text: /**
   * Copyright (c) 2013-2014 Tomas Dzetkulic
   * Copyright (c) 2013-2014 Pavol Rusnak
   * Copyright (c)      2015 Jochen Hoenicke
   *
   * 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 <stdint.h>
  #include <stdlib.h>
  #include <string.h>
  #include <assert.h>
  
  #include "address.h"
  #include "bignum.h"
  #include "rand.h"
  #include "hmac.h"
  #include "ecdsa.h"
  #include "base58.h"
  #include "secp256k1.h"
  #include "rfc6979.h"
  #include "memzero.h"
  
  // Set cp2 = cp1
  void point_copy(const curve_point *cp1, curve_point *cp2)
  {
  	*cp2 = *cp1;
  }
  
  // cp2 = cp1 + cp2
  void point_add(const ecdsa_curve *curve, const curve_point *cp1, curve_point *cp2)
  {
  	bignum256 lambda, inv, xr, yr;
  
  	if (point_is_infinity(cp1)) {
  		return;
  	}
  	if (point_is_infinity(cp2)) {
  		point_copy(cp1, cp2);
  		return;
  	}
  	if (point_is_equal(cp1, cp2)) {
  		point_double(curve, cp2);
  		return;
  	}
  	if (point_is_negative_of(cp1, cp2)) {
  		point_set_infinity(cp2);
  		return;
  	}
  
  	bn_subtractmod(&(cp2->x), &(cp1->x), &inv, &curve->prime);
  	bn_inverse(&inv, &curve->prime);
  	bn_subtractmod(&(cp2->y), &(cp1->y), &lambda, &curve->prime);
  	bn_multiply(&inv, &lambda, &curve->prime);
  
  	// xr = lambda^2 - x1 - x2
  	xr = lambda;
  	bn_multiply(&xr, &xr, &curve->prime);
  	yr = cp1->x;
  	bn_addmod(&yr, &(cp2->x), &curve->prime);
  	bn_subtractmod(&xr, &yr, &xr, &curve->prime);
  	bn_fast_mod(&xr, &curve->prime);
  	bn_mod(&xr, &curve->prime);
  
  	// yr = lambda (x1 - xr) - y1
  	bn_subtractmod(&(cp1->x), &xr, &yr, &curve->prime);
  	bn_multiply(&lambda, &yr, &curve->prime);
  	bn_subtractmod(&yr, &(cp1->y), &yr, &curve->prime);
  	bn_fast_mod(&yr, &curve->prime);
  	bn_mod(&yr, &curve->prime);
  
  	cp2->x = xr;
  	cp2->y = yr;
  }
  
  // cp = cp + cp
  void point_double(const ecdsa_curve *curve, curve_point *cp)
  {
  	bignum256 lambda, xr, yr;
  
  	if (point_is_infinity(cp)) {
  		return;
  	}
  	if (bn_is_zero(&(cp->y))) {
  		point_set_infinity(cp);
  		return;
  	}
  
  	// lambda = (3 x^2 + a) / (2 y)
  	lambda = cp->y;
  	bn_mult_k(&lambda, 2, &curve->prime);
  	bn_inverse(&lambda, &curve->prime);
  
  	xr = cp->x;
  	bn_multiply(&xr, &xr, &curve->prime);
  	bn_mult_k(&xr, 3, &curve->prime);
  	bn_subi(&xr, -curve->a, &curve->prime);
  	bn_multiply(&xr, &lambda, &curve->prime);
  
  	// xr = lambda^2 - 2*x
  	xr = lambda;
  	bn_multiply(&xr, &xr, &curve->prime);
  	yr = cp->x;
  	bn_lshift(&yr);
  	bn_subtractmod(&xr, &yr, &xr, &curve->prime);
  	bn_fast_mod(&xr, &curve->prime);
  	bn_mod(&xr, &curve->prime);
  
  	// yr = lambda (x - xr) - y
  	bn_subtractmod(&(cp->x), &xr, &yr, &curve->prime);
  	bn_multiply(&lambda, &yr, &curve->prime);
  	bn_subtractmod(&yr, &(cp->y), &yr, &curve->prime);
  	bn_fast_mod(&yr, &curve->prime);
  	bn_mod(&yr, &curve->prime);
  
  	cp->x = xr;
  	cp->y = yr;
  }
  
  // set point to internal representation of point at infinity
  void point_set_infinity(curve_point *p)
  {
  	bn_zero(&(p->x));
  	bn_zero(&(p->y));
  }
  
  // return true iff p represent point at infinity
  // both coords are zero in internal representation
  int point_is_infinity(const curve_point *p)
  {
  	return bn_is_zero(&(p->x)) && bn_is_zero(&(p->y));
  }
  
  // return true iff both points are equal
  int point_is_equal(const curve_point *p, const curve_point *q)
  {
  	return bn_is_equal(&(p->x), &(q->x)) && bn_is_equal(&(p->y), &(q->y));
  }
  
  // returns true iff p == -q
  // expects p and q be valid points on curve other than point at infinity
  int point_is_negative_of(const curve_point *p, const curve_point *q)
  {
  	// if P == (x, y), then -P would be (x, -y) on this curve
  	if (!bn_is_equal(&(p->x), &(q->x))) {
  		return 0;
  	}
  
  	// we shouldn't hit this for a valid point
  	if (bn_is_zero(&(p->y))) {
  		return 0;
  	}
  
  	return !bn_is_equal(&(p->y), &(q->y));
  }
  
  // Negate a (modulo prime) if cond is 0xffffffff, keep it if cond is 0.
  // The timing of this function does not depend on cond.
  void conditional_negate(uint32_t cond, bignum256 *a, const bignum256 *prime)
  {
  	int j;
  	uint32_t tmp = 1;
  	assert(a->val[8] < 0x20000);
  	for (j = 0; j < 8; j++) {
  		tmp += 0x3fffffff + 2*prime->val[j] - a->val[j];
  		a->val[j] = ((tmp & 0x3fffffff) & cond) | (a->val[j] & ~cond);
  		tmp >>= 30;
  	}
  	tmp += 0x3fffffff + 2*prime->val[j] - a->val[j];
  	a->val[j] = ((tmp & 0x3fffffff) & cond) | (a->val[j] & ~cond);
  	assert(a->val[8] < 0x20000);
  }
  
  typedef struct jacobian_curve_point {
  	bignum256 x, y, z;
  } jacobian_curve_point;
  
  // generate random K for signing/side-channel noise
  static void generate_k_random(bignum256 *k, const bignum256 *prime) {
  	do {
  		int i;
  		for (i = 0; i < 8; i++) {
  			k->val[i] = random32() & 0x3FFFFFFF;
  		}
  		k->val[8] = random32() & 0xFFFF;
  		// check that k is in range and not zero.
  	} while (bn_is_zero(k) || !bn_is_less(k, prime));
  }
  
  void curve_to_jacobian(const curve_point *p, jacobian_curve_point *jp, const bignum256 *prime) {
  	// randomize z coordinate
  	generate_k_random(&jp->z, prime);
  
  	jp->x = jp->z;
  	bn_multiply(&jp->z, &jp->x, prime);
  	// x = z^2
  	jp->y = jp->x;
  	bn_multiply(&jp->z, &jp->y, prime);
  	// y = z^3
  
  	bn_multiply(&p->x, &jp->x, prime);
  	bn_multiply(&p->y, &jp->y, prime);
  }
  
  void jacobian_to_curve(const jacobian_curve_point *jp, curve_point *p, const bignum256 *prime) {
  	p->y = jp->z;
  	bn_inverse(&p->y, prime);
  	// p->y = z^-1
  	p->x = p->y;
  	bn_multiply(&p->x, &p->x, prime);
  	// p->x = z^-2
  	bn_multiply(&p->x, &p->y, prime);
  	// p->y = z^-3
  	bn_multiply(&jp->x, &p->x, prime);
  	// p->x = jp->x * z^-2
  	bn_multiply(&jp->y, &p->y, prime);
  	// p->y = jp->y * z^-3
  	bn_mod(&p->x, prime);
  	bn_mod(&p->y, prime);
  }
  
  void point_jacobian_add(const curve_point *p1, jacobian_curve_point *p2, const ecdsa_curve *curve) {
  	bignum256 r, h, r2;
  	bignum256 hcby, hsqx;
  	bignum256 xz, yz, az;
  	int is_doubling;
  	const bignum256 *prime = &curve->prime;
  	int a = curve->a;
  
  	assert (-3 <= a && a <= 0);
  
  	/* First we bring p1 to the same denominator:
  	 * x1' := x1 * z2^2
  	 * y1' := y1 * z2^3
  	 */
  	/*
  	 * lambda  = ((y1' - y2)/z2^3) / ((x1' - x2)/z2^2)
  	 *         = (y1' - y2) / (x1' - x2) z2
  	 * x3/z3^2 = lambda^2 - (x1' + x2)/z2^2
  	 * y3/z3^3 = 1/2 lambda * (2x3/z3^2 - (x1' + x2)/z2^2) + (y1'+y2)/z2^3
  	 *
  	 * For the special case x1=x2, y1=y2 (doubling) we have
  	 * lambda = 3/2 ((x2/z2^2)^2 + a) / (y2/z2^3)
  	 *        = 3/2 (x2^2 + a*z2^4) / y2*z2)
  	 *
  	 * to get rid of fraction we write lambda as
  	 * lambda = r / (h*z2)
  	 * with  r = is_doubling ? 3/2 x2^2 + az2^4 : (y1 - y2)
  	 *       h = is_doubling ?      y1+y2       : (x1 - x2)
  	 *
  	 * With z3 = h*z2  (the denominator of lambda)
  	 * we get x3 = lambda^2*z3^2 - (x1' + x2)/z2^2*z3^2
  	 *           = r^2 - h^2 * (x1' + x2)
  	 *    and y3 = 1/2 r * (2x3 - h^2*(x1' + x2)) + h^3*(y1' + y2)
  	 */
  
  	/* h = x1 - x2
  	 * r = y1 - y2
  	 * x3 = r^2 - h^3 - 2*h^2*x2
  	 * y3 = r*(h^2*x2 - x3) - h^3*y2
  	 * z3 = h*z2
  	 */
  
  	xz = p2->z;
  	bn_multiply(&xz, &xz, prime); // xz = z2^2
  	yz = p2->z;
  	bn_multiply(&xz, &yz, prime); // yz = z2^3
  	
  	if (a != 0) {
  		az  = xz;
  		bn_multiply(&az, &az, prime);   // az = z2^4
  		bn_mult_k(&az, -a, prime);      // az = -az2^4
  	}
  	
  	bn_multiply(&p1->x, &xz, prime);        // xz = x1' = x1*z2^2;
  	h = xz;
  	bn_subtractmod(&h, &p2->x, &h, prime);
  	bn_fast_mod(&h, prime);
  	// h = x1' - x2;
  
  	bn_add(&xz, &p2->x);
  	// xz = x1' + x2
  
  	// check for h == 0 % prime.  Note that h never normalizes to
  	// zero, since h = x1' + 2*prime - x2 > 0 and a positive
  	// multiple of prime is always normalized to prime by
  	// bn_fast_mod.
  	is_doubling = bn_is_equal(&h, prime);
  
  	bn_multiply(&p1->y, &yz, prime);        // yz = y1' = y1*z2^3;
  	bn_subtractmod(&yz, &p2->y, &r, prime);
  	// r = y1' - y2;
  
  	bn_add(&yz, &p2->y);
  	// yz = y1' + y2
  
  	r2 = p2->x;
  	bn_multiply(&r2, &r2, prime);
  	bn_mult_k(&r2, 3, prime);
  	
  	if (a != 0) {
  		// subtract -a z2^4, i.e, add a z2^4
  		bn_subtractmod(&r2, &az, &r2, prime);
  	}
  	bn_cmov(&r, is_doubling, &r2, &r);
  	bn_cmov(&h, is_doubling, &yz, &h);
  	
  
  	// hsqx = h^2
  	hsqx = h;
  	bn_multiply(&hsqx, &hsqx, prime);
  
  	// hcby = h^3
  	hcby = h;
  	bn_multiply(&hsqx, &hcby, prime);
  
  	// hsqx = h^2 * (x1 + x2)
  	bn_multiply(&xz, &hsqx, prime);
  
  	// hcby = h^3 * (y1 + y2)
  	bn_multiply(&yz, &hcby, prime);
  
  	// z3 = h*z2
  	bn_multiply(&h, &p2->z, prime);
  
  	// x3 = r^2 - h^2 (x1 + x2)
  	p2->x = r;
  	bn_multiply(&p2->x, &p2->x, prime);
  	bn_subtractmod(&p2->x, &hsqx, &p2->x, prime);
  	bn_fast_mod(&p2->x, prime);
  
  	// y3 = 1/2 (r*(h^2 (x1 + x2) - 2x3) - h^3 (y1 + y2))
  	bn_subtractmod(&hsqx, &p2->x, &p2->y, prime);
  	bn_subtractmod(&p2->y, &p2->x, &p2->y, prime);
  	bn_multiply(&r, &p2->y, prime);
  	bn_subtractmod(&p2->y, &hcby, &p2->y, prime);
  	bn_mult_half(&p2->y, prime);
  	bn_fast_mod(&p2->y, prime);
  }
  
  void point_jacobian_double(jacobian_curve_point *p, const ecdsa_curve *curve) {
  	bignum256 az4, m, msq, ysq, xysq;
  	const bignum256 *prime = &curve->prime;
  
  	assert (-3 <= curve->a && curve->a <= 0);
  	/* usual algorithm:
  	 *
  	 * lambda  = (3((x/z^2)^2 + a) / 2y/z^3) = (3x^2 + az^4)/2yz
  	 * x3/z3^2 = lambda^2 - 2x/z^2
  	 * y3/z3^3 = lambda * (x/z^2 - x3/z3^2) - y/z^3
  	 *
  	 * to get rid of fraction we set
  	 *  m = (3 x^2 + az^4) / 2
  	 * Hence,
  	 *  lambda = m / yz = m / z3
  	 *
  	 * With z3 = yz  (the denominator of lambda)
  	 * we get x3 = lambda^2*z3^2 - 2*x/z^2*z3^2
  	 *           = m^2 - 2*xy^2
  	 *    and y3 = (lambda * (x/z^2 - x3/z3^2) - y/z^3) * z3^3
  	 *           = m * (xy^2 - x3) - y^4
  	 */
  
  	/* m = (3*x^2 + a z^4) / 2
  	 * x3 = m^2 - 2*xy^2
  	 * y3 = m*(xy^2 - x3) - 8y^4
  	 * z3 = y*z
  	 */
  
  	m = p->x;
  	bn_multiply(&m, &m, prime);
  	bn_mult_k(&m, 3, prime);
  
  	az4 = p->z;
  	bn_multiply(&az4, &az4, prime);
  	bn_multiply(&az4, &az4, prime);
  	bn_mult_k(&az4, -curve->a, prime);
  	bn_subtractmod(&m, &az4, &m, prime);
  	bn_mult_half(&m, prime);
  
  	// msq = m^2
  	msq = m;
  	bn_multiply(&msq, &msq, prime);
  	// ysq = y^2
  	ysq = p->y;
  	bn_multiply(&ysq, &ysq, prime);
  	// xysq = xy^2
  	xysq = p->x;
  	bn_multiply(&ysq, &xysq, prime);
  
  	// z3 = yz
  	bn_multiply(&p->y, &p->z, prime);
  
  	// x3 = m^2 - 2*xy^2
  	p->x = xysq;
  	bn_lshift(&p->x);
  	bn_fast_mod(&p->x, prime);
  	bn_subtractmod(&msq, &p->x, &p->x, prime);
  	bn_fast_mod(&p->x, prime);
  
  	// y3 = m*(xy^2 - x3) - y^4
  	bn_subtractmod(&xysq, &p->x, &p->y, prime);
  	bn_multiply(&m, &p->y, prime);
  	bn_multiply(&ysq, &ysq, prime);
  	bn_subtractmod(&p->y, &ysq, &p->y, prime);
  	bn_fast_mod(&p->y, prime);
  }
  
  // res = k * p
  void point_multiply(const ecdsa_curve *curve, const bignum256 *k, const curve_point *p, curve_point *res)
  {
  	// this algorithm is loosely based on
  	//  Katsuyuki Okeya and Tsuyoshi Takagi, The Width-w NAF Method Provides
  	//  Small Memory and Fast Elliptic Scalar Multiplications Secure against
  	//  Side Channel Attacks.
  	assert (bn_is_less(k, &curve->order));
  
  	int i, j;
  	static CONFIDENTIAL bignum256 a;
  	uint32_t *aptr;
  	uint32_t abits;
  	int ashift;
  	uint32_t is_even = (k->val[0] & 1) - 1;
  	uint32_t bits, sign, nsign;
  	static CONFIDENTIAL jacobian_curve_point jres;
  	curve_point pmult[8];
  	const bignum256 *prime = &curve->prime;
  
  	// is_even = 0xffffffff if k is even, 0 otherwise.
  
  	// add 2^256.
  	// make number odd: subtract curve->order if even
  	uint32_t tmp = 1;
  	uint32_t is_non_zero = 0;
  	for (j = 0; j < 8; j++) {
  		is_non_zero |= k->val[j];
  		tmp += 0x3fffffff + k->val[j] - (curve->order.val[j] & is_even);
  		a.val[j] = tmp & 0x3fffffff;
  		tmp >>= 30;
  	}
  	is_non_zero |= k->val[j];
  	a.val[j] = tmp + 0xffff + k->val[j] - (curve->order.val[j] & is_even);
  	assert((a.val[0] & 1) != 0);
  
  	// special case 0*p:  just return zero. We don't care about constant time.
  	if (!is_non_zero) {
  		point_set_infinity(res);
  		return;
  	}
  
  	// Now a = k + 2^256 (mod curve->order) and a is odd.
  	//
  	// The idea is to bring the new a into the form.
  	// sum_{i=0..64} a[i] 16^i,  where |a[i]| < 16 and a[i] is odd.
  	// a[0] is odd, since a is odd.  If a[i] would be even, we can
  	// add 1 to it and subtract 16 from a[i-1].  Afterwards,
  	// a[64] = 1, which is the 2^256 that we added before.
  	//
  	// Since k = a - 2^256 (mod curve->order), we can compute
  	//   k*p = sum_{i=0..63} a[i] 16^i * p
  	//
  	// We compute |a[i]| * p in advance for all possible
  	// values of |a[i]| * p.  pmult[i] = (2*i+1) * p
  	// We compute p, 3*p, ..., 15*p and store it in the table pmult.
  	// store p^2 temporarily in pmult[7]
  	pmult[7] = *p;
  	point_double(curve, &pmult[7]);
  	// compute 3*p, etc by repeatedly adding p^2.
  	pmult[0] = *p;
  	for (i = 1; i < 8; i++) {
  		pmult[i] = pmult[7];
  		point_add(curve, &pmult[i-1], &pmult[i]);
  	}
  
  	// now compute  res = sum_{i=0..63} a[i] * 16^i * p step by step,
  	// starting with i = 63.
  	// initialize jres = |a[63]| * p.
  	// Note that a[i] = a>>(4*i) & 0xf if (a&0x10) != 0
  	// and - (16 - (a>>(4*i) & 0xf)) otherwise.   We can compute this as
  	//   ((a ^ (((a >> 4) & 1) - 1)) & 0xf) >> 1
  	// since a is odd.
  	aptr = &a.val[8];
  	abits = *aptr;
  	ashift = 12;
  	bits = abits >> ashift;
  	sign = (bits >> 4) - 1;
  	bits ^= sign;
  	bits &= 15;
  	curve_to_jacobian(&pmult[bits>>1], &jres, prime);
  	for (i = 62; i >= 0; i--) {
  		// sign = sign(a[i+1])  (0xffffffff for negative, 0 for positive)
  		// invariant jres = (-1)^sign sum_{j=i+1..63} (a[j] * 16^{j-i-1} * p)
  		// abits >> (ashift - 4) = lowbits(a >> (i*4))
  
  		point_jacobian_double(&jres, curve);
  		point_jacobian_double(&jres, curve);
  		point_jacobian_double(&jres, curve);
  		point_jacobian_double(&jres, curve);
  
  		// get lowest 5 bits of a >> (i*4).
  		ashift -= 4;
  		if (ashift < 0) {
  			// the condition only depends on the iteration number and
  			// leaks no private information to a side-channel.
  			bits = abits << (-ashift);
  			abits = *(--aptr);
  			ashift += 30;
  			bits |= abits >> ashift;
  		} else {
  			bits = abits >> ashift;
  		}
  		bits &= 31;
  		nsign = (bits >> 4) - 1;
  		bits ^= nsign;
  		bits &= 15;
  
  		// negate last result to make signs of this round and the
  		// last round equal.
  		conditional_negate(sign ^ nsign, &jres.z, prime);
  
  		// add odd factor
  		point_jacobian_add(&pmult[bits >> 1], &jres, curve);
  		sign = nsign;
  	}
  	conditional_negate(sign, &jres.z, prime);
  	jacobian_to_curve(&jres, res, prime);
  	memzero(&a, sizeof(a));
  	memzero(&jres, sizeof(jres));
  }
  
  #if USE_PRECOMPUTED_CP
  
  // res = k * G
  // k must be a normalized number with 0 <= k < curve->order
  void scalar_multiply(const ecdsa_curve *curve, const bignum256 *k, curve_point *res)
  {
  	assert (bn_is_less(k, &curve->order));
  
  	int i, j;
  	static CONFIDENTIAL bignum256 a;
  	uint32_t is_even = (k->val[0] & 1) - 1;
  	uint32_t lowbits;
  	static CONFIDENTIAL jacobian_curve_point jres;
  	const bignum256 *prime = &curve->prime;
  
  	// is_even = 0xffffffff if k is even, 0 otherwise.
  
  	// add 2^256.
  	// make number odd: subtract curve->order if even
  	uint32_t tmp = 1;
  	uint32_t is_non_zero = 0;
  	for (j = 0; j < 8; j++) {
  		is_non_zero |= k->val[j];
  		tmp += 0x3fffffff + k->val[j] - (curve->order.val[j] & is_even);
  		a.val[j] = tmp & 0x3fffffff;
  		tmp >>= 30;
  	}
  	is_non_zero |= k->val[j];
  	a.val[j] = tmp + 0xffff + k->val[j] - (curve->order.val[j] & is_even);
  	assert((a.val[0] & 1) != 0);
  
  	// special case 0*G:  just return zero. We don't care about constant time.
  	if (!is_non_zero) {
  		point_set_infinity(res);
  		return;
  	}
  
  	// Now a = k + 2^256 (mod curve->order) and a is odd.
  	//
  	// The idea is to bring the new a into the form.
  	// sum_{i=0..64} a[i] 16^i,  where |a[i]| < 16 and a[i] is odd.
  	// a[0] is odd, since a is odd.  If a[i] would be even, we can
  	// add 1 to it and subtract 16 from a[i-1].  Afterwards,
  	// a[64] = 1, which is the 2^256 that we added before.
  	//
  	// Since k = a - 2^256 (mod curve->order), we can compute
  	//   k*G = sum_{i=0..63} a[i] 16^i * G
  	//
  	// We have a big table curve->cp that stores all possible
  	// values of |a[i]| 16^i * G.
  	// curve->cp[i][j] = (2*j+1) * 16^i * G
  
  	// now compute  res = sum_{i=0..63} a[i] * 16^i * G step by step.
  	// initial res = |a[0]| * G.  Note that a[0] = a & 0xf if (a&0x10) != 0
  	// and - (16 - (a & 0xf)) otherwise.   We can compute this as
  	//   ((a ^ (((a >> 4) & 1) - 1)) & 0xf) >> 1
  	// since a is odd.
  	lowbits = a.val[0] & ((1 << 5) - 1);
  	lowbits ^= (lowbits >> 4) - 1;
  	lowbits &= 15;
  	curve_to_jacobian(&curve->cp[0][lowbits >> 1], &jres, prime);
  	for (i = 1; i < 64; i ++) {
  		// invariant res = sign(a[i-1]) sum_{j=0..i-1} (a[j] * 16^j * G)
  
  		// shift a by 4 places.
  		for (j = 0; j < 8; j++) {
  			a.val[j] = (a.val[j] >> 4) | ((a.val[j + 1] & 0xf) << 26);
  		}
  		a.val[j] >>= 4;
  		// a = old(a)>>(4*i)
  		// a is even iff sign(a[i-1]) = -1
  
  		lowbits = a.val[0] & ((1 << 5) - 1);
  		lowbits ^= (lowbits >> 4) - 1;
  		lowbits &= 15;
  		// negate last result to make signs of this round and the
  		// last round equal.
  		conditional_negate((lowbits & 1) - 1, &jres.y, prime);
  
  		// add odd factor
  		point_jacobian_add(&curve->cp[i][lowbits >> 1], &jres, curve);
  	}
  	conditional_negate(((a.val[0] >> 4) & 1) - 1, &jres.y, prime);
  	jacobian_to_curve(&jres, res, prime);
  	memzero(&a, sizeof(a));
  	memzero(&jres, sizeof(jres));
  }
  
  #else
  
  void scalar_multiply(const ecdsa_curve *curve, const bignum256 *k, curve_point *res)
  {
  	point_multiply(curve, k, &curve->G, res);
  }
  
  #endif
  
  int ecdh_multiply(const ecdsa_curve *curve, const uint8_t *priv_key, const uint8_t *pub_key, uint8_t *session_key)
  {
  	curve_point point;
  	if (!ecdsa_read_pubkey(curve, pub_key, &point)) {
  		return 1;
  	}
  
  	bignum256 k;
  	bn_read_be(priv_key, &k);
  	point_multiply(curve, &k, &point, &point);
  	memzero(&k, sizeof(k));
  
  	session_key[0] = 0x04;
  	bn_write_be(&point.x, session_key + 1);
  	bn_write_be(&point.y, session_key + 33);
  	memzero(&point, sizeof(point));
  
  	return 0;
  }
  
  
  // msg is a data to be signed
  // msg_len is the message length
  int ecdsa_sign(const ecdsa_curve *curve, HasherType hasher_sign, const uint8_t *priv_key, const uint8_t *msg, uint32_t msg_len, uint8_t *sig, uint8_t *pby, int (*is_canonical)(uint8_t by, uint8_t sig[64]))
  {
  	uint8_t hash[32];
  	hasher_Raw(hasher_sign, msg, msg_len, hash);
  	int res = ecdsa_sign_digest(curve, priv_key, hash, sig, pby, is_canonical);
  	memzero(hash, sizeof(hash));
  	return res;
  
  }
  
  // uses secp256k1 curve
  // priv_key is a 32 byte big endian stored number
  // sig is 64 bytes long array for the signature
  // digest is 32 bytes of digest
  // is_canonical is an optional function that checks if the signature
  // conforms to additional coin-specific rules.
  int ecdsa_sign_digest(const ecdsa_curve *curve, const uint8_t *priv_key, const uint8_t *digest, uint8_t *sig, uint8_t *pby, int (*is_canonical)(uint8_t by, uint8_t sig[64]))
  {
  	int i;
  	curve_point R;
  	bignum256 k, z, randk;
  	bignum256 *s = &R.y;
  	uint8_t by; // signature recovery byte
  
  #if USE_RFC6979
  	rfc6979_state rng;
  	init_rfc6979(priv_key, digest, &rng);
  #endif
  
  	bn_read_be(digest, &z);
  
  	for (i = 0; i < 10000; i++) {
  
  #if USE_RFC6979
  		// generate K deterministically
  		generate_k_rfc6979(&k, &rng);
  		// if k is too big or too small, we don't like it
  		if (bn_is_zero(&k) || !bn_is_less(&k, &curve->order)) {
  			continue;
  		}
  #else
  		// generate random number k
  		generate_k_random(&k, &curve->order);
  #endif
  
  		// compute k*G
  		scalar_multiply(curve, &k, &R);
  		by = R.y.val[0] & 1;
  		// r = (rx mod n)
  		if (!bn_is_less(&R.x, &curve->order)) {
  			bn_subtract(&R.x, &curve->order, &R.x);
  			by |= 2;
  		}
  		// if r is zero, we retry
  		if (bn_is_zero(&R.x)) {
  			continue;
  		}
  
  		// randomize operations to counter side-channel attacks
  		generate_k_random(&randk, &curve->order);
  		bn_multiply(&randk, &k, &curve->order); // k*rand
  		bn_inverse(&k, &curve->order);         // (k*rand)^-1
  		bn_read_be(priv_key, s);               // priv
  		bn_multiply(&R.x, s, &curve->order);   // R.x*priv
  		bn_add(s, &z);                         // R.x*priv + z
  		bn_multiply(&k, s, &curve->order);     // (k*rand)^-1 (R.x*priv + z)
  		bn_multiply(&randk, s, &curve->order);  // k^-1 (R.x*priv + z)
  		bn_mod(s, &curve->order);
  		// if s is zero, we retry
  		if (bn_is_zero(s)) {
  			continue;
  		}
  
  		// if S > order/2 => S = -S
  		if (bn_is_less(&curve->order_half, s)) {
  			bn_subtract(&curve->order, s, s);
  			by ^= 1;
  		}
  		// we are done, R.x and s is the result signature
  		bn_write_be(&R.x, sig);
  		bn_write_be(s, sig + 32);
  
  		// check if the signature is acceptable or retry
  		if (is_canonical && !is_canonical(by, sig)) {
  			continue;
  		}
  
  		if (pby) {
  			*pby = by;
  		}
  
  		memzero(&k, sizeof(k));
  		memzero(&randk, sizeof(randk));
  #if USE_RFC6979
  		memzero(&rng, sizeof(rng));
  #endif
  		return 0;
  	}
  
  	// Too many retries without a valid signature
  	// -> fail with an error
  	memzero(&k, sizeof(k));
  	memzero(&randk, sizeof(randk));
  #if USE_RFC6979
  	memzero(&rng, sizeof(rng));
  #endif
  	return -1;
  }
  
  void ecdsa_get_public_key33(const ecdsa_curve *curve, const uint8_t *priv_key, uint8_t *pub_key)
  {
  	curve_point R;
  	bignum256 k;
  
  	bn_read_be(priv_key, &k);
  	// compute k*G
  	scalar_multiply(curve, &k, &R);
  	pub_key[0] = 0x02 | (R.y.val[0] & 0x01);
  	bn_write_be(&R.x, pub_key + 1);
  	memzero(&R, sizeof(R));
  	memzero(&k, sizeof(k));
  }
  
  void ecdsa_get_public_key65(const ecdsa_curve *curve, const uint8_t *priv_key, uint8_t *pub_key)
  {
  	curve_point R;
  	bignum256 k;
  
  	bn_read_be(priv_key, &k);
  	// compute k*G
  	scalar_multiply(curve, &k, &R);
  	pub_key[0] = 0x04;
  	bn_write_be(&R.x, pub_key + 1);
  	bn_write_be(&R.y, pub_key + 33);
  	memzero(&R, sizeof(R));
  	memzero(&k, sizeof(k));
  }
  
  int ecdsa_uncompress_pubkey(const ecdsa_curve *curve, const uint8_t *pub_key, uint8_t *uncompressed)
  {
  	curve_point pub;
  
  	if (!ecdsa_read_pubkey(curve, pub_key, &pub)) {
  		return 0;
  	}
  
  	uncompressed[0] = 4;
  	bn_write_be(&pub.x, uncompressed + 1);
  	bn_write_be(&pub.y, uncompressed + 33);
  
  	return 1;
  }
  
  void ecdsa_get_pubkeyhash(const uint8_t *pub_key, HasherType hasher_pubkey, uint8_t *pubkeyhash)
  {
  	uint8_t h[HASHER_DIGEST_LENGTH];
  	if (pub_key[0] == 0x04) {  // uncompressed format
  		hasher_Raw(hasher_pubkey, pub_key, 65, h);
  	} else if (pub_key[0] == 0x00) { // point at infinity
  		hasher_Raw(hasher_pubkey, pub_key,  1, h);
  	} else { // expecting compressed format
  		hasher_Raw(hasher_pubkey, pub_key, 33, h);
  	}
  	memcpy(pubkeyhash, h, 20);
  	memzero(h, sizeof(h));
  }
  
  void ecdsa_get_address_raw(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, uint8_t *addr_raw)
  {
  	size_t prefix_len = address_prefix_bytes_len(version);
  	address_write_prefix_bytes(version, addr_raw);
  	ecdsa_get_pubkeyhash(pub_key, hasher_pubkey, addr_raw + prefix_len);
  }
  
  void ecdsa_get_address(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, HasherType hasher_base58, char *addr, int addrsize)
  {
  	uint8_t raw[MAX_ADDR_RAW_SIZE];
  	size_t prefix_len = address_prefix_bytes_len(version);
  	ecdsa_get_address_raw(pub_key, version, hasher_pubkey, raw);
  	base58_encode_check(raw, 20 + prefix_len, hasher_base58, addr, addrsize);
  	// not as important to clear this one, but we might as well
  	memzero(raw, sizeof(raw));
  }
  
  void ecdsa_get_address_segwit_p2sh_raw(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, uint8_t *addr_raw)
  {
  	uint8_t buf[32 + 2];
  	buf[0] = 0; // version byte
  	buf[1] = 20; // push 20 bytes
  	ecdsa_get_pubkeyhash(pub_key, hasher_pubkey, buf + 2);
  	size_t prefix_len = address_prefix_bytes_len(version);
  	address_write_prefix_bytes(version, addr_raw);
  	hasher_Raw(hasher_pubkey, buf, 22, addr_raw + prefix_len);
  }
  
  void ecdsa_get_address_segwit_p2sh(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, HasherType hasher_base58, char *addr, int addrsize)
  {
  	uint8_t raw[MAX_ADDR_RAW_SIZE];
  	size_t prefix_len = address_prefix_bytes_len(version);
  	ecdsa_get_address_segwit_p2sh_raw(pub_key, version, hasher_pubkey, raw);
  	base58_encode_check(raw, prefix_len + 20, hasher_base58, addr, addrsize);
  	memzero(raw, sizeof(raw));
  }
  
  void ecdsa_get_wif(const uint8_t *priv_key, uint32_t version, HasherType hasher_base58, char *wif, int wifsize)
  {
  	uint8_t wif_raw[MAX_WIF_RAW_SIZE];
  	size_t prefix_len = address_prefix_bytes_len(version);
  	address_write_prefix_bytes(version, wif_raw);
  	memcpy(wif_raw + prefix_len, priv_key, 32);
  	wif_raw[prefix_len + 32] = 0x01;
  	base58_encode_check(wif_raw, prefix_len + 32 + 1, hasher_base58, wif, wifsize);
  	// private keys running around our stack can cause trouble
  	memzero(wif_raw, sizeof(wif_raw));
  }
  
  int ecdsa_address_decode(const char *addr, uint32_t version, HasherType hasher_base58, uint8_t *out)
  {
  	if (!addr) return 0;
  	int prefix_len = address_prefix_bytes_len(version);
  	return base58_decode_check(addr, hasher_base58, out, 20 + prefix_len) == 20 + prefix_len
  		&& address_check_prefix(out, version);
  }
  
  void uncompress_coords(const ecdsa_curve *curve, uint8_t odd, const bignum256 *x, bignum256 *y)
  {
  	// y^2 = x^3 + a*x + b
  	memcpy(y, x, sizeof(bignum256));         // y is x
  	bn_multiply(x, y, &curve->prime);        // y is x^2
  	bn_subi(y, -curve->a, &curve->prime);    // y is x^2 + a
  	bn_multiply(x, y, &curve->prime);        // y is x^3 + ax
  	bn_add(y, &curve->b);                    // y is x^3 + ax + b
  	bn_sqrt(y, &curve->prime);               // y = sqrt(y)
  	if ((odd & 0x01) != (y->val[0] & 1)) {
  		bn_subtract(&curve->prime, y, y);   // y = -y
  	}
  }
  
  int ecdsa_read_pubkey(const ecdsa_curve *curve, const uint8_t *pub_key, curve_point *pub)
  {
  	if (!curve) {
  		curve = &secp256k1;
  	}
  	if (pub_key[0] == 0x04) {
  		bn_read_be(pub_key + 1, &(pub->x));
  		bn_read_be(pub_key + 33, &(pub->y));
  		return ecdsa_validate_pubkey(curve, pub);
  	}
  	if (pub_key[0] == 0x02 || pub_key[0] == 0x03) { // compute missing y coords
  		bn_read_be(pub_key + 1, &(pub->x));
  		uncompress_coords(curve, pub_key[0], &(pub->x), &(pub->y));
  		return ecdsa_validate_pubkey(curve, pub);
  	}
  	// error
  	return 0;
  }
  
  // Verifies that:
  //   - pub is not the point at infinity.
  //   - pub->x and pub->y are in range [0,p-1].
  //   - pub is on the curve.
  
  int ecdsa_validate_pubkey(const ecdsa_curve *curve, const curve_point *pub)
  {
  	bignum256 y_2, x3_ax_b;
  
  	if (point_is_infinity(pub)) {
  		return 0;
  	}
  
  	if (!bn_is_less(&(pub->x), &curve->prime) || !bn_is_less(&(pub->y), &curve->prime)) {
  		return 0;
  	}
  
  	memcpy(&y_2, &(pub->y), sizeof(bignum256));
  	memcpy(&x3_ax_b, &(pub->x), sizeof(bignum256));
  
  	// y^2
  	bn_multiply(&(pub->y), &y_2, &curve->prime);
  	bn_mod(&y_2, &curve->prime);
  
  	// x^3 + ax + b
  	bn_multiply(&(pub->x), &x3_ax_b, &curve->prime);  // x^2
  	bn_subi(&x3_ax_b, -curve->a, &curve->prime);      // x^2 + a
  	bn_multiply(&(pub->x), &x3_ax_b, &curve->prime);  // x^3 + ax
  	bn_addmod(&x3_ax_b, &curve->b, &curve->prime);    // x^3 + ax + b
  	bn_mod(&x3_ax_b, &curve->prime);
  
  	if (!bn_is_equal(&x3_ax_b, &y_2)) {
  		return 0;
  	}
  
  	return 1;
  }
  
  // uses secp256k1 curve
  // pub_key - 65 bytes uncompressed key
  // signature - 64 bytes signature
  // msg is a data that was signed
  // msg_len is the message length
  
  int ecdsa_verify(const ecdsa_curve *curve, HasherType hasher_sign, const uint8_t *pub_key, const uint8_t *sig, const uint8_t *msg, uint32_t msg_len)
  {
  	uint8_t hash[32];
  	hasher_Raw(hasher_sign, msg, msg_len, hash);
  	int res = ecdsa_verify_digest(curve, pub_key, sig, hash);
  	memzero(hash, sizeof(hash));
  	return res;
  }
  
  // Compute public key from signature and recovery id.
  // returns 0 if the key is successfully recovered
  int ecdsa_recover_pub_from_sig (const ecdsa_curve *curve, uint8_t *pub_key, const uint8_t *sig, const uint8_t *digest, int recid)
  {
  	bignum256 r, s, e;
  	curve_point cp, cp2;
  
  	// read r and s
  	bn_read_be(sig, &r);
  	bn_read_be(sig + 32, &s);
  	if (!bn_is_less(&r, &curve->order) || bn_is_zero(&r)) {
  		return 1;
  	}
  	if (!bn_is_less(&s, &curve->order) || bn_is_zero(&s)) {
  		return 1;
  	}
  	// cp = R = k * G (k is secret nonce when signing)
  	memcpy(&cp.x, &r, sizeof(bignum256));
  	if (recid & 2) {
  		bn_add(&cp.x, &curve->order);
  		if (!bn_is_less(&cp.x, &curve->prime)) {
  			return 1;
  		}
  	}
  	// compute y from x
  	uncompress_coords(curve, recid & 1, &cp.x, &cp.y);
  	if (!ecdsa_validate_pubkey(curve, &cp)) {
  		return 1;
  	}
  	// e = -digest
  	bn_read_be(digest, &e);
  	bn_subtractmod(&curve->order, &e, &e, &curve->order);
  	bn_fast_mod(&e, &curve->order);
  	bn_mod(&e, &curve->order);
  	// r := r^-1
  	bn_inverse(&r, &curve->order);
  	// cp := s * R = s * k *G
  	point_multiply(curve, &s, &cp, &cp);
  	// cp2 := -digest * G
  	scalar_multiply(curve, &e, &cp2);
  	// cp := (s * k - digest) * G = (r*priv) * G = r * Pub
  	point_add(curve, &cp2, &cp);
  	// cp := r^{-1} * r * Pub = Pub
  	point_multiply(curve, &r, &cp, &cp);
  	pub_key[0] = 0x04;
  	bn_write_be(&cp.x, pub_key + 1);
  	bn_write_be(&cp.y, pub_key + 33);
  	return 0;
  }
  
  // returns 0 if verification succeeded
  int ecdsa_verify_digest(const ecdsa_curve *curve, const uint8_t *pub_key, const uint8_t *sig, const uint8_t *digest)
  {
  	curve_point pub, res;
  	bignum256 r, s, z;
  
  	if (!ecdsa_read_pubkey(curve, pub_key, &pub)) {
  		return 1;
  	}
  
  	bn_read_be(sig, &r);
  	bn_read_be(sig + 32, &s);
  
  	bn_read_be(digest, &z);
  
  	if (bn_is_zero(&r) || bn_is_zero(&s) ||
  		(!bn_is_less(&r, &curve->order)) ||
  		(!bn_is_less(&s, &curve->order))) return 2;
  
  	bn_inverse(&s, &curve->order); // s^-1
  	bn_multiply(&s, &z, &curve->order); // z*s^-1
  	bn_mod(&z, &curve->order);
  	bn_multiply(&r, &s, &curve->order); // r*s^-1
  	bn_mod(&s, &curve->order);
  
  	int result = 0;
  	if (bn_is_zero(&z)) {
  		// our message hashes to zero
  		// I don't expect this to happen any time soon
  		result = 3;
  	} else {
  		scalar_multiply(curve, &z, &res);
  	}
  
  	if (result == 0) {
  		// both pub and res can be infinity, can have y = 0 OR can be equal -> false negative
  		point_multiply(curve, &s, &pub, &pub);
  		point_add(curve, &pub, &res);
  		bn_mod(&(res.x), &curve->order);
  		// signature does not match
  		if (!bn_is_equal(&res.x, &r)) {
  			result = 5;
  		}
  	}
  
  	memzero(&pub, sizeof(pub));
  	memzero(&res, sizeof(res));
  	memzero(&r, sizeof(r));
  	memzero(&s, sizeof(s));
  	memzero(&z, sizeof(z));
  
  	// all OK
  	return result;
  }
  
  int ecdsa_sig_to_der(const uint8_t *sig, uint8_t *der)
  {
  	int i;
  	uint8_t *p = der, *len, *len1, *len2;
  	*p = 0x30; p++;                        // sequence
  	*p = 0x00; len = p; p++;               // len(sequence)
  
  	*p = 0x02; p++;                        // integer
  	*p = 0x00; len1 = p; p++;              // len(integer)
  
  	// process R
  	i = 0;
  	while (sig[i] == 0 && i < 32) { i++; } // skip leading zeroes
  	if (sig[i] >= 0x80) { // put zero in output if MSB set
  		*p = 0x00; p++; *len1 = *len1 + 1;
  	}
  	while (i < 32) { // copy bytes to output
  		*p = sig[i]; p++; *len1 = *len1 + 1; i++;
  	}
  
  	*p = 0x02; p++;                        // integer
  	*p = 0x00; len2 = p; p++;              // len(integer)
  
  	// process S
  	i = 32;
  	while (sig[i] == 0 && i < 64) { i++; } // skip leading zeroes
  	if (sig[i] >= 0x80) { // put zero in output if MSB set
  		*p = 0x00; p++; *len2 = *len2 + 1;
  	}
  	while (i < 64) { // copy bytes to output
  		*p = sig[i]; p++; *len2 = *len2 + 1; i++;
  	}
  
  	*len = *len1 + *len2 + 4;
  	return *len + 2;
  }
  