text: /*
   * Copyright (c) 2017 Thomas Pornin <pornin@bolet.org>
   *
   * 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_NO_ARITH_SHIFT is undefined, or defined to 0, then we _assume_
   * that right-shifting a signed negative integer copies the sign bit
   * (arithmetic right-shift). This is "implementation-defined behaviour",
   * i.e. it is not undefined, but it may differ between compilers. Each
   * compiler is supposed to document its behaviour in that respect. GCC
   * explicitly defines that an arithmetic right shift is used. We expect
   * all other compilers to do the same, because underlying CPU offer an
   * arithmetic right shift opcode that could not be used otherwise.
   */
  #if BR_NO_ARITH_SHIFT
  #define ARSH(x, n)    (((uint32_t)(x) >> (n)) \
                        | ((-((uint32_t)(x) >> 31)) << (32 - (n))))
  #define ARSHW(x, n)   (((uint64_t)(x) >> (n)) \
                        | ((-((uint64_t)(x) >> 63)) << (64 - (n))))
  #else
  #define ARSH(x, n)    ((*(int32_t *)&(x)) >> (n))
  #define ARSHW(x, n)   ((*(int64_t *)&(x)) >> (n))
  #endif
  
  /*
   * Convert an integer from unsigned big-endian encoding to a sequence of
   * 30-bit words in little-endian order. The final "partial" word is
   * returned.
   */
  static uint32_t
  be8_to_le30(uint32_t *dst, const unsigned char *src, size_t len)
  {
  	uint32_t acc;
  	int acc_len;
  
  	acc = 0;
  	acc_len = 0;
  	while (len -- > 0) {
  		uint32_t b;
  
  		b = src[len];
  		if (acc_len < 22) {
  			acc |= b << acc_len;
  			acc_len += 8;
  		} else {
  			*dst ++ = (acc | (b << acc_len)) & 0x3FFFFFFF;
  			acc = b >> (30 - acc_len);
  			acc_len -= 22;
  		}
  	}
  	return acc;
  }
  
  /*
   * Convert an integer (30-bit words, little-endian) to unsigned
   * big-endian encoding. The total encoding length is provided; all
   * the destination bytes will be filled.
   */
  static void
  le30_to_be8(unsigned char *dst, size_t len, const uint32_t *src)
  {
  	uint32_t acc;
  	int acc_len;
  
  	acc = 0;
  	acc_len = 0;
  	while (len -- > 0) {
  		if (acc_len < 8) {
  			uint32_t w;
  
  			w = *src ++;
  			dst[len] = (unsigned char)(acc | (w << acc_len));
  			acc = w >> (8 - acc_len);
  			acc_len += 22;
  		} else {
  			dst[len] = (unsigned char)acc;
  			acc >>= 8;
  			acc_len -= 8;
  		}
  	}
  }
  
  /*
   * Multiply two integers. Source integers are represented as arrays of
   * nine 30-bit words, for values up to 2^270-1. Result is encoded over
   * 18 words of 30 bits each.
   */
  static void
  mul9(uint32_t *d, const uint32_t *a, const uint32_t *b)
  {
  	/*
  	 * Maximum intermediate result is no more than
  	 * 10376293531797946367, which fits in 64 bits. Reason:
  	 *
  	 *   10376293531797946367 = 9 * (2^30-1)^2 + 9663676406
  	 *   10376293531797946367 < 9663676407 * 2^30
  	 *
  	 * Thus, adding together 9 products of 30-bit integers, with
  	 * a carry of at most 9663676406, yields an integer that fits
  	 * on 64 bits and generates a carry of at most 9663676406.
  	 */
  	uint64_t t[17];
  	uint64_t cc;
  	int i;
  
  	t[ 0] = MUL31(a[0], b[0]);
  	t[ 1] = MUL31(a[0], b[1])
  		+ MUL31(a[1], b[0]);
  	t[ 2] = MUL31(a[0], b[2])
  		+ MUL31(a[1], b[1])
  		+ MUL31(a[2], b[0]);
  	t[ 3] = MUL31(a[0], b[3])
  		+ MUL31(a[1], b[2])
  		+ MUL31(a[2], b[1])
  		+ MUL31(a[3], b[0]);
  	t[ 4] = MUL31(a[0], b[4])
  		+ MUL31(a[1], b[3])
  		+ MUL31(a[2], b[2])
  		+ MUL31(a[3], b[1])
  		+ MUL31(a[4], b[0]);
  	t[ 5] = MUL31(a[0], b[5])
  		+ MUL31(a[1], b[4])
  		+ MUL31(a[2], b[3])
  		+ MUL31(a[3], b[2])
  		+ MUL31(a[4], b[1])
  		+ MUL31(a[5], b[0]);
  	t[ 6] = MUL31(a[0], b[6])
  		+ MUL31(a[1], b[5])
  		+ MUL31(a[2], b[4])
  		+ MUL31(a[3], b[3])
  		+ MUL31(a[4], b[2])
  		+ MUL31(a[5], b[1])
  		+ MUL31(a[6], b[0]);
  	t[ 7] = MUL31(a[0], b[7])
  		+ MUL31(a[1], b[6])
  		+ MUL31(a[2], b[5])
  		+ MUL31(a[3], b[4])
  		+ MUL31(a[4], b[3])
  		+ MUL31(a[5], b[2])
  		+ MUL31(a[6], b[1])
  		+ MUL31(a[7], b[0]);
  	t[ 8] = MUL31(a[0], b[8])
  		+ MUL31(a[1], b[7])
  		+ MUL31(a[2], b[6])
  		+ MUL31(a[3], b[5])
  		+ MUL31(a[4], b[4])
  		+ MUL31(a[5], b[3])
  		+ MUL31(a[6], b[2])
  		+ MUL31(a[7], b[1])
  		+ MUL31(a[8], b[0]);
  	t[ 9] = MUL31(a[1], b[8])
  		+ MUL31(a[2], b[7])
  		+ MUL31(a[3], b[6])
  		+ MUL31(a[4], b[5])
  		+ MUL31(a[5], b[4])
  		+ MUL31(a[6], b[3])
  		+ MUL31(a[7], b[2])
  		+ MUL31(a[8], b[1]);
  	t[10] = MUL31(a[2], b[8])
  		+ MUL31(a[3], b[7])
  		+ MUL31(a[4], b[6])
  		+ MUL31(a[5], b[5])
  		+ MUL31(a[6], b[4])
  		+ MUL31(a[7], b[3])
  		+ MUL31(a[8], b[2]);
  	t[11] = MUL31(a[3], b[8])
  		+ MUL31(a[4], b[7])
  		+ MUL31(a[5], b[6])
  		+ MUL31(a[6], b[5])
  		+ MUL31(a[7], b[4])
  		+ MUL31(a[8], b[3]);
  	t[12] = MUL31(a[4], b[8])
  		+ MUL31(a[5], b[7])
  		+ MUL31(a[6], b[6])
  		+ MUL31(a[7], b[5])
  		+ MUL31(a[8], b[4]);
  	t[13] = MUL31(a[5], b[8])
  		+ MUL31(a[6], b[7])
  		+ MUL31(a[7], b[6])
  		+ MUL31(a[8], b[5]);
  	t[14] = MUL31(a[6], b[8])
  		+ MUL31(a[7], b[7])
  		+ MUL31(a[8], b[6]);
  	t[15] = MUL31(a[7], b[8])
  		+ MUL31(a[8], b[7]);
  	t[16] = MUL31(a[8], b[8]);
  
  	/*
  	 * Propagate carries.
  	 */
  	cc = 0;
  	for (i = 0; i < 17; i ++) {
  		uint64_t w;
  
  		w = t[i] + cc;
  		d[i] = (uint32_t)w & 0x3FFFFFFF;
  		cc = w >> 30;
  	}
  	d[17] = (uint32_t)cc;
  }
  
  /*
   * Square a 270-bit integer, represented as an array of nine 30-bit words.
   * Result uses 18 words of 30 bits each.
   */
  static void
  square9(uint32_t *d, const uint32_t *a)
  {
  	uint64_t t[17];
  	uint64_t cc;
  	int i;
  
  	t[ 0] = MUL31(a[0], a[0]);
  	t[ 1] = ((MUL31(a[0], a[1])) << 1);
  	t[ 2] = MUL31(a[1], a[1])
  		+ ((MUL31(a[0], a[2])) << 1);
  	t[ 3] = ((MUL31(a[0], a[3])
  		+ MUL31(a[1], a[2])) << 1);
  	t[ 4] = MUL31(a[2], a[2])
  		+ ((MUL31(a[0], a[4])
  		+ MUL31(a[1], a[3])) << 1);
  	t[ 5] = ((MUL31(a[0], a[5])
  		+ MUL31(a[1], a[4])
  		+ MUL31(a[2], a[3])) << 1);
  	t[ 6] = MUL31(a[3], a[3])
  		+ ((MUL31(a[0], a[6])
  		+ MUL31(a[1], a[5])
  		+ MUL31(a[2], a[4])) << 1);
  	t[ 7] = ((MUL31(a[0], a[7])
  		+ MUL31(a[1], a[6])
  		+ MUL31(a[2], a[5])
  		+ MUL31(a[3], a[4])) << 1);
  	t[ 8] = MUL31(a[4], a[4])
  		+ ((MUL31(a[0], a[8])
  		+ MUL31(a[1], a[7])
  		+ MUL31(a[2], a[6])
  		+ MUL31(a[3], a[5])) << 1);
  	t[ 9] = ((MUL31(a[1], a[8])
  		+ MUL31(a[2], a[7])
  		+ MUL31(a[3], a[6])
  		+ MUL31(a[4], a[5])) << 1);
  	t[10] = MUL31(a[5], a[5])
  		+ ((MUL31(a[2], a[8])
  		+ MUL31(a[3], a[7])
  		+ MUL31(a[4], a[6])) << 1);
  	t[11] = ((MUL31(a[3], a[8])
  		+ MUL31(a[4], a[7])
  		+ MUL31(a[5], a[6])) << 1);
  	t[12] = MUL31(a[6], a[6])
  		+ ((MUL31(a[4], a[8])
  		+ MUL31(a[5], a[7])) << 1);
  	t[13] = ((MUL31(a[5], a[8])
  		+ MUL31(a[6], a[7])) << 1);
  	t[14] = MUL31(a[7], a[7])
  		+ ((MUL31(a[6], a[8])) << 1);
  	t[15] = ((MUL31(a[7], a[8])) << 1);
  	t[16] = MUL31(a[8], a[8]);
  
  	/*
  	 * Propagate carries.
  	 */
  	cc = 0;
  	for (i = 0; i < 17; i ++) {
  		uint64_t w;
  
  		w = t[i] + cc;
  		d[i] = (uint32_t)w & 0x3FFFFFFF;
  		cc = w >> 30;
  	}
  	d[17] = (uint32_t)cc;
  }
  
  /*
   * Base field modulus for P-256.
   */
  static const uint32_t F256[] = {
  
  	0x3FFFFFFF, 0x3FFFFFFF, 0x3FFFFFFF, 0x0000003F, 0x00000000,
  	0x00000000, 0x00001000, 0x3FFFC000, 0x0000FFFF
  };
  
  /*
   * The 'b' curve equation coefficient for P-256.
   */
  static const uint32_t P256_B[] = {
  
  	0x27D2604B, 0x2F38F0F8, 0x053B0F63, 0x0741AC33, 0x1886BC65,
  	0x2EF555DA, 0x293E7B3E, 0x0D762A8E, 0x00005AC6
  };
  
  /*
   * Addition in the field. Source operands shall fit on 257 bits; output
   * will be lower than twice the modulus.
   */
  static void
  add_f256(uint32_t *d, const uint32_t *a, const uint32_t *b)
  {
  	uint32_t w, cc;
  	int i;
  
  	cc = 0;
  	for (i = 0; i < 9; i ++) {
  		w = a[i] + b[i] + cc;
  		d[i] = w & 0x3FFFFFFF;
  		cc = w >> 30;
  	}
  	w >>= 16;
  	d[8] &= 0xFFFF;
  	d[3] -= w << 6;
  	d[6] -= w << 12;
  	d[7] += w << 14;
  	cc = w;
  	for (i = 0; i < 9; i ++) {
  		w = d[i] + cc;
  		d[i] = w & 0x3FFFFFFF;
  		cc = ARSH(w, 30);
  	}
  }
  
  /*
   * Subtraction in the field. Source operands shall be smaller than twice
   * the modulus; the result will fulfil the same property.
   */
  static void
  sub_f256(uint32_t *d, const uint32_t *a, const uint32_t *b)
  {
  	uint32_t w, cc;
  	int i;
  
  	/*
  	 * We really compute a - b + 2*p to make sure that the result is
  	 * positive.
  	 */
  	w = a[0] - b[0] - 0x00002;
  	d[0] = w & 0x3FFFFFFF;
  	w = a[1] - b[1] + ARSH(w, 30);
  	d[1] = w & 0x3FFFFFFF;
  	w = a[2] - b[2] + ARSH(w, 30);
  	d[2] = w & 0x3FFFFFFF;
  	w = a[3] - b[3] + ARSH(w, 30) + 0x00080;
  	d[3] = w & 0x3FFFFFFF;
  	w = a[4] - b[4] + ARSH(w, 30);
  	d[4] = w & 0x3FFFFFFF;
  	w = a[5] - b[5] + ARSH(w, 30);
  	d[5] = w & 0x3FFFFFFF;
  	w = a[6] - b[6] + ARSH(w, 30) + 0x02000;
  	d[6] = w & 0x3FFFFFFF;
  	w = a[7] - b[7] + ARSH(w, 30) - 0x08000;
  	d[7] = w & 0x3FFFFFFF;
  	w = a[8] - b[8] + ARSH(w, 30) + 0x20000;
  	d[8] = w & 0xFFFF;
  	w >>= 16;
  	d[8] &= 0xFFFF;
  	d[3] -= w << 6;
  	d[6] -= w << 12;
  	d[7] += w << 14;
  	cc = w;
  	for (i = 0; i < 9; i ++) {
  		w = d[i] + cc;
  		d[i] = w & 0x3FFFFFFF;
  		cc = ARSH(w, 30);
  	}
  }
  
  /*
   * Compute a multiplication in F256. Source operands shall be less than
   * twice the modulus.
   */
  static void
  mul_f256(uint32_t *d, const uint32_t *a, const uint32_t *b)
  {
  	uint32_t t[18];
  	uint64_t s[18];
  	uint64_t cc, x;
  	uint32_t z;
  	int i;
  
  	mul9(t, a, b);
  
  	/*
  	 * Modular reduction: each high word in added/subtracted where
  	 * necessary.
  	 *
  	 * The modulus is:
  	 *    p = 2^256 - 2^224 + 2^192 + 2^96 - 1
  	 * Therefore:
  	 *    2^256 = 2^224 - 2^192 - 2^96 + 1 mod p
  	 *
  	 * For a word x at bit offset n (n >= 256), we have:
  	 *    x*2^n = x*2^(n-32) - x*2^(n-64)
  	 *            - x*2^(n - 160) + x*2^(n-256) mod p
  	 *
  	 * Thus, we can nullify the high word if we reinject it at some
  	 * proper emplacements.
  	 *
  	 * We use 64-bit intermediate words to allow for carries to
  	 * accumulate easily, before performing the final propagation.
  	 */
  	for (i = 0; i < 18; i ++) {
  		s[i] = t[i];
  	}
  
  	for (i = 17; i >= 9; i --) {
  		uint64_t x;
  
  		x = s[i];
  		s[i - 1] += ARSHW(x, 2);
  		s[i - 2] += (x << 28) & 0x3FFFFFFF;
  		s[i - 2] -= ARSHW(x, 4);
  		s[i - 3] -= (x << 26) & 0x3FFFFFFF;
  		s[i - 5] -= ARSHW(x, 10);
  		s[i - 6] -= (x << 20) & 0x3FFFFFFF;
  		s[i - 8] += ARSHW(x, 16);
  		s[i - 9] += (x << 14) & 0x3FFFFFFF;
  	}
  
  	/*
  	 * Carry propagation must be signed. Moreover, we may have overdone
  	 * it a bit, and obtain a negative result.
  	 *
  	 * The loop above ran 9 times; each time, each word was augmented
  	 * by at most one extra word (in absolute value). Thus, the top
  	 * word must in fine fit in 39 bits, so the carry below will fit
  	 * on 9 bits.
  	 */
  	cc = 0;
  	for (i = 0; i < 9; i ++) {
  		x = s[i] + cc;
  		d[i] = (uint32_t)x & 0x3FFFFFFF;
  		cc = ARSHW(x, 30);
  	}
  
  	/*
  	 * All nine words fit on 30 bits, but there may be an extra
  	 * carry for a few bits (at most 9), and that carry may be
  	 * negative. Moreover, we want the result to fit on 257 bits.
  	 * The two lines below ensure that the word in d[] has length
  	 * 256 bits, and the (signed) carry (beyond 2^256) is in cc. The
  	 * significant length of cc is less than 24 bits, so we will be
  	 * able to switch to 32-bit operations.
  	 */
  	cc = ARSHW(x, 16);
  	d[8] &= 0xFFFF;
  
  	/*
  	 * Subtract cc*p.
  	 */
  	z = (uint32_t)cc;
  	d[3] -= z << 6;
  	d[6] -= (z << 12) & 0x3FFFFFFF;
  	d[7] -= ARSH(z, 18);
  	d[7] += (z << 14) & 0x3FFFFFFF;
  	d[8] += ARSH(z, 16);
  	for (i = 0; i < 9; i ++) {
  		uint32_t w;
  
  		w = d[i] + z;
  		d[i] = w & 0x3FFFFFFF;
  		z = ARSH(w, 30);
  	}
  }
  
  /*
   * Compute a square in F256. Source operand shall be less than
   * twice the modulus.
   */
  static void
  square_f256(uint32_t *d, const uint32_t *a)
  {
  	uint32_t t[18];
  	uint64_t s[18];
  	uint64_t cc, x;
  	uint32_t z;
  	int i;
  
  	square9(t, a);
  
  	/*
  	 * Modular reduction: each high word in added/subtracted where
  	 * necessary.
  	 *
  	 * The modulus is:
  	 *    p = 2^256 - 2^224 + 2^192 + 2^96 - 1
  	 * Therefore:
  	 *    2^256 = 2^224 - 2^192 - 2^96 + 1 mod p
  	 *
  	 * For a word x at bit offset n (n >= 256), we have:
  	 *    x*2^n = x*2^(n-32) - x*2^(n-64)
  	 *            - x*2^(n - 160) + x*2^(n-256) mod p
  	 *
  	 * Thus, we can nullify the high word if we reinject it at some
  	 * proper emplacements.
  	 *
  	 * We use 64-bit intermediate words to allow for carries to
  	 * accumulate easily, before performing the final propagation.
  	 */
  	for (i = 0; i < 18; i ++) {
  		s[i] = t[i];
  	}
  
  	for (i = 17; i >= 9; i --) {
  		uint64_t x;
  
  		x = s[i];
  		s[i - 1] += ARSHW(x, 2);
  		s[i - 2] += (x << 28) & 0x3FFFFFFF;
  		s[i - 2] -= ARSHW(x, 4);
  		s[i - 3] -= (x << 26) & 0x3FFFFFFF;
  		s[i - 5] -= ARSHW(x, 10);
  		s[i - 6] -= (x << 20) & 0x3FFFFFFF;
  		s[i - 8] += ARSHW(x, 16);
  		s[i - 9] += (x << 14) & 0x3FFFFFFF;
  	}
  
  	/*
  	 * Carry propagation must be signed. Moreover, we may have overdone
  	 * it a bit, and obtain a negative result.
  	 *
  	 * The loop above ran 9 times; each time, each word was augmented
  	 * by at most one extra word (in absolute value). Thus, the top
  	 * word must in fine fit in 39 bits, so the carry below will fit
  	 * on 9 bits.
  	 */
  	cc = 0;
  	for (i = 0; i < 9; i ++) {
  		x = s[i] + cc;
  		d[i] = (uint32_t)x & 0x3FFFFFFF;
  		cc = ARSHW(x, 30);
  	}
  
  	/*
  	 * All nine words fit on 30 bits, but there may be an extra
  	 * carry for a few bits (at most 9), and that carry may be
  	 * negative. Moreover, we want the result to fit on 257 bits.
  	 * The two lines below ensure that the word in d[] has length
  	 * 256 bits, and the (signed) carry (beyond 2^256) is in cc. The
  	 * significant length of cc is less than 24 bits, so we will be
  	 * able to switch to 32-bit operations.
  	 */
  	cc = ARSHW(x, 16);
  	d[8] &= 0xFFFF;
  
  	/*
  	 * Subtract cc*p.
  	 */
  	z = (uint32_t)cc;
  	d[3] -= z << 6;
  	d[6] -= (z << 12) & 0x3FFFFFFF;
  	d[7] -= ARSH(z, 18);
  	d[7] += (z << 14) & 0x3FFFFFFF;
  	d[8] += ARSH(z, 16);
  	for (i = 0; i < 9; i ++) {
  		uint32_t w;
  
  		w = d[i] + z;
  		d[i] = w & 0x3FFFFFFF;
  		z = ARSH(w, 30);
  	}
  }
  
  /*
   * Perform a "final reduction" in field F256 (field for curve P-256).
   * The source value must be less than twice the modulus. If the value
   * is not lower than the modulus, then the modulus is subtracted and
   * this function returns 1; otherwise, it leaves it untouched and it
   * returns 0.
   */
  static uint32_t
  reduce_final_f256(uint32_t *d)
  {
  	uint32_t t[9];
  	uint32_t cc;
  	int i;
  
  	cc = 0;
  	for (i = 0; i < 9; i ++) {
  		uint32_t w;
  
  		w = d[i] - F256[i] - cc;
  		cc = w >> 31;
  		t[i] = w & 0x3FFFFFFF;
  	}
  	cc ^= 1;
  	CCOPY(cc, d, t, sizeof t);
  	return cc;
  }
  
  /*
   * Jacobian coordinates for a point in P-256: affine coordinates (X,Y)
   * are such that:
   *   X = x / z^2
   *   Y = y / z^3
   * For the point at infinity, z = 0.
   * Each point thus admits many possible representations.
   *
   * Coordinates are represented in arrays of 32-bit integers, each holding
   * 30 bits of data. Values may also be slightly greater than the modulus,
   * but they will always be lower than twice the modulus.
   */
  typedef struct {
  	uint32_t x[9];
  	uint32_t y[9];
  	uint32_t z[9];
  } p256_jacobian;
  
  /*
   * Convert a point to affine coordinates:
   *  - If the point is the point at infinity, then all three coordinates
   *    are set to 0.
   *  - Otherwise, the 'z' coordinate is set to 1, and the 'x' and 'y'
   *    coordinates are the 'X' and 'Y' affine coordinates.
   * The coordinates are guaranteed to be lower than the modulus.
   */
  static void
  p256_to_affine(p256_jacobian *P)
  {
  	uint32_t t1[9], t2[9];
  	int i;
  
  	/*
  	 * Invert z with a modular exponentiation: the modulus is
  	 * p = 2^256 - 2^224 + 2^192 + 2^96 - 1, and the exponent is
  	 * p-2. Exponent bit pattern (from high to low) is:
  	 *  - 32 bits of value 1
  	 *  - 31 bits of value 0
  	 *  - 1 bit of value 1
  	 *  - 96 bits of value 0
  	 *  - 94 bits of value 1
  	 *  - 1 bit of value 0
  	 *  - 1 bit of value 1
  	 * Thus, we precompute z^(2^31-1) to speed things up.
  	 *
  	 * If z = 0 (point at infinity) then the modular exponentiation
  	 * will yield 0, which leads to the expected result (all three
  	 * coordinates set to 0).
  	 */
  
  	/*
  	 * A simple square-and-multiply for z^(2^31-1). We could save about
  	 * two dozen multiplications here with an addition chain, but
  	 * this would require a bit more code, and extra stack buffers.
  	 */
  	memcpy(t1, P->z, sizeof P->z);
  	for (i = 0; i < 30; i ++) {
  		square_f256(t1, t1);
  		mul_f256(t1, t1, P->z);
  	}
  
  	/*
  	 * Square-and-multiply. Apart from the squarings, we have a few
  	 * multiplications to set bits to 1; we multiply by the original z
  	 * for setting 1 bit, and by t1 for setting 31 bits.
  	 */
  	memcpy(t2, P->z, sizeof P->z);
  	for (i = 1; i < 256; i ++) {
  		square_f256(t2, t2);
  		switch (i) {
  		case 31:
  		case 190:
  		case 221:
  		case 252:
  			mul_f256(t2, t2, t1);
  			break;
  		case 63:
  		case 253:
  		case 255:
  			mul_f256(t2, t2, P->z);
  			break;
  		}
  	}
  
  	/*
  	 * Now that we have 1/z, multiply x by 1/z^2 and y by 1/z^3.
  	 */
  	mul_f256(t1, t2, t2);
  	mul_f256(P->x, t1, P->x);
  	mul_f256(t1, t1, t2);
  	mul_f256(P->y, t1, P->y);
  	reduce_final_f256(P->x);
  	reduce_final_f256(P->y);
  
  	/*
  	 * Multiply z by 1/z. If z = 0, then this will yield 0, otherwise
  	 * this will set z to 1.
  	 */
  	mul_f256(P->z, P->z, t2);
  	reduce_final_f256(P->z);
  }
  
  /*
   * Double a point in P-256. This function works for all valid points,
   * including the point at infinity.
   */
  static void
  p256_double(p256_jacobian *Q)
  {
  	/*
  	 * Doubling formulas are:
  	 *
  	 *   s = 4*x*y^2
  	 *   m = 3*(x + z^2)*(x - z^2)
  	 *   x' = m^2 - 2*s
  	 *   y' = m*(s - x') - 8*y^4
  	 *   z' = 2*y*z
  	 *
  	 * These formulas work for all points, including points of order 2
  	 * and points at infinity:
  	 *   - If y = 0 then z' = 0. But there is no such point in P-256
  	 *     anyway.
  	 *   - If z = 0 then z' = 0.
  	 */
  	uint32_t t1[9], t2[9], t3[9], t4[9];
  
  	/*
  	 * Compute z^2 in t1.
  	 */
  	square_f256(t1, Q->z);
  
  	/*
  	 * Compute x-z^2 in t2 and x+z^2 in t1.
  	 */
  	add_f256(t2, Q->x, t1);
  	sub_f256(t1, Q->x, t1);
  
  	/*
  	 * Compute 3*(x+z^2)*(x-z^2) in t1.
  	 */
  	mul_f256(t3, t1, t2);
  	add_f256(t1, t3, t3);
  	add_f256(t1, t3, t1);
  
  	/*
  	 * Compute 4*x*y^2 (in t2) and 2*y^2 (in t3).
  	 */
  	square_f256(t3, Q->y);
  	add_f256(t3, t3, t3);
  	mul_f256(t2, Q->x, t3);
  	add_f256(t2, t2, t2);
  
  	/*
  	 * Compute x' = m^2 - 2*s.
  	 */
  	square_f256(Q->x, t1);
  	sub_f256(Q->x, Q->x, t2);
  	sub_f256(Q->x, Q->x, t2);
  
  	/*
  	 * Compute z' = 2*y*z.
  	 */
  	mul_f256(t4, Q->y, Q->z);
  	add_f256(Q->z, t4, t4);
  
  	/*
  	 * Compute y' = m*(s - x') - 8*y^4. Note that we already have
  	 * 2*y^2 in t3.
  	 */
  	sub_f256(t2, t2, Q->x);
  	mul_f256(Q->y, t1, t2);
  	square_f256(t4, t3);
  	add_f256(t4, t4, t4);
  	sub_f256(Q->y, Q->y, t4);
  }
  
  /*
   * Add point P2 to point P1.
   *
   * This function computes the wrong result in the following cases:
   *
   *   - If P1 == 0 but P2 != 0
   *   - If P1 != 0 but P2 == 0
   *   - If P1 == P2
   *
   * In all three cases, P1 is set to the point at infinity.
   *
   * Returned value is 0 if one of the following occurs:
   *
   *   - P1 and P2 have the same Y coordinate
   *   - P1 == 0 and P2 == 0
   *   - The Y coordinate of one of the points is 0 and the other point is
   *     the point at infinity.
   *
   * The third case cannot actually happen with valid points, since a point
   * with Y == 0 is a point of order 2, and there is no point of order 2 on
   * curve P-256.
   *
   * Therefore, assuming that P1 != 0 and P2 != 0 on input, then the caller
   * can apply the following:
   *
   *   - If the result is not the point at infinity, then it is correct.
   *   - Otherwise, if the returned value is 1, then this is a case of
   *     P1+P2 == 0, so the result is indeed the point at infinity.
   *   - Otherwise, P1 == P2, so a "double" operation should have been
   *     performed.
   */
  static uint32_t
  p256_add(p256_jacobian *P1, const p256_jacobian *P2)
  {
  	/*
  	 * Addtions formulas are:
  	 *
  	 *   u1 = x1 * z2^2
  	 *   u2 = x2 * z1^2
  	 *   s1 = y1 * z2^3
  	 *   s2 = y2 * z1^3
  	 *   h = u2 - u1
  	 *   r = s2 - s1
  	 *   x3 = r^2 - h^3 - 2 * u1 * h^2
  	 *   y3 = r * (u1 * h^2 - x3) - s1 * h^3
  	 *   z3 = h * z1 * z2
  	 */
  	uint32_t t1[9], t2[9], t3[9], t4[9], t5[9], t6[9], t7[9];
  	uint32_t ret;
  	int i;
  
  	/*
  	 * Compute u1 = x1*z2^2 (in t1) and s1 = y1*z2^3 (in t3).
  	 */
  	square_f256(t3, P2->z);
  	mul_f256(t1, P1->x, t3);
  	mul_f256(t4, P2->z, t3);
  	mul_f256(t3, P1->y, t4);
  
  	/*
  	 * Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
  	 */
  	square_f256(t4, P1->z);
  	mul_f256(t2, P2->x, t4);
  	mul_f256(t5, P1->z, t4);
  	mul_f256(t4, P2->y, t5);
  
  	/*
  	 * Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
  	 * We need to test whether r is zero, so we will do some extra
  	 * reduce.
  	 */
  	sub_f256(t2, t2, t1);
  	sub_f256(t4, t4, t3);
  	reduce_final_f256(t4);
  	ret = 0;
  	for (i = 0; i < 9; i ++) {
  		ret |= t4[i];
  	}
  	ret = (ret | -ret) >> 31;
  
  	/*
  	 * Compute u1*h^2 (in t6) and h^3 (in t5);
  	 */
  	square_f256(t7, t2);
  	mul_f256(t6, t1, t7);
  	mul_f256(t5, t7, t2);
  
  	/*
  	 * Compute x3 = r^2 - h^3 - 2*u1*h^2.
  	 */
  	square_f256(P1->x, t4);
  	sub_f256(P1->x, P1->x, t5);
  	sub_f256(P1->x, P1->x, t6);
  	sub_f256(P1->x, P1->x, t6);
  
  	/*
  	 * Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
  	 */
  	sub_f256(t6, t6, P1->x);
  	mul_f256(P1->y, t4, t6);
  	mul_f256(t1, t5, t3);
  	sub_f256(P1->y, P1->y, t1);
  
  	/*
  	 * Compute z3 = h*z1*z2.
  	 */
  	mul_f256(t1, P1->z, P2->z);
  	mul_f256(P1->z, t1, t2);
  
  	return ret;
  }
  
  /*
   * Add point P2 to point P1. This is a specialised function for the
   * case when P2 is a non-zero point in affine coordinate.
   *
   * This function computes the wrong result in the following cases:
   *
   *   - If P1 == 0
   *   - If P1 == P2
   *
   * In both cases, P1 is set to the point at infinity.
   *
   * Returned value is 0 if one of the following occurs:
   *
   *   - P1 and P2 have the same Y coordinate
   *   - The Y coordinate of P2 is 0 and P1 is the point at infinity.
   *
   * The second case cannot actually happen with valid points, since a point
   * with Y == 0 is a point of order 2, and there is no point of order 2 on
   * curve P-256.
   *
   * Therefore, assuming that P1 != 0 on input, then the caller
   * can apply the following:
   *
   *   - If the result is not the point at infinity, then it is correct.
   *   - Otherwise, if the returned value is 1, then this is a case of
   *     P1+P2 == 0, so the result is indeed the point at infinity.
   *   - Otherwise, P1 == P2, so a "double" operation should have been
   *     performed.
   */
  static uint32_t
  p256_add_mixed(p256_jacobian *P1, const p256_jacobian *P2)
  {
  	/*
  	 * Addtions formulas are:
  	 *
  	 *   u1 = x1
  	 *   u2 = x2 * z1^2
  	 *   s1 = y1
  	 *   s2 = y2 * z1^3
  	 *   h = u2 - u1
  	 *   r = s2 - s1
  	 *   x3 = r^2 - h^3 - 2 * u1 * h^2
  	 *   y3 = r * (u1 * h^2 - x3) - s1 * h^3
  	 *   z3 = h * z1
  	 */
  	uint32_t t1[9], t2[9], t3[9], t4[9], t5[9], t6[9], t7[9];
  	uint32_t ret;
  	int i;
  
  	/*
  	 * Compute u1 = x1 (in t1) and s1 = y1 (in t3).
  	 */
  	memcpy(t1, P1->x, sizeof t1);
  	memcpy(t3, P1->y, sizeof t3);
  
  	/*
  	 * Compute u2 = x2*z1^2 (in t2) and s2 = y2*z1^3 (in t4).
  	 */
  	square_f256(t4, P1->z);
  	mul_f256(t2, P2->x, t4);
  	mul_f256(t5, P1->z, t4);
  	mul_f256(t4, P2->y, t5);
  
  	/*
  	 * Compute h = h2 - u1 (in t2) and r = s2 - s1 (in t4).
  	 * We need to test whether r is zero, so we will do some extra
  	 * reduce.
  	 */
  	sub_f256(t2, t2, t1);
  	sub_f256(t4, t4, t3);
  	reduce_final_f256(t4);
  	ret = 0;
  	for (i = 0; i < 9; i ++) {
  		ret |= t4[i];
  	}
  	ret = (ret | -ret) >> 31;
  
  	/*
  	 * Compute u1*h^2 (in t6) and h^3 (in t5);
  	 */
  	square_f256(t7, t2);
  	mul_f256(t6, t1, t7);
  	mul_f256(t5, t7, t2);
  
  	/*
  	 * Compute x3 = r^2 - h^3 - 2*u1*h^2.
  	 */
  	square_f256(P1->x, t4);
  	sub_f256(P1->x, P1->x, t5);
  	sub_f256(P1->x, P1->x, t6);
  	sub_f256(P1->x, P1->x, t6);
  
  	/*
  	 * Compute y3 = r*(u1*h^2 - x3) - s1*h^3.
  	 */
  	sub_f256(t6, t6, P1->x);
  	mul_f256(P1->y, t4, t6);
  	mul_f256(t1, t5, t3);
  	sub_f256(P1->y, P1->y, t1);
  
  	/*
  	 * Compute z3 = h*z1*z2.
  	 */
  	mul_f256(P1->z, P1->z, t2);
  
  	return ret;
  }
  
  /*
   * Decode a P-256 point. This function does not support the point at
   * infinity. Returned value is 0 if the point is invalid, 1 otherwise.
   */
  static uint32_t
  p256_decode(p256_jacobian *P, const void *src, size_t len)
  {
  	const unsigned char *buf;
  	uint32_t tx[9], ty[9], t1[9], t2[9];
  	uint32_t bad;
  	int i;
  
  	if (len != 65) {
  		return 0;
  	}
  	buf = src;
  
  	/*
  	 * First byte must be 0x04 (uncompressed format). We could support
  	 * "hybrid format" (first byte is 0x06 or 0x07, and encodes the
  	 * least significant bit of the Y coordinate), but it is explicitly
  	 * forbidden by RFC 5480 (section 2.2).
  	 */
  	bad = NEQ(buf[0], 0x04);
  
  	/*
  	 * Decode the coordinates, and check that they are both lower
  	 * than the modulus.
  	 */
  	tx[8] = be8_to_le30(tx, buf + 1, 32);
  	ty[8] = be8_to_le30(ty, buf + 33, 32);
  	bad |= reduce_final_f256(tx);
  	bad |= reduce_final_f256(ty);
  
  	/*
  	 * Check curve equation.
  	 */
  	square_f256(t1, tx);
  	mul_f256(t1, tx, t1);
  	square_f256(t2, ty);
  	sub_f256(t1, t1, tx);
  	sub_f256(t1, t1, tx);
  	sub_f256(t1, t1, tx);
  	add_f256(t1, t1, P256_B);
  	sub_f256(t1, t1, t2);
  	reduce_final_f256(t1);
  	for (i = 0; i < 9; i ++) {
  		bad |= t1[i];
  	}
  
  	/*
  	 * Copy coordinates to the point structure.
  	 */
  	memcpy(P->x, tx, sizeof tx);
  	memcpy(P->y, ty, sizeof ty);
  	memset(P->z, 0, sizeof P->z);
  	P->z[0] = 1;
  	return NEQ(bad, 0) ^ 1;
  }
  
  /*
   * Encode a point into a buffer. This function assumes that the point is
   * valid, in affine coordinates, and not the point at infinity.
   */
  static void
  p256_encode(void *dst, const p256_jacobian *P)
  {
  	unsigned char *buf;
  
  	buf = dst;
  	buf[0] = 0x04;
  	le30_to_be8(buf + 1, 32, P->x);
  	le30_to_be8(buf + 33, 32, P->y);
  }
  
  /*
   * Multiply a curve point by an integer. The integer is assumed to be
   * lower than the curve order, and the base point must not be the point
   * at infinity.
   */
  static void
  p256_mul(p256_jacobian *P, const unsigned char *x, size_t xlen)
  {
  	/*
  	 * qz is a flag that is initially 1, and remains equal to 1
  	 * as long as the point is the point at infinity.
  	 *
  	 * We use a 2-bit window to handle multiplier bits by pairs.
  	 * The precomputed window really is the points P2 and P3.
  	 */
  	uint32_t qz;
  	p256_jacobian P2, P3, Q, T, U;
  
  	/*
  	 * Compute window values.
  	 */
  	P2 = *P;
  	p256_double(&P2);
  	P3 = *P;
  	p256_add(&P3, &P2);
  
  	/*
  	 * We start with Q = 0. We process multiplier bits 2 by 2.
  	 */
  	memset(&Q, 0, sizeof Q);
  	qz = 1;
  	while (xlen -- > 0) {
  		int k;
  
  		for (k = 6; k >= 0; k -= 2) {
  			uint32_t bits;
  			uint32_t bnz;
  
  			p256_double(&Q);
  			p256_double(&Q);
  			T = *P;
  			U = Q;
  			bits = (*x >> k) & (uint32_t)3;
  			bnz = NEQ(bits, 0);
  			CCOPY(EQ(bits, 2), &T, &P2, sizeof T);
  			CCOPY(EQ(bits, 3), &T, &P3, sizeof T);
  			p256_add(&U, &T);
  			CCOPY(bnz & qz, &Q, &T, sizeof Q);
  			CCOPY(bnz & ~qz, &Q, &U, sizeof Q);
  			qz &= ~bnz;
  		}
  		x ++;
  	}
  	*P = Q;
  }
  
  /*
   * Precomputed window: k*G points, where G is the curve generator, and k
   * is an integer from 1 to 15 (inclusive). The X and Y coordinates of
   * the point are encoded as 9 words of 30 bits each (little-endian
   * order).
   */
  static const uint32_t Gwin[15][18] = {
  
  	{ 0x1898C296, 0x1284E517, 0x1EB33A0F, 0x00DF604B,
  	  0x2440F277, 0x339B958E, 0x04247F8B, 0x347CB84B,
  	  0x00006B17, 0x37BF51F5, 0x2ED901A0, 0x3315ECEC,
  	  0x338CD5DA, 0x0F9E162B, 0x1FAD29F0, 0x27F9B8EE,
  	  0x10B8BF86, 0x00004FE3 },
  
  	{ 0x07669978, 0x182D23F1, 0x3F21B35A, 0x225A789D,
  	  0x351AC3C0, 0x08E00C12, 0x34F7E8A5, 0x1EC62340,
  	  0x00007CF2, 0x227873D1, 0x3812DE74, 0x0E982299,
  	  0x1F6B798F, 0x3430DBBA, 0x366B1A7D, 0x2D040293,
  	  0x154436E3, 0x00000777 },
  
  	{ 0x06E7FD6C, 0x2D05986F, 0x3ADA985F, 0x31ADC87B,
  	  0x0BF165E6, 0x1FBE5475, 0x30A44C8F, 0x3934698C,
  	  0x00005ECB, 0x227D5032, 0x29E6C49E, 0x04FB83D9,
  	  0x0AAC0D8E, 0x24A2ECD8, 0x2C1B3869, 0x0FF7E374,
  	  0x19031266, 0x00008734 },
  
  	{ 0x2B030852, 0x024C0911, 0x05596EF5, 0x07F8B6DE,
  	  0x262BD003, 0x3779967B, 0x08FBBA02, 0x128D4CB4,
  	  0x0000E253, 0x184ED8C6, 0x310B08FC, 0x30EE0055,
  	  0x3F25B0FC, 0x062D764E, 0x3FB97F6A, 0x33CC719D,
  	  0x15D69318, 0x0000E0F1 },
  
  	{ 0x03D033ED, 0x05552837, 0x35BE5242, 0x2320BF47,
  	  0x268FDFEF, 0x13215821, 0x140D2D78, 0x02DE9454,
  	  0x00005159, 0x3DA16DA4, 0x0742ED13, 0x0D80888D,
  	  0x004BC035, 0x0A79260D, 0x06FCDAFE, 0x2727D8AE,
  	  0x1F6A2412, 0x0000E0C1 },
  
  	{ 0x3C2291A9, 0x1AC2ABA4, 0x3B215B4C, 0x131D037A,
  	  0x17DDE302, 0x0C90B2E2, 0x0602C92D, 0x05CA9DA9,
  	  0x0000B01A, 0x0FC77FE2, 0x35F1214E, 0x07E16BDF,
  	  0x003DDC07, 0x2703791C, 0x3038B7EE, 0x3DAD56FE,
  	  0x041D0C8D, 0x0000E85C },
  
  	{ 0x3187B2A3, 0x0018A1C0, 0x00FEF5B3, 0x3E7E2E2A,
  	  0x01FB607E, 0x2CC199F0, 0x37B4625B, 0x0EDBE82F,
  	  0x00008E53, 0x01F400B4, 0x15786A1B, 0x3041B21C,
  	  0x31CD8CF2, 0x35900053, 0x1A7E0E9B, 0x318366D0,
  	  0x076F780C, 0x000073EB },
  
  	{ 0x1B6FB393, 0x13767707, 0x3CE97DBB, 0x348E2603,
  	  0x354CADC1, 0x09D0B4EA, 0x1B053404, 0x1DE76FBA,
  	  0x000062D9, 0x0F09957E, 0x295029A8, 0x3E76A78D,
  	  0x3B547DAE, 0x27CEE0A2, 0x0575DC45, 0x1D8244FF,
  	  0x332F647A, 0x0000AD5A },
  
  	{ 0x10949EE0, 0x1E7A292E, 0x06DF8B3D, 0x02B2E30B,
  	  0x31F8729E, 0x24E35475, 0x30B71878, 0x35EDBFB7,
  	  0x0000EA68, 0x0DD048FA, 0x21688929, 0x0DE823FE,
  	  0x1C53FAA9, 0x0EA0C84D, 0x052A592A, 0x1FCE7870,
  	  0x11325CB2, 0x00002A27 },
  
  	{ 0x04C5723F, 0x30D81A50, 0x048306E4, 0x329B11C7,
  	  0x223FB545, 0x085347A8, 0x2993E591, 0x1B5ACA8E,
  	  0x0000CEF6, 0x04AF0773, 0x28D2EEA9, 0x2751EEEC,
  	  0x037B4A7F, 0x3B4C1059, 0x08F37674, 0x2AE906E1,
  	  0x18A88A6A, 0x00008786 },
  
  	{ 0x34BC21D1, 0x0CCE474D, 0x15048BF4, 0x1D0BB409,
  	  0x021CDA16, 0x20DE76C3, 0x34C59063, 0x04EDE20E,
  	  0x00003ED1, 0x282A3740, 0x0BE3BBF3, 0x29889DAE,
  	  0x03413697, 0x34C68A09, 0x210EBE93, 0x0C8A224C,
  	  0x0826B331, 0x00009099 },
  
  	{ 0x0624E3C4, 0x140317BA, 0x2F82C99D, 0x260C0A2C,
  	  0x25D55179, 0x194DCC83, 0x3D95E462, 0x356F6A05,
  	  0x0000741D, 0x0D4481D3, 0x2657FC8B, 0x1BA5CA71,
  	  0x3AE44B0D, 0x07B1548E, 0x0E0D5522, 0x05FDC567,
  	  0x2D1AA70E, 0x00000770 },
  
  	{ 0x06072C01, 0x23857675, 0x1EAD58A9, 0x0B8A12D9,
  	  0x1EE2FC79, 0x0177CB61, 0x0495A618, 0x20DEB82B,
  	  0x0000177C, 0x2FC7BFD8, 0x310EEF8B, 0x1FB4DF39,
  	  0x3B8530E8, 0x0F4E7226, 0x0246B6D0, 0x2A558A24,
  	  0x163353AF, 0x000063BB },
  
  	{ 0x24D2920B, 0x1C249DCC, 0x2069C5E5, 0x09AB2F9E,
  	  0x36DF3CF1, 0x1991FD0C, 0x062B97A7, 0x1E80070E,
  	  0x000054E7, 0x20D0B375, 0x2E9F20BD, 0x35090081,
  	  0x1C7A9DDC, 0x22E7C371, 0x087E3016, 0x03175421,
  	  0x3C6ECA7D, 0x0000F599 },
  
  	{ 0x259B9D5F, 0x0D9A318F, 0x23A0EF16, 0x00EBE4B7,
  	  0x088265AE, 0x2CDE2666, 0x2BAE7ADF, 0x1371A5C6,
  	  0x0000F045, 0x0D034F36, 0x1F967378, 0x1B5FA3F4,
  	  0x0EC8739D, 0x1643E62A, 0x1653947E, 0x22D1F4E6,
  	  0x0FB8D64B, 0x0000B5B9 }
  };
  
  /*
   * Lookup one of the Gwin[] values, by index. This is constant-time.
   */
  static void
  lookup_Gwin(p256_jacobian *T, uint32_t idx)
  {
  	uint32_t xy[18];
  	uint32_t k;
  	size_t u;
  
  	memset(xy, 0, sizeof xy);
  	for (k = 0; k < 15; k ++) {
  		uint32_t m;
  
  		m = -EQ(idx, k + 1);
  		for (u = 0; u < 18; u ++) {
  			xy[u] |= m & Gwin[k][u];
  		}
  	}
  	memcpy(T->x, &xy[0], sizeof T->x);
  	memcpy(T->y, &xy[9], sizeof T->y);
  	memset(T->z, 0, sizeof T->z);
  	T->z[0] = 1;
  }
  
  /*
   * Multiply the generator by an integer. The integer is assumed non-zero
   * and lower than the curve order.
   */
  static void
  p256_mulgen(p256_jacobian *P, const unsigned char *x, size_t xlen)
  {
  	/*
  	 * qz is a flag that is initially 1, and remains equal to 1
  	 * as long as the point is the point at infinity.
  	 *
  	 * We use a 4-bit window to handle multiplier bits by groups
  	 * of 4. The precomputed window is constant static data, with
  	 * points in affine coordinates; we use a constant-time lookup.
  	 */
  	p256_jacobian Q;
  	uint32_t qz;
  
  	memset(&Q, 0, sizeof Q);
  	qz = 1;
  	while (xlen -- > 0) {
  		int k;
  		unsigned bx;
  
  		bx = *x ++;
  		for (k = 0; k < 2; k ++) {
  			uint32_t bits;
  			uint32_t bnz;
  			p256_jacobian T, U;
  
  			p256_double(&Q);
  			p256_double(&Q);
  			p256_double(&Q);
  			p256_double(&Q);
  			bits = (bx >> 4) & 0x0F;
  			bnz = NEQ(bits, 0);
  			lookup_Gwin(&T, bits);
  			U = Q;
  			p256_add_mixed(&U, &T);
  			CCOPY(bnz & qz, &Q, &T, sizeof Q);
  			CCOPY(bnz & ~qz, &Q, &U, sizeof Q);
  			qz &= ~bnz;
  			bx <<= 4;
  		}
  	}
  	*P = Q;
  }
  
  static const unsigned char P256_G[] = {
  	0x04, 0x6B, 0x17, 0xD1, 0xF2, 0xE1, 0x2C, 0x42, 0x47, 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, 0x8E, 0xE7, 0xEB, 0x4A, 0x7C, 0x0F, 0x9E, 0x16, 0x2B,
  	0xCE, 0x33, 0x57, 0x6B, 0x31, 0x5E, 0xCE, 0xCB, 0xB6, 0x40,
  	0x68, 0x37, 0xBF, 0x51, 0xF5
  };
  
  static const unsigned char P256_N[] = {
  	0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, 0xFF, 0xFF,
  	0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xBC, 0xE6, 0xFA, 0xAD,
  	0xA7, 0x17, 0x9E, 0x84, 0xF3, 0xB9, 0xCA, 0xC2, 0xFC, 0x63,
  	0x25, 0x51
  };
  
  static const unsigned char *
  api_generator(int curve, size_t *len)
  {
  	(void)curve;
  	*len = sizeof P256_G;
  	return P256_G;
  }
  
  static const unsigned char *
  api_order(int curve, size_t *len)
  {
  	(void)curve;
  	*len = sizeof P256_N;
  	return P256_N;
  }
  
  static size_t
  api_xoff(int curve, size_t *len)
  {
  	(void)curve;
  	*len = 32;
  	return 1;
  }
  
  static uint32_t
  api_mul(unsigned char *G, size_t Glen,
  	const unsigned char *x, size_t xlen, int curve)
  {
  	uint32_t r;
  	p256_jacobian P;
  
  	(void)curve;
  	r = p256_decode(&P, G, Glen);
  	p256_mul(&P, x, xlen);
  	if (Glen >= 65) {
  		p256_to_affine(&P);
  		p256_encode(G, &P);
  	}
  	return r;
  }
  
  static size_t
  api_mulgen(unsigned char *R,
  	const unsigned char *x, size_t xlen, int curve)
  {
  	p256_jacobian P;
  
  	(void)curve;
  	p256_mulgen(&P, x, xlen);
  	p256_to_affine(&P);
  	p256_encode(R, &P);
  	return 65;
  
  	/*
  	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)
  {
  	p256_jacobian P, Q;
  	uint32_t r, t, z;
  	int i;
  
  	(void)curve;
  	r = p256_decode(&P, A, len);
  	p256_mul(&P, x, xlen);
  	if (B == NULL) {
  		p256_mulgen(&Q, y, ylen);
  	} else {
  		r &= p256_decode(&Q, B, len);
  		p256_mul(&Q, y, ylen);
  	}
  
  	/*
  	 * The final addition may fail in case both points are equal.
  	 */
  	t = p256_add(&P, &Q);
  	reduce_final_f256(P.z);
  	z = 0;
  	for (i = 0; i < 9; i ++) {
  		z |= P.z[i];
  	}
  	z = EQ(z, 0);
  	p256_double(&Q);
  
  	/*
  	 * If z is 1 then either P+Q = 0 (t = 1) or P = Q (t = 0). So we
  	 * have the following:
  	 *
  	 *   z = 0, t = 0   return P (normal addition)
  	 *   z = 0, t = 1   return P (normal addition)
  	 *   z = 1, t = 0   return Q (a 'double' case)
  	 *   z = 1, t = 1   report an error (P+Q = 0)
  	 */
  	CCOPY(z & ~t, &P, &Q, sizeof Q);
  	p256_to_affine(&P);
  	p256_encode(A, &P);
  	r &= ~(z & t);
  	return r;
  }
  
  /* see bearssl_ec.h */
  const br_ec_impl br_ec_p256_m31 = {
  	(uint32_t)0x00800000,
  	&api_generator,
  	&api_order,
  	&api_xoff,
  	&api_mul,
  	&api_mulgen,
  	&api_muladd
  };
  