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 <assert.h>
  #include <stdint.h>
  #include <stdlib.h>
  #include <string.h>
  
  #include <TrezorCrypto/address.h>
  #include <TrezorCrypto/base58.h>
  #include <TrezorCrypto/bignum.h>
  #include <TrezorCrypto/ecdsa.h>
  #include <TrezorCrypto/hmac.h>
  #include <TrezorCrypto/memzero.h>
  #include <TrezorCrypto/rand.h>
  #include <TrezorCrypto/rfc6979.h>
  #include <TrezorCrypto/secp256k1.h>
  #include <TrezorCrypto/schnorr.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 = {0}, inv = {0}, xr = {0}, yr = {0};
  
    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;
    }
  
    // lambda = (y2 - y1) / (x2 - x1)
    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 = {0}, xr = {0}, yr = {0};
  
    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_fast_mod(&lambda, &curve->prime);
    bn_mod(&lambda, &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));
  }
  
  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 = 0;
      for (i = 0; i < 8; i++) {
        k->val[i] = random32() & ((1u << BN_BITS_PER_LIMB) - 1);
      }
      k->val[8] = random32() & ((1u << BN_BITS_LAST_LIMB) - 1);
      // 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 = {0}, h = {0}, r2 = {0};
    bignum256 hcby = {0}, hsqx = {0};
    bignum256 xz = {0}, yz = {0}, az = {0};
    int is_doubling = 0;
    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 = {0}, m = {0}, msq = {0}, ysq = {0}, xysq = {0};
    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 = 0, j = 0;
    CONFIDENTIAL bignum256 a;
    uint32_t *aptr = NULL;
    uint32_t abits = 0;
    int ashift = 0;
    uint32_t is_even = (k->val[0] & 1) - 1;
    uint32_t bits = {0}, sign = {0}, nsign = {0};
    CONFIDENTIAL jacobian_curve_point jres;
    curve_point pmult[8] = {0};
    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 += (BN_BASE - 1) + k->val[j] - (curve->order.val[j] & is_even);
      a.val[j] = tmp & (BN_BASE - 1);
      tmp >>= BN_BITS_PER_LIMB;
    }
    is_non_zero |= k->val[j];
    a.val[j] = tmp + 0xffffff + 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 = 256 - (BN_BITS_PER_LIMB * 8) - 4;
    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 += BN_BITS_PER_LIMB;
        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.
      bn_cnegate((sign ^ nsign) & 1, &jres.z, prime);
  
      // add odd factor
      point_jacobian_add(&pmult[bits >> 1], &jres, curve);
      sign = nsign;
    }
    bn_cnegate(sign & 1, &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 = {0}, j = {0};
    CONFIDENTIAL bignum256 a;
    uint32_t is_even = (k->val[0] & 1) - 1;
    uint32_t lowbits = 0;
    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 += (BN_BASE - 1) + k->val[j] - (curve->order.val[j] & is_even);
      a.val[j] = tmp & (BN_BASE - 1);
      tmp >>= BN_BITS_PER_LIMB;
    }
    is_non_zero |= k->val[j];
    a.val[j] = tmp + 0xffffff + 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) << (BN_BITS_PER_LIMB - 4));
      }
      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.
      bn_cnegate(~lowbits & 1, &jres.y, prime);
  
      // add odd factor
      point_jacobian_add(&curve->cp[i][lowbits >> 1], &jres, curve);
    }
    bn_cnegate(~(a.val[0] >> 4) & 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 = {0};
    if (!ecdsa_read_pubkey(curve, pub_key, &point)) {
      return 1;
    }
  
    bignum256 k = {0};
    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] = {0};
    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 = 0;
    curve_point R = {0};
    bignum256 k = {0}, z = {0}, randk = {0};
    bignum256 *s = &R.y;
    uint8_t by;  // signature recovery byte
  
  #if USE_RFC6979
    rfc6979_state rng = {0};
    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 = {0};
    bignum256 k = {0};
  
    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 = {0};
    bignum256 k = {0};
  
    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 = {0};
  
    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] = {0};
    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] = {0};
    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] = {0};
    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] = {0};
    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] = {0};
    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 compress_coords(const curve_point *cp, uint8_t *compressed) {
    compressed[0] = bn_is_odd(&cp->y) ? 0x03 : 0x02;
    bn_write_be(&cp->x, compressed + 1);
  }
  
  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.
  // We assume that all curves using this code have cofactor 1, so there is no
  // need to verify that pub is a scalar multiple of G.
  int ecdsa_validate_pubkey(const ecdsa_curve *curve, const curve_point *pub) {
    bignum256 y_2 = {0}, x3_ax_b = {0};
  
    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] = {0};
    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 = {0}, s = {0}, e = {0};
    curve_point cp = {0}, cp2 = {0};
  
    // 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_mod(&e, &curve->order);
    bn_subtract(&curve->order, &e, &e);
    // r = r^-1
    bn_inverse(&r, &curve->order);
    // e = -digest * r^-1
    bn_multiply(&r, &e, &curve->order);
    bn_mod(&e, &curve->order);
    // s = s * r^-1
    bn_multiply(&r, &s, &curve->order);
    bn_mod(&s, &curve->order);
    // cp = s * r^-1 * k * G
    point_multiply(curve, &s, &cp, &cp);
    // cp2 = -digest * r^-1 * G
    scalar_multiply(curve, &e, &cp2);
    // cp = (s * r^-1 * k - digest * r^-1) * G = Pub
    point_add(curve, &cp2, &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 = {0}, res = {0};
    bignum256 r = {0}, s = {0}, z = {0};
    int result = 0;
  
    if (!ecdsa_read_pubkey(curve, pub_key, &pub)) {
      result = 1;
    }
  
    if (result == 0) {
      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))) {
        result = 2;
      }
      if (bn_is_zero(&z)) {
        // The digest was all-zero. The probability of this happening by chance is
        // infinitesimal, but it could be induced by a fault injection. In this
        // case the signature (r,s) can be forged by taking r := (t * Q).x mod n
        // and s := r * t^-1 mod n for any t in [1, n-1]. We fail verification,
        // because there is no guarantee that the signature was created by the
        // owner of the private key.
        result = 3;
      }
    }
  
    if (result == 0) {
      bn_inverse(&s, &curve->order);       // s = s^-1
      bn_multiply(&s, &z, &curve->order);  // z = z * s  [u1 = z * s^-1 mod n]
      bn_mod(&z, &curve->order);
    }
  
    if (result == 0) {
      bn_multiply(&r, &s, &curve->order);  // s = r * s  [u2 = r * s^-1 mod n]
      bn_mod(&s, &curve->order);
      scalar_multiply(curve, &z, &res);       // res = z * G    [= u1 * G]
      point_multiply(curve, &s, &pub, &pub);  // pub = s * pub  [= u2 * Q]
      point_add(curve, &pub, &res);  // res = pub + res  [R = u1 * G + u2 * Q]
      if (point_is_infinity(&res)) {
        // R == Infinity
        result = 4;
      }
    }
  
    if (result == 0) {
      bn_mod(&(res.x), &curve->order);
      if (!bn_is_equal(&res.x, &r)) {
        // R.x != r
        // signature does not match
        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 = 0;
    uint8_t *p = der, *len = NULL, *len1 = NULL, *len2 = NULL;
    *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;
  }
  
  // Parse a DER-encoded signature. We don't check whether the encoded integers
  // satisfy DER requirements regarding leading zeros.
  int ecdsa_sig_from_der(const uint8_t *der, size_t der_len, uint8_t sig[64]) {
    memzero(sig, 64);
  
    // Check sequence header.
    if (der_len < 2 || der_len > 72 || der[0] != 0x30 || der[1] != der_len - 2) {
      return 1;
    }
  
    // Read two DER-encoded integers.
    size_t pos = 2;
    for (int i = 0; i < 2; ++i) {
      // Check integer header.
      if (der_len < pos + 2 || der[pos] != 0x02) {
        return 1;
      }
  
      // Locate the integer.
      size_t int_len = der[pos + 1];
      pos += 2;
      if (pos + int_len > der_len) {
        return 1;
      }
  
      // Skip a possible leading zero.
      if (int_len != 0 && der[pos] == 0) {
        int_len--;
        pos++;
      }
  
      // Copy the integer to the output, making sure it fits.
      if (int_len > 32) {
        return 1;
      }
      memcpy(sig + 32 * (i + 1) - int_len, der + pos, int_len);
  
      // Move on to the next one.
      pos += int_len;
    }
  
    // Check that there are no trailing elements in the sequence.
    if (pos != der_len) {
      return 1;
    }
  
    return 0;
  }
  
  // [wallet-core]
  int zil_schnorr_sign(const ecdsa_curve *curve, const uint8_t *priv_key, const uint8_t *msg, const uint32_t msg_len, uint8_t *sig)
  {
  	int i;
  	bignum256 k;
  
  	uint8_t hash[32];
  	sha256_Raw(msg, msg_len, hash);
  
  	rfc6979_state rng;
  	init_rfc6979(priv_key, hash, &rng);
  
  	for (i = 0; i < 10000; i++) {
  		// 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;
  		}
  
  		schnorr_sign_pair sign;
  		if (schnorr_sign(curve, priv_key, &k, msg, msg_len, &sign) != 0) {
  			continue;
  		}
  
  		// we're done
  		memcpy(sig, sign.r, 32);
  		memcpy(sig + 32, sign.s, 32);
  
  		memzero(&k, sizeof(k));
  		memzero(&rng, sizeof(rng));
  		memzero(&sign, sizeof(sign));
  		return 0;
  	}
  
  	// Too many retries without a valid signature
  	// -> fail with an error
  	memzero(&k, sizeof(k));
  	memzero(&rng, sizeof(rng));
  	return -1;
  }
  
  // [wallet-core]
  int zil_schnorr_verify(const ecdsa_curve *curve, const uint8_t *pub_key, const uint8_t *sig, const uint8_t *msg, const uint32_t msg_len)
  {
  	schnorr_sign_pair sign;
    
  	memcpy(sign.r, sig, 32);
  	memcpy(sign.s, sig + 32, 32);
  
  	return schnorr_verify(curve, pub_key, msg, msg_len, &sign);
  }
  