC_word_t *uv,
                              const uECC_word_t *mod,
                              wordcount_t num_words)
{
    uECC_word_t carry = 0;

    if (!EVEN(uv)) {
        carry = uECC_vli_add(uv, uv, mod, num_words);
    }

    uECC_vli_rshift1(uv, num_words);

    if (carry) {
        uv[num_words - 1] |= HIGH_BIT_SET;
    }
}

void uECC_vli_modInv(uECC_word_t *result, const uECC_word_t *input,
                     const uECC_word_t *mod, wordcount_t num_words)
{
    uECC_word_t a[NUM_ECC_WORDS], b[NUM_ECC_WORDS];
    uECC_word_t u[NUM_ECC_WORDS], v[NUM_ECC_WORDS];
    cmpresult_t cmpResult;

    if (uECC_vli_isZero(input, num_words)) {
        uECC_vli_clear(result, num_words);
        return;
    }

    uECC_vli_set(a, input, num_words);
    uECC_vli_set(b, mod, num_words);
    uECC_vli_clear(u, num_words);
    u[0] = 1;
    uECC_vli_clear(v, num_words);

    while ((cmpResult = uECC_vli_cmp_unsafe(a, b, num_words)) != 0) {
        if (EVEN(a)) {
            uECC_vli_rshift1(a, num_words);
            vli_modInv_update(u, mod, num_words);
        } else if (EVEN(b)) {
            uECC_vli_rshift1(b, num_words);
            vli_modInv_update(v, mod, num_words);
        } else if (cmpResult > 0) {
            uECC_vli_sub(a, a, b, num_words);
            uECC_vli_rshift1(a, num_words);

            if (uECC_vli_cmp_unsafe(u, v, num_words) < 0) {
                uECC_vli_add(u, u, mod, num_words);
            }

            uECC_vli_sub(u, u, v, num_words);
            vli_modInv_update(u, mod, num_words);
        } else {
            uECC_vli_sub(b, b, a, num_words);
            uECC_vli_rshift1(b, num_words);

            if (uECC_vli_cmp_unsafe(v, u, num_words) < 0) {
                uECC_vli_add(v, v, mod, num_words);
            }

            uECC_vli_sub(v, v, u, num_words);
            vli_modInv_update(v, mod, num_words);
        }
    }

    uECC_vli_set(result, u, num_words);
}

/* ------ Point operations ------ */

void double_jacobian_default(uECC_word_t *X1, uECC_word_t *Y1,
                             uECC_word_t *Z1, uECC_Curve curve)
{
    /* t1 = X, t2 = Y, t3 = Z */
    uECC_word_t t4[NUM_ECC_WORDS];
    uECC_word_t t5[NUM_ECC_WORDS];
    wordcount_t num_words = curve->num_words;

    if (uECC_vli_isZero(Z1, num_words)) {
        return;
    }

    uECC_vli_modSquare_fast(t4, Y1, curve);   /* t4 = y1^2 */
    uECC_vli_modMult_fast(t5, X1, t4, curve); /* t5 = x1*y1^2 = A */
    uECC_vli_modSquare_fast(t4, t4, curve);   /* t4 = y1^4 */
    uECC_vli_modMult_fast(Y1, Y1, Z1, curve); /* t2 = y1*z1 = z3 */
    uECC_vli_modSquare_fast(Z1, Z1, curve);   /* t3 = z1^2 */
    uECC_vli_modAdd(X1, X1, Z1, curve->p, num_words); /* t1 = x1 + z1^2 */
    uECC_vli_modAdd(Z1, Z1, Z1, curve->p, num_words); /* t3 = 2*z1^2 */
    uECC_vli_modSub(Z1, X1, Z1, curve->p, num_words); /* t3 = x1 - z1^2 */
    uECC_vli_modMult_fast(X1, X1, Z1, curve); /* t1 = x1^2 - z1^4 */
    uECC_vli_modAdd(Z1, X1, X1, curve->p, num_words); /* t3 = 2*(x1^2 - z1^4) */
    uECC_vli_modAdd(X1, X1, Z1, curve->p, num_words); /* t1 = 3*(x1^2 - z1^4) */

    if (uECC_vli_testBit(X1, 0)) {
        uECC_word_t l_carry = uECC_vli_add(X1, X1, curve->p, num_words);
        uECC_vli_rshift1(X1, num_words);
        X1[num_words - 1] |= l_carry << (uECC_WORD_BITS - 1);
    } else {
        uECC_vli_rshift1(X1, num_words);
    }

    /* t1 = 3/2*(x1^2 - z1^4) = B */
    uECC_vli_modSquare_fast(Z1, X1, curve); /* t3 = B^2 */
    uECC_vli_modSub(Z1, Z1, t5, curve->p, num_words); /* t3 = B^2 - A */
    uECC_vli_modSub(Z1, Z1, t5, curve->p, num_words); /* t3 = B^2 - 2A = x3 */
    uECC_vli_modSub(t5, t5, Z1, curve->p, num_words); /* t5 = A - x3 */
    uECC_vli_modMult_fast(X1, X1, t5, curve); /* t1 = B * (A - x3) */
    /* t4 = B * (A - x3) - y1^4 = y3: */
    uECC_vli_modSub(t4, X1, t4, curve->p, num_words);
    uECC_vli_set(X1, Z1, num_words);
    uECC_vli_set(Z1, Y1, num_words);
    uECC_vli_set(Y1, t4, num_words);
}

void x_side_default(uECC_word_t *result,
                    const uECC_word_t *x,
                    uECC_Curve curve)
{
    uECC_word_t _3[NUM_ECC_WORDS] = {3}; // 3:-a
    wordcount_t num_words = curve->num_words;
    uECC_vli_modSquare_fast(result, x, curve); /* r = x^2 */
    uECC_vli_modSub(result, result, _3, curve->p, num_words); /* r = x^2 - 3 */
    uECC_vli_modMult_fast(result, result, x, curve); /* r = x^3 - 3x */
    /* r = x^3 - 3x + b: */
    uECC_vli_modAdd(result, result, curve->b, curve->p, num_words);
}

uECC_Curve uECC_secp256r1(void)
{
    return &curve_secp256r1;
}

void vli_mmod_fast_secp256r1(unsigned int *result, unsigned int *product)
{
    unsigned int tmp[NUM_ECC_WORDS];
    int carry;
    /* t */
    uECC_vli_set(result, product, NUM_ECC_WORDS);
    /* s1 */
    tmp[0] = tmp[1] = tmp[2] = 0; // 2:array element
    tmp[3] = product[11]; // 3:array element, 11:array element
    tmp[4] = product[12]; // 4:array element, 12:array element
    tmp[5] = product[13]; // 5:array element, 13:array element
    tmp[6] = product[14]; // 6:array element, 14:array element
    tmp[7] = product[15]; // 7:array element, 15:array element
    carry = uECC_vli_add(tmp, tmp, tmp, NUM_ECC_WORDS);
    carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);
    /* s2 */
    tmp[3] = product[12]; // 3:array element, 12:array element
    tmp[4] = product[13]; // 4:array element, 13:array element
    tmp[5] = product[14]; // 5:array element, 14:array element
    tmp[6] = product[15]; // 6:array element, 15:array element
    tmp[7] = 0; // 7:array element
    carry += uECC_vli_add(tmp, tmp, tmp, NUM_ECC_WORDS);
    carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);
    /* s3 */
    tmp[0] = product[8]; // 8:array element
    tmp[1] = product[9]; // 9:array element
    tmp[2] = product[10]; // 2:array element, 10:array element
    tmp[3] = tmp[4] = tmp[5] = 0; // 3:array element, 4:array element, 5:array element
    tmp[6] = product[14]; // 6:array element, 14:array element
    tmp[7] = product[15]; // 7:array element, 15:array element
    carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);
    /* s4 */
    tmp[0] = product[9]; // 9:array element
    tmp[1] = product[10]; // 10:array element
    tmp[2] = product[11]; // 2:array element, 11:array element
    tmp[3] = product[13]; // 3:array element, 13:array element
    tmp[4] = product[14]; // 4:array element, 14:array element
    tmp[5] = product[15]; // 5:array element, 15:array element
    tmp[6] = product[13]; // 6:array element, 13:array element
    tmp[7] = product[8]; // 7:array element, 8:array element
    carry += uECC_vli_add(result, result, tmp, NUM_ECC_WORDS);
    /* d1 */
    tmp[0] = product[11]; // 11:array element
    tmp[1] = product[12]; // 12:array element
    tmp[2] = product[13]; // 2:array element, 13:array element
    tmp[3] = tmp[4] = tmp[5] = 0; // 3:array element, 4:array element, 5:array element
    tmp[6] = product[8]; // 6:array element, 8:array element
    tmp[7] = product[10]; // 7:array element, 10:array element
    carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);
    /* d2 */
    tmp[0] = product[12]; // 12:array element
    tmp[1] = product[13]; // 13:array element
    tmp[2] = product[14]; // 2:array element, 14:array element
    tmp[3] = product[15]; // 3:array element, 15:array element
    tmp[4] = tmp[5] = 0; // 4:array element, 5:array element
    tmp[6] = product[9]; // 6:array element, 9:array element
    tmp[7] = product[11]; // 7:array element, 11:array element
    carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);
    /* d3 */
    tmp[0] = product[13]; // 13:array element
    tmp[1] = product[14]; // 14:array element
    tmp[2] = product[15]; // 2:array element, 15:array element
    tmp[3] = product[8]; // 3:array element, 8:array element
    tmp[4] = product[9]; // 4:array element, 9:array element
    tmp[5] = product[10]; // 5:array element, 10:array element
    tmp[6] = 0; // 6:array element
    tmp[7] = product[12]; // 7:array element, 12:array element
    carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);
    /* d4 */
    tmp[0] = product[14]; // 14:array element
    tmp[1] = product[15]; // 1:array element, 15:array element
    tmp[2] = 0; // 2:array element
    tmp[3] = product[9]; // 3:array element, 9:array element
    tmp[4] = product[10]; // 4:array element, 10:array element
    tmp[5] = product[11]; // 5:array element, 11:array element
    tmp[6] = 0; // 6:array element
    tmp[7] = product[13]; // 7:array element, 13:array element
    carry -= uECC_vli_sub(result, result, tmp, NUM_ECC_WORDS);
    if (carry < 0) {
        do {
            carry += uECC_vli_add(result, result, curve_secp256r1.p, NUM_ECC_WORDS);
        } while (carry < 0);
    } else  {
        while (carry ||
                uECC_vli_cmp_unsafe(curve_secp256r1.p, result, NUM_ECC_WORDS) != 1) {
            carry -= uECC_vli_sub(result, result, curve_secp256r1.p, NUM_ECC_WORDS);
        }
    }
}

uECC_word_t EccPoint_isZero(const uECC_word_t *point, uECC_Curve curve)
{
    return uECC_vli_isZero(point, curve->num_words * 2); // 2:byte alignment
}

void apply_z(uECC_word_t *X1, uECC_word_t *Y1, const uECC_word_t *const Z,
             uECC_Curve curve)
{
    uECC_word_t t1[NUM_ECC_WORDS];
    uECC_vli_modSquare_fast(t1, Z, curve);    /* z^2 */
    uECC_vli_modMult_fast(X1, X1, t1, curve); /* x1 * z^2 */
    uECC_vli_modMult_fast(t1, t1, Z, curve);  /* z^3 */
    uECC_vli_modMult_fast(Y1, Y1, t1, curve); /* y1 * z^3 */
}

/* P = (x1, y1) => 2P, (x2, y2) => P' */
static void XYcZ_initial_double(uECC_word_t *X1, uECC_word_t *Y1,
                                uECC_word_t *X2, uECC_word_t *Y2,
                                const uECC_word_t *const initial_Z,
                                uECC_Curve curve)
{
    uECC_word_t z[NUM_ECC_WORDS];
    wordcount_t num_words = curve->num_words;

    if (initial_Z) {
        uECC_vli_set(z, initial_Z, num_words);
    } else {
        uECC_vli_clear(z, num_words);
        z[0] = 1;
    }

    uECC_vli_set(X2, X1, num_words);
    uECC_vli_set(Y2, Y1, num_words);
    apply_z(X1, Y1, z, curve);
    curve->double_jacobian(X1, Y1, z, curve);
    apply_z(X2, Y2, z, curve);
}

void XYcZ_add(uECC_word_t *X1, uECC_word_t *Y1,
              uECC_word_t *X2, uECC_word_t *Y2,
              uECC_Curve curve)
{
    /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
    uECC_word_t t5[NUM_ECC_WORDS];
    wordcount_t num_words = curve->num_words;
    uECC_vli_modSub(t5, X2, X1, curve->p, num_words); /* t5 = x2 - x1 */
    uECC_vli_modSquare_fast(t5, t5, curve); /* t5 = (x2 - x1)^2 = A */
    uECC_vli_modMult_fast(X1, X1, t5, curve); /* t1 = x1*A = B */
    uECC_vli_modMult_fast(X2, X2, t5, curve); /* t3 = x2*A = C */
    uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */
    uECC_vli_modSquare_fast(t5, Y2, curve); /* t5 = (y2 - y1)^2 = D */
    uECC_vli_modSub(t5, t5, X1, curve->p, num_words); /* t5 = D - B */
    uECC_vli_modSub(t5, t5, X2, curve->p, num_words); /* t5 = D - B - C = x3 */
    uECC_vli_modSub(X2, X2, X1, curve->p, num_words); /* t3 = C - B */
    uECC_vli_modMult_fast(Y1, Y1, X2, curve); /* t2 = y1*(C - B) */
    uECC_vli_modSub(X2, X1, t5, curve->p, num_words); /* t3 = B - x3 */
    uECC_vli_modMult_fast(Y2, Y2, X2, curve); /* t4 = (y2 - y1)*(B - x3) */
    uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y3 */
    uECC_vli_set(X2, t5, num_words);
}

/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
   Output P + Q = (x3, y3, Z3), P - Q = (x3', y3', Z3)
   or P => P - Q, Q => P + Q
 */
static void XYcZ_addC(uECC_word_t *X1, uECC_word_t *Y1,
                      uECC_word_t *X2, uECC_word_t *Y2,
                      uECC_Curve curve)
{
    /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
    uECC_word_t t5[NUM_ECC_WORDS];
    uECC_word_t t6[NUM_ECC_WORDS];
    uECC_word_t t7[NUM_ECC_WORDS];
    wordcount_t num_words = curve->num_words;
    uECC_vli_modSub(t5, X2, X1, curve->p, num_words); /* t5 = x2 - x1 */
    uECC_vli_modSquare_fast(t5, t5, curve); /* t5 = (x2 - x1)^2 = A */
    uECC_vli_modMult_fast(X1, X1, t5, curve); /* t1 = x1*A = B */
    uECC_vli_modMult_fast(X2, X2, t5, curve); /* t3 = x2*A = C */
    uECC_vli_modAdd(t5, Y2, Y1, curve->p, num_words); /* t5 = y2 + y1 */
    uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */
    uECC_vli_modSub(t6, X2, X1, curve->p, num_words); /* t6 = C - B */
    uECC_vli_modMult_fast(Y1, Y1, t6, curve); /* t2 = y1 * (C - B) = E */
    uECC_vli_modAdd(t6, X1, X2, curve->p, num_words); /* t6 = B + C */
    uECC_vli_modSquare_fast(X2, Y2, curve); /* t3 = (y2 - y1)^2 = D */
    uECC_vli_modSub(X2, X2, t6, curve->p, num_words); /* t3 = D - (B + C) = x3 */
    uECC_vli_modSub(t7, X1, X2, curve->p, num_words); /* t7 = B - x3 */
    uECC_vli_modMult_fast(Y2, Y2, t7, curve); /* t4 = (y2 - y1)*(B - x3) */
    /* t4 = (y2 - y1)*(B - x3) - E = y3: */
    uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words);
    uECC_vli_modSquare_fast(t7, t5, curve); /* t7 = (y2 + y1)^2 = F */
    uECC_vli_modSub(t7, t7, t6, curve->p, num_words); /* t7 = F - (B + C) = x3' */
    uECC_vli_modSub(t6, t7, X1, curve->p, num_words); /* t6 = x3' - B */
    uECC_vli_modMult_fast(t6, t6, t5, curve); /* t6 = (y2+y1)*(x3' - B) */
    /* t2 = (y2+y1)*(x3' - B) - E = y3': */
    uECC_vli_modSub(Y1, t6, Y1, curve->p, num_words);
    uECC_vli_set(X1, t7, num_words);
}

void EccPoint_mult(uECC_word_t *result, const uECC_word_t *point,
                   const uECC_word_t *scalar,
                   const uECC_word_t *initial_Z,
                   bitcount_t num_bits, uECC_Curve curve)
{
    /* R0 and R1 */
    uECC_word_t Rx[2][NUM_ECC_WORDS]; // 2:array element
    uECC_word_t Ry[2][NUM_ECC_WORDS]; // 2:array element
    uECC_word_t z[NUM_ECC_WORDS];
    bitcount_t i;
    uECC_word_t nb;
    wordcount_t num_words = curve->num_words;
    uECC_vli_set(Rx[1], point, num_words);
    uECC_vli_set(Ry[1], point + num_words, num_words);
    XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], initial_Z, curve);

    for (i = num_bits - 2; i > 0; --i) { // 2:byte alignment
        nb = !uECC_vli_testBit(scalar, i);
        XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], curve);
        XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], curve);
    }

    nb = !uECC_vli_testBit(scalar, 0);
    XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], curve);
    /* Find final 1/Z value. */
    uECC_vli_modSub(z, Rx[1], Rx[0], curve->p, num_words); /* X1 - X0 */
    uECC_vli_modMult_fast(z, z, Ry[1 - nb], curve); /* Yb * (X1 - X0) */
    uECC_vli_modMult_fast(z, z, point, curve); /* xP * Yb * (X1 - X0) */
    uECC_vli_modInv(z, z, curve->p, num_words); /* 1 / (xP * Yb * (X1 - X0)) */
    /* yP / (xP * Yb * (X1 - X0)) */
    uECC_vli_modMult_fast(z, z, point + num_words, curve);
    /* Xb * yP / (xP * Yb * (X1 - X0)) */
    uECC_vli_modMult_fast(z, z, Rx[1 - nb], curve);
    /* End 1/Z calculation */
    XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], curve);
    apply_z(Rx[0], Ry[0], z, curve);
    uECC_vli_set(result, Rx[0], num_words);
    uECC_vli_set(result + num_words, Ry[0], num_words);
}

uECC_word_t regularize_k(const uECC_word_t *const k, uECC_word_t *k0,
                         uECC_word_t *k1, uECC_Curve curve)
{
    wordcount_t num_n_words = BITS_TO_WORDS(curve->num_n_bits);
    bitcount_t num_n_bits = curve->num_n_bits;
    uECC_word_t carry = uECC_vli_add(k0, k, curve->n, num_n_words) ||
                        (num_n_bits < ((bitcount_t)num_n_words * uECC_WORD_SIZE * 8) && // 8:byte alignment
                         uECC_vli_testBit(k0, num_n_bits));
    uECC_vli_add(k1, k0, curve->n, num_n_words);
    return carry;
}

uECC_word_t EccPoint_compute_public_key(uECC_word_t *result,
                                        uECC_word_t *private_key,
                                        uECC_Curve curve)
{
    uECC_word_t tmp1[NUM_ECC_WORDS];
    uECC_word_t tmp2[NUM_ECC_WORDS];
    uECC_word_t *p2[2] = {tmp1, tmp2}; // 2:array element
    uECC_word_t carry;
    /* Regularize the bitcount for the private key so that attackers cannot
     * use a side channel attack to learn the number of leading zeros. */
    carry = regularize_k(private_key, tmp1, tmp2, curve);
    EccPoint_mult(result, curve->G, p2[!carry], 0, curve->num_n_bits + 1, curve);

    if (EccPoint_isZero(result, curve)) {
        return 0;
    }

    return 1;
}

/* Converts an integer in uECC native format to big-endian bytes. */
void uECC_vli_nativeToBytes(uint8_t *bytes, int num_bytes,
                            const unsigned int *native)
{
    wordcount_t i;

    for (i = 0; i < num_bytes; ++i) {
        unsigned b = num_bytes - 1 - i;
        bytes[i] = native[b / uECC_WORD_SIZE] >> (8 * (b % uECC_WORD_SIZE));
    }
}

/* Converts big-endian bytes to an integer in uECC native format. */
void uECC_vli_bytesToNative(unsigned int *native, const uint8_t *bytes,
                            int num_bytes)
{
    wordcount_t i;
    uECC_vli_clear(native, (num_bytes + (uECC_WORD_SIZE - 1)) / uECC_WORD_SIZE);

    for (i = 0; i < num_bytes; ++i) {
        unsigned b = num_bytes - 1 - i;
        native[b / uECC_WORD_SIZE] |=
                        (uECC_word_t)bytes[i] << (8 * (b % uECC_WORD_SIZE)); // 8:byte alignment
    }
}

int uECC_generate_random_int(uECC_word_t *random, const uECC_word_t *top,
                             wordcount_t num_words)
{
    uECC_word_t mask = (uECC_word_t) -1;
    uECC_word_t tries;
    bitcount_t num_bits = uECC_vli_numBits(top, num_words);

    if (!g_rng_function) {
        return 0;
    }

    for (tries = 0; tries < uECC_RNG_MAX_TRIES; ++tries) {
        if (!g_rng_function((uint8_t *)random, num_words * uECC_WORD_SIZE)) {
            return 0;
        }

        random[num_words - 1] &=
                        mask >> ((bitcount_t)(num_words * uECC_WORD_SIZE * 8 - num_bits)); // 8:byte alignment

        if (!uECC_vli_isZero(random, num_words) &&
                uECC_vli_cmp(top, random, num_words) == 1) {
            return 1;
        }
    }

    return 0;
}

int uECC_valid_point(const uECC_word_t *point, uECC_Curve curve)
{
    uECC_word_t tmp1[NUM_ECC_WORDS];
    uECC_word_t tmp2[NUM_ECC_WORDS];
    wordcount_t num_words = curve->num_words;

    /* The point at infinity is invalid. */
    if (EccPoint_isZero(point, curve)) {
        return -1;
    }

    /* x and y must be smaller than p. */
    if (uECC_vli_cmp_unsafe(curve->p, point, num_words) != 1 ||
            uECC_vli_cmp_unsafe(curve->p, point + num_words, num_words) != 1) {
        return -2; // -2:return value
    }

    uECC_vli_modSquare_fast(tmp1, point + num_words, curve);
    curve->x_side(tmp2, point, curve); /* tmp2 = x^3 + ax + b */

    /* Make sure that y^2 == x^3 + ax + b */
    if (uECC_vli_equal(tmp1, tmp2, num_words) != 0) {
        return -3; // -3:return value
    }
    return 0;
}

int uECC_valid_public_key(const uint8_t *public_key, uECC_Curve curve)
{
    uECC_word_t _public[NUM_ECC_WORDS * 2]; // 2:byte alignment
    uECC_vli_bytesToNative(_public, public_key, curve->num_bytes);
    uECC_vli_bytesToNative(
        _public + curve->num_words,
        public_key + curve->num_bytes,
        curve->num_bytes);

    if (uECC_vli_cmp_unsafe(_public, curve->G, NUM_ECC_WORDS * 2) == 0) { // 2:byte alignment
        return -4; // -4:return value
    }

    return uECC_valid_point(_public, curve);
}

int uECC_compute_public_key(const uint8_t *private_key, uint8_t *public_key,
                            uECC_Curve curve)
{
    uECC_word_t _private[NUM_ECC_WORDS];
    uECC_word_t _public[NUM_ECC_WORDS * 2]; // 2:byte alignment
    uECC_vli_bytesToNative(
        _private,
        private_key,
        BITS_TO_BYTES(curve->num_n_bits));

    /* Make sure the private key is in the range [1, n-1]. */
    if (uECC_vli_isZero(_private, BITS_TO_WORDS(curve->num_n_bits))) {
        return 0;
    }

    if (uECC_vli_cmp(curve->n, _private, BITS_TO_WORDS(curve->num_n_bits)) != 1) {
        return 0;
    }

    /* Compute public key. */
    if (!EccPoint_compute_public_key(_public, _private, curve)) {
        return 0;
    }

    uECC_vli_nativeToBytes(public_key, curve->num_bytes, _public);
    uECC_vli_nativeToBytes(
        public_key +
        curve->num_bytes, curve->num_bytes, _public + curve->num_words);
    return 1;
}