text: /* Copyright 2008, Google Inc.
   * All rights reserved.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions are
   * met:
   *
   *     * Redistributions of source code must retain the above copyright
   * notice, this list of conditions and the following disclaimer.
   *     * Redistributions in binary form must reproduce the above
   * copyright notice, this list of conditions and the following disclaimer
   * in the documentation and/or other materials provided with the
   * distribution.
   *     * Neither the name of Google Inc. nor the names of its
   * contributors may be used to endorse or promote products derived from
   * this software without specific prior written permission.
   *
   * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
   * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
   * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
   * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
   * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
   * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
   * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
   * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
   * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
   * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
   * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
   *
   * curve25519-donna: Curve25519 elliptic curve, public key function
   *
   * http://code.google.com/p/curve25519-donna/
   *
   * Adam Langley <agl@imperialviolet.org>
   *
   * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
   *
   * More information about curve25519 can be found here
   *   http://cr.yp.to/ecdh.html
   *
   * djb's sample implementation of curve25519 is written in a special assembly
   * language called qhasm and uses the floating point registers.
   *
   * This is, almost, a clean room reimplementation from the curve25519 paper. It
   * uses many of the tricks described therein. Only the crecip function is taken
   * from the sample implementation. */
  
  #include <string.h>
  #include <stdint.h>
  #include <crypto/ecc.h>
  
  #define X25519_PUBKEY_LEN 32
  
  #ifdef _MSC_VER
  #define inline __inline
  #endif
  
  //typedef uint8_t u8;
  //typedef int32_t s32;
  typedef int64_t limb;
  
  /* Field element representation:
   *
   * Field elements are written as an array of signed, 64-bit limbs, least
   * significant first. The value of the field element is:
   *   x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
   *
   * i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
  
  /* Sum two numbers: output += in */
  static void fsum(limb *output, const limb *in) {
    unsigned i;
    for (i = 0; i < 10; i += 2) {
      output[0+i] = output[0+i] + in[0+i];
      output[1+i] = output[1+i] + in[1+i];
    }
  }
  
  /* Find the difference of two numbers: output = in - output
   * (note the order of the arguments!). */
  static void fdifference(limb *output, const limb *in) {
    unsigned i;
    for (i = 0; i < 10; ++i) {
      output[i] = in[i] - output[i];
    }
  }
  
  /* Multiply a number by a scalar: output = in * scalar */
  static void fscalar_product(limb *output, const limb *in, const limb scalar) {
    unsigned i;
    for (i = 0; i < 10; ++i) {
      output[i] = in[i] * scalar;
    }
  }
  
  /* Multiply two numbers: output = in2 * in
   *
   * output must be distinct to both inputs. The inputs are reduced coefficient
   * form, the output is not.
   *
   * output[x] <= 14 * the largest product of the input limbs. */
  static void fproduct(limb *output, const limb *in2, const limb *in) {
    output[0] =       ((limb) ((s32) in2[0])) * ((s32) in[0]);
    output[1] =       ((limb) ((s32) in2[0])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[0]);
    output[2] =  2 *  ((limb) ((s32) in2[1])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[0]);
    output[3] =       ((limb) ((s32) in2[1])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[0]);
    output[4] =       ((limb) ((s32) in2[2])) * ((s32) in[2]) +
                 2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[1])) +
                      ((limb) ((s32) in2[0])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[0]);
    output[5] =       ((limb) ((s32) in2[2])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[0]);
    output[6] =  2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[1])) +
                      ((limb) ((s32) in2[2])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[0]);
    output[7] =       ((limb) ((s32) in2[3])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[0]);
    output[8] =       ((limb) ((s32) in2[4])) * ((s32) in[4]) +
                 2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[1])) +
                      ((limb) ((s32) in2[2])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[0]);
    output[9] =       ((limb) ((s32) in2[4])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[2]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[1]) +
                      ((limb) ((s32) in2[0])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[0]);
    output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[1])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[1])) +
                      ((limb) ((s32) in2[4])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[2]);
    output[11] =      ((limb) ((s32) in2[5])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[4]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[3]) +
                      ((limb) ((s32) in2[2])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[2]);
    output[12] =      ((limb) ((s32) in2[6])) * ((s32) in[6]) +
                 2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[3])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[3])) +
                      ((limb) ((s32) in2[4])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[4]);
    output[13] =      ((limb) ((s32) in2[6])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[7])) * ((s32) in[6]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[5]) +
                      ((limb) ((s32) in2[4])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[4]);
    output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[5])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[5])) +
                      ((limb) ((s32) in2[6])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[6]);
    output[15] =      ((limb) ((s32) in2[7])) * ((s32) in[8]) +
                      ((limb) ((s32) in2[8])) * ((s32) in[7]) +
                      ((limb) ((s32) in2[6])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[6]);
    output[16] =      ((limb) ((s32) in2[8])) * ((s32) in[8]) +
                 2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[7]));
    output[17] =      ((limb) ((s32) in2[8])) * ((s32) in[9]) +
                      ((limb) ((s32) in2[9])) * ((s32) in[8]);
    output[18] = 2 *  ((limb) ((s32) in2[9])) * ((s32) in[9]);
  }
  
  /* Reduce a long form to a short form by taking the input mod 2^255 - 19.
   *
   * On entry: |output[i]| < 14*2^54
   * On exit: |output[0..8]| < 280*2^54 */
  static void freduce_degree(limb *output) {
    /* Each of these shifts and adds ends up multiplying the value by 19.
     *
     * For output[0..8], the absolute entry value is < 14*2^54 and we add, at
     * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
    output[8] += output[18] << 4;
    output[8] += output[18] << 1;
    output[8] += output[18];
    output[7] += output[17] << 4;
    output[7] += output[17] << 1;
    output[7] += output[17];
    output[6] += output[16] << 4;
    output[6] += output[16] << 1;
    output[6] += output[16];
    output[5] += output[15] << 4;
    output[5] += output[15] << 1;
    output[5] += output[15];
    output[4] += output[14] << 4;
    output[4] += output[14] << 1;
    output[4] += output[14];
    output[3] += output[13] << 4;
    output[3] += output[13] << 1;
    output[3] += output[13];
    output[2] += output[12] << 4;
    output[2] += output[12] << 1;
    output[2] += output[12];
    output[1] += output[11] << 4;
    output[1] += output[11] << 1;
    output[1] += output[11];
    output[0] += output[10] << 4;
    output[0] += output[10] << 1;
    output[0] += output[10];
  }
  
  #if (-1 & 3) != 3
  #error "This code only works on a two's complement system"
  #endif
  
  /* return v / 2^26, using only shifts and adds.
   *
   * On entry: v can take any value. */
  static inline limb
  div_by_2_26(const limb v)
  {
    /* High word of v; no shift needed. */
    const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
    /* Set to all 1s if v was negative; else set to 0s. */
    const int32_t sign = ((int32_t) highword) >> 31;
    /* Set to 0x3ffffff if v was negative; else set to 0. */
    const int32_t roundoff = ((uint32_t) sign) >> 6;
    /* Should return v / (1<<26) */
    return (v + roundoff) >> 26;
  }
  
  /* return v / (2^25), using only shifts and adds.
   *
   * On entry: v can take any value. */
  static inline limb
  div_by_2_25(const limb v)
  {
    /* High word of v; no shift needed*/
    const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
    /* Set to all 1s if v was negative; else set to 0s. */
    const int32_t sign = ((int32_t) highword) >> 31;
    /* Set to 0x1ffffff if v was negative; else set to 0. */
    const int32_t roundoff = ((uint32_t) sign) >> 7;
    /* Should return v / (1<<25) */
    return (v + roundoff) >> 25;
  }
  
  /* Reduce all coefficients of the short form input so that |x| < 2^26.
   *
   * On entry: |output[i]| < 280*2^54 */
  static void freduce_coefficients(limb *output) {
    unsigned i;
  
    output[10] = 0;
  
    for (i = 0; i < 10; i += 2) {
      limb over = div_by_2_26(output[i]);
      /* The entry condition (that |output[i]| < 280*2^54) means that over is, at
       * most, 280*2^28 in the first iteration of this loop. This is added to the
       * next limb and we can approximate the resulting bound of that limb by
       * 281*2^54. */
      output[i] -= over << 26;
      output[i+1] += over;
  
      /* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
       * 281*2^29. When this is added to the next limb, the resulting bound can
       * be approximated as 281*2^54.
       *
       * For subsequent iterations of the loop, 281*2^54 remains a conservative
       * bound and no overflow occurs. */
      over = div_by_2_25(output[i+1]);
      output[i+1] -= over << 25;
      output[i+2] += over;
    }
    /* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
    output[0] += output[10] << 4;
    output[0] += output[10] << 1;
    output[0] += output[10];
  
    output[10] = 0;
  
    /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
     * So |over| will be no more than 2^16. */
    {
      limb over = div_by_2_26(output[0]);
      output[0] -= over << 26;
      output[1] += over;
    }
  
    /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
     * bound on |output[1]| is sufficient to meet our needs. */
  }
  
  /* A helpful wrapper around fproduct: output = in * in2.
   *
   * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
   *
   * output must be distinct to both inputs. The output is reduced degree
   * (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
  static void
  fmul(limb *output, const limb *in, const limb *in2) {
    limb t[19];
    fproduct(t, in, in2);
    /* |t[i]| < 14*2^54 */
    freduce_degree(t);
    freduce_coefficients(t);
    /* |t[i]| < 2^26 */
    memcpy(output, t, sizeof(limb) * 10);
  }
  
  /* Square a number: output = in**2
   *
   * output must be distinct from the input. The inputs are reduced coefficient
   * form, the output is not.
   *
   * output[x] <= 14 * the largest product of the input limbs. */
  static void fsquare_inner(limb *output, const limb *in) {
    output[0] =       ((limb) ((s32) in[0])) * ((s32) in[0]);
    output[1] =  2 *  ((limb) ((s32) in[0])) * ((s32) in[1]);
    output[2] =  2 * (((limb) ((s32) in[1])) * ((s32) in[1]) +
                      ((limb) ((s32) in[0])) * ((s32) in[2]));
    output[3] =  2 * (((limb) ((s32) in[1])) * ((s32) in[2]) +
                      ((limb) ((s32) in[0])) * ((s32) in[3]));
    output[4] =       ((limb) ((s32) in[2])) * ((s32) in[2]) +
                 4 *  ((limb) ((s32) in[1])) * ((s32) in[3]) +
                 2 *  ((limb) ((s32) in[0])) * ((s32) in[4]);
    output[5] =  2 * (((limb) ((s32) in[2])) * ((s32) in[3]) +
                      ((limb) ((s32) in[1])) * ((s32) in[4]) +
                      ((limb) ((s32) in[0])) * ((s32) in[5]));
    output[6] =  2 * (((limb) ((s32) in[3])) * ((s32) in[3]) +
                      ((limb) ((s32) in[2])) * ((s32) in[4]) +
                      ((limb) ((s32) in[0])) * ((s32) in[6]) +
                 2 *  ((limb) ((s32) in[1])) * ((s32) in[5]));
    output[7] =  2 * (((limb) ((s32) in[3])) * ((s32) in[4]) +
                      ((limb) ((s32) in[2])) * ((s32) in[5]) +
                      ((limb) ((s32) in[1])) * ((s32) in[6]) +
                      ((limb) ((s32) in[0])) * ((s32) in[7]));
    output[8] =       ((limb) ((s32) in[4])) * ((s32) in[4]) +
                 2 * (((limb) ((s32) in[2])) * ((s32) in[6]) +
                      ((limb) ((s32) in[0])) * ((s32) in[8]) +
                 2 * (((limb) ((s32) in[1])) * ((s32) in[7]) +
                      ((limb) ((s32) in[3])) * ((s32) in[5])));
    output[9] =  2 * (((limb) ((s32) in[4])) * ((s32) in[5]) +
                      ((limb) ((s32) in[3])) * ((s32) in[6]) +
                      ((limb) ((s32) in[2])) * ((s32) in[7]) +
                      ((limb) ((s32) in[1])) * ((s32) in[8]) +
                      ((limb) ((s32) in[0])) * ((s32) in[9]));
    output[10] = 2 * (((limb) ((s32) in[5])) * ((s32) in[5]) +
                      ((limb) ((s32) in[4])) * ((s32) in[6]) +
                      ((limb) ((s32) in[2])) * ((s32) in[8]) +
                 2 * (((limb) ((s32) in[3])) * ((s32) in[7]) +
                      ((limb) ((s32) in[1])) * ((s32) in[9])));
    output[11] = 2 * (((limb) ((s32) in[5])) * ((s32) in[6]) +
                      ((limb) ((s32) in[4])) * ((s32) in[7]) +
                      ((limb) ((s32) in[3])) * ((s32) in[8]) +
                      ((limb) ((s32) in[2])) * ((s32) in[9]));
    output[12] =      ((limb) ((s32) in[6])) * ((s32) in[6]) +
                 2 * (((limb) ((s32) in[4])) * ((s32) in[8]) +
                 2 * (((limb) ((s32) in[5])) * ((s32) in[7]) +
                      ((limb) ((s32) in[3])) * ((s32) in[9])));
    output[13] = 2 * (((limb) ((s32) in[6])) * ((s32) in[7]) +
                      ((limb) ((s32) in[5])) * ((s32) in[8]) +
                      ((limb) ((s32) in[4])) * ((s32) in[9]));
    output[14] = 2 * (((limb) ((s32) in[7])) * ((s32) in[7]) +
                      ((limb) ((s32) in[6])) * ((s32) in[8]) +
                 2 *  ((limb) ((s32) in[5])) * ((s32) in[9]));
    output[15] = 2 * (((limb) ((s32) in[7])) * ((s32) in[8]) +
                      ((limb) ((s32) in[6])) * ((s32) in[9]));
    output[16] =      ((limb) ((s32) in[8])) * ((s32) in[8]) +
                 4 *  ((limb) ((s32) in[7])) * ((s32) in[9]);
    output[17] = 2 *  ((limb) ((s32) in[8])) * ((s32) in[9]);
    output[18] = 2 *  ((limb) ((s32) in[9])) * ((s32) in[9]);
  }
  
  /* fsquare sets output = in^2.
   *
   * On entry: The |in| argument is in reduced coefficients form and |in[i]| <
   * 2^27.
   *
   * On exit: The |output| argument is in reduced coefficients form (indeed, one
   * need only provide storage for 10 limbs) and |out[i]| < 2^26. */
  static void
  fsquare(limb *output, const limb *in) {
    limb t[19];
    fsquare_inner(t, in);
    /* |t[i]| < 14*2^54 because the largest product of two limbs will be <
     * 2^(27+27) and fsquare_inner adds together, at most, 14 of those
     * products. */
    freduce_degree(t);
    freduce_coefficients(t);
    /* |t[i]| < 2^26 */
    memcpy(output, t, sizeof(limb) * 10);
  }
  
  /* Take a little-endian, 32-byte number and expand it into polynomial form */
  static void
  fexpand(limb *output, const u8 *input) {
  #define F(n,start,shift,mask) \
    output[n] = ((((limb) input[start + 0]) | \
                  ((limb) input[start + 1]) << 8 | \
                  ((limb) input[start + 2]) << 16 | \
                  ((limb) input[start + 3]) << 24) >> shift) & mask;
    F(0, 0, 0, 0x3ffffff);
    F(1, 3, 2, 0x1ffffff);
    F(2, 6, 3, 0x3ffffff);
    F(3, 9, 5, 0x1ffffff);
    F(4, 12, 6, 0x3ffffff);
    F(5, 16, 0, 0x1ffffff);
    F(6, 19, 1, 0x3ffffff);
    F(7, 22, 3, 0x1ffffff);
    F(8, 25, 4, 0x3ffffff);
    F(9, 28, 6, 0x1ffffff);
  #undef F
  }
  
  #if (-32 >> 1) != -16
  #error "This code only works when >> does sign-extension on negative numbers"
  #endif
  
  /* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
  static s32 s32_eq(s32 a, s32 b) {
    a = ~(a ^ b);
    a &= a << 16;
    a &= a << 8;
    a &= a << 4;
    a &= a << 2;
    a &= a << 1;
    return a >> 31;
  }
  
  /* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
   * both non-negative. */
  static s32 inline s32_gte(s32 a, s32 b) {
    a -= b;
    /* a >= 0 iff a >= b. */
    return ~(a >> 31);
  }
  
  /* Take a fully reduced polynomial form number and contract it into a
   * little-endian, 32-byte array.
   *
   * On entry: |input_limbs[i]| < 2^26 */
  static inline void
  fcontract(u8 *output, limb *input_limbs) {
    int i;
    int j;
    s32 input[10];
    s32 mask;
  
    /* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
    for (i = 0; i < 10; i++) {
      input[i] = input_limbs[i];
    }
  
    for (j = 0; j < 2; ++j) {
      for (i = 0; i < 9; ++i) {
        if ((i & 1) == 1) {
          /* This calculation is a time-invariant way to make input[i]
           * non-negative by borrowing from the next-larger limb. */
          const s32 mask = input[i] >> 31;
          const s32 carry = -((input[i] & mask) >> 25);
          input[i] = input[i] + (carry << 25);
          input[i+1] = input[i+1] - carry;
        } else {
          const s32 mask = input[i] >> 31;
          const s32 carry = -((input[i] & mask) >> 26);
          input[i] = input[i] + (carry << 26);
          input[i+1] = input[i+1] - carry;
        }
      }
  
      /* There's no greater limb for input[9] to borrow from, but we can multiply
       * by 19 and borrow from input[0], which is valid mod 2^255-19. */
      {
        const s32 mask = input[9] >> 31;
        const s32 carry = -((input[9] & mask) >> 25);
        input[9] = input[9] + (carry << 25);
        input[0] = input[0] - (carry * 19);
      }
  
      /* After the first iteration, input[1..9] are non-negative and fit within
       * 25 or 26 bits, depending on position. However, input[0] may be
       * negative. */
    }
  
    /* The first borrow-propagation pass above ended with every limb
       except (possibly) input[0] non-negative.
  
       If input[0] was negative after the first pass, then it was because of a
       carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
       one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
  
       In the second pass, each limb is decreased by at most one. Thus the second
       borrow-propagation pass could only have wrapped around to decrease
       input[0] again if the first pass left input[0] negative *and* input[1]
       through input[9] were all zero.  In that case, input[1] is now 2^25 - 1,
       and this last borrow-propagation step will leave input[1] non-negative. */
    {
      const s32 mask = input[0] >> 31;
      const s32 carry = -((input[0] & mask) >> 26);
      input[0] = input[0] + (carry << 26);
      input[1] = input[1] - carry;
    }
  
    /* All input[i] are now non-negative. However, there might be values between
     * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
    for (j = 0; j < 2; j++) {
      for (i = 0; i < 9; i++) {
        if ((i & 1) == 1) {
          const s32 carry = input[i] >> 25;
          input[i] &= 0x1ffffff;
          input[i+1] += carry;
        } else {
          const s32 carry = input[i] >> 26;
          input[i] &= 0x3ffffff;
          input[i+1] += carry;
        }
      }
  
      {
        const s32 carry = input[9] >> 25;
        input[9] &= 0x1ffffff;
        input[0] += 19*carry;
      }
    }
  
    /* If the first carry-chain pass, just above, ended up with a carry from
     * input[9], and that caused input[0] to be out-of-bounds, then input[0] was
     * < 2^26 + 2*19, because the carry was, at most, two.
     *
     * If the second pass carried from input[9] again then input[0] is < 2*19 and
     * the input[9] -> input[0] carry didn't push input[0] out of bounds. */
  
    /* It still remains the case that input might be between 2^255-19 and 2^255.
     * In this case, input[1..9] must take their maximum value and input[0] must
     * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
    mask = s32_gte(input[0], 0x3ffffed);
    for (i = 1; i < 10; i++) {
      if ((i & 1) == 1) {
        mask &= s32_eq(input[i], 0x1ffffff);
      } else {
        mask &= s32_eq(input[i], 0x3ffffff);
      }
    }
  
    /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
     * this conditionally subtracts 2^255-19. */
    input[0] -= mask & 0x3ffffed;
  
    for (i = 1; i < 10; i++) {
      if ((i & 1) == 1) {
        input[i] -= mask & 0x1ffffff;
      } else {
        input[i] -= mask & 0x3ffffff;
      }
    }
  
    input[1] <<= 2;
    input[2] <<= 3;
    input[3] <<= 5;
    input[4] <<= 6;
    input[6] <<= 1;
    input[7] <<= 3;
    input[8] <<= 4;
    input[9] <<= 6;
  #define F(i, s) \
    output[s+0] |=  input[i] & 0xff; \
    output[s+1]  = (input[i] >> 8) & 0xff; \
    output[s+2]  = (input[i] >> 16) & 0xff; \
    output[s+3]  = (input[i] >> 24) & 0xff;
    output[0] = 0;
    output[16] = 0;
    F(0,0);
    F(1,3);
    F(2,6);
    F(3,9);
    F(4,12);
    F(5,16);
    F(6,19);
    F(7,22);
    F(8,25);
    F(9,28);
  #undef F
  }
  
  /* Input: Q, Q', Q-Q'
   * Output: 2Q, Q+Q'
   *
   *   x2 z3: long form
   *   x3 z3: long form
   *   x z: short form, destroyed
   *   xprime zprime: short form, destroyed
   *   qmqp: short form, preserved
   *
   * On entry and exit, the absolute value of the limbs of all inputs and outputs
   * are < 2^26. */
  static inline void fmonty(limb *x2, limb *z2,  /* output 2Q */
                     limb *x3, limb *z3,  /* output Q + Q' */
                     limb *x, limb *z,    /* input Q */
                     limb *xprime, limb *zprime,  /* input Q' */
                     const limb *qmqp /* input Q - Q' */) {
    limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
          zzprime[19], zzzprime[19], xxxprime[19];
  
    memcpy(origx, x, 10 * sizeof(limb));
    fsum(x, z);
    /* |x[i]| < 2^27 */
    fdifference(z, origx);  /* does x - z */
    /* |z[i]| < 2^27 */
  
    memcpy(origxprime, xprime, sizeof(limb) * 10);
    fsum(xprime, zprime);
    /* |xprime[i]| < 2^27 */
    fdifference(zprime, origxprime);
    /* |zprime[i]| < 2^27 */
    fproduct(xxprime, xprime, z);
    /* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
     * 2^(27+27) and fproduct adds together, at most, 14 of those products.
     * (Approximating that to 2^58 doesn't work out.) */
    fproduct(zzprime, x, zprime);
    /* |zzprime[i]| < 14*2^54 */
    freduce_degree(xxprime);
    freduce_coefficients(xxprime);
    /* |xxprime[i]| < 2^26 */
    freduce_degree(zzprime);
    freduce_coefficients(zzprime);
    /* |zzprime[i]| < 2^26 */
    memcpy(origxprime, xxprime, sizeof(limb) * 10);
    fsum(xxprime, zzprime);
    /* |xxprime[i]| < 2^27 */
    fdifference(zzprime, origxprime);
    /* |zzprime[i]| < 2^27 */
    fsquare(xxxprime, xxprime);
    /* |xxxprime[i]| < 2^26 */
    fsquare(zzzprime, zzprime);
    /* |zzzprime[i]| < 2^26 */
    fproduct(zzprime, zzzprime, qmqp);
    /* |zzprime[i]| < 14*2^52 */
    freduce_degree(zzprime);
    freduce_coefficients(zzprime);
    /* |zzprime[i]| < 2^26 */
    memcpy(x3, xxxprime, sizeof(limb) * 10);
    memcpy(z3, zzprime, sizeof(limb) * 10);
  
    fsquare(xx, x);
    /* |xx[i]| < 2^26 */
    fsquare(zz, z);
    /* |zz[i]| < 2^26 */
    fproduct(x2, xx, zz);
    /* |x2[i]| < 14*2^52 */
    freduce_degree(x2);
    freduce_coefficients(x2);
    /* |x2[i]| < 2^26 */
    fdifference(zz, xx);  // does zz = xx - zz
    /* |zz[i]| < 2^27 */
    memset(zzz + 10, 0, sizeof(limb) * 9);
    fscalar_product(zzz, zz, 121665);
    /* |zzz[i]| < 2^(27+17) */
    /* No need to call freduce_degree here:
       fscalar_product doesn't increase the degree of its input. */
    freduce_coefficients(zzz);
    /* |zzz[i]| < 2^26 */
    fsum(zzz, xx);
    /* |zzz[i]| < 2^27 */
    fproduct(z2, zz, zzz);
    /* |z2[i]| < 14*2^(26+27) */
    freduce_degree(z2);
    freduce_coefficients(z2);
    /* |z2|i| < 2^26 */
  }
  
  /* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
   * them unchanged if 'iswap' is 0.  Runs in data-invariant time to avoid
   * side-channel attacks.
   *
   * NOTE that this function requires that 'iswap' be 1 or 0; other values give
   * wrong results.  Also, the two limb arrays must be in reduced-coefficient,
   * reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
   * and all all values in a[0..9],b[0..9] must have magnitude less than
   * INT32_MAX. */
  static inline void
  swap_conditional(limb a[19], limb b[19], limb iswap) {
    unsigned i;
    const s32 swap = (s32) -iswap;
  
    for (i = 0; i < 10; ++i) {
      const s32 x = swap & ( ((s32)a[i]) ^ ((s32)b[i]) );
      a[i] = ((s32)a[i]) ^ x;
      b[i] = ((s32)b[i]) ^ x;
    }
  }
  
  /* Calculates nQ where Q is the x-coordinate of a point on the curve
   *
   *   resultx/resultz: the x coordinate of the resulting curve point (short form)
   *   n: a little endian, 32-byte number
   *   q: a point of the curve (short form) */
  static void
  cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
    limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
    limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
    limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
    limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
  
    unsigned i, j;
  
    memcpy(nqpqx, q, sizeof(limb) * 10);
  
    for (i = 0; i < 32; ++i) {
      u8 byte = n[31 - i];
      for (j = 0; j < 8; ++j) {
        const limb bit = byte >> 7;
  
        swap_conditional(nqx, nqpqx, bit);
        swap_conditional(nqz, nqpqz, bit);
        fmonty(nqx2, nqz2,
               nqpqx2, nqpqz2,
               nqx, nqz,
               nqpqx, nqpqz,
               q);
        swap_conditional(nqx2, nqpqx2, bit);
        swap_conditional(nqz2, nqpqz2, bit);
  
        t = nqx;
        nqx = nqx2;
        nqx2 = t;
        t = nqz;
        nqz = nqz2;
        nqz2 = t;
        t = nqpqx;
        nqpqx = nqpqx2;
        nqpqx2 = t;
        t = nqpqz;
        nqpqz = nqpqz2;
        nqpqz2 = t;
  
        byte <<= 1;
      }
    }
  
    memcpy(resultx, nqx, sizeof(limb) * 10);
    memcpy(resultz, nqz, sizeof(limb) * 10);
  }
  
  // -----------------------------------------------------------------------------
  // Shamelessly copied from djb's code
  // -----------------------------------------------------------------------------
  static inline void
  crecip(limb *out, const limb *z) {
    limb z2[10];
    limb z9[10];
    limb z11[10];
    limb z2_5_0[10];
    limb z2_10_0[10];
    limb z2_20_0[10];
    limb z2_50_0[10];
    limb z2_100_0[10];
    limb t0[10];
    limb t1[10];
    int i;
  
    /* 2 */ fsquare(z2,z);
    /* 4 */ fsquare(t1,z2);
    /* 8 */ fsquare(t0,t1);
    /* 9 */ fmul(z9,t0,z);
    /* 11 */ fmul(z11,z9,z2);
    /* 22 */ fsquare(t0,z11);
    /* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9);
  
    /* 2^6 - 2^1 */ fsquare(t0,z2_5_0);
    /* 2^7 - 2^2 */ fsquare(t1,t0);
    /* 2^8 - 2^3 */ fsquare(t0,t1);
    /* 2^9 - 2^4 */ fsquare(t1,t0);
    /* 2^10 - 2^5 */ fsquare(t0,t1);
    /* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0);
  
    /* 2^11 - 2^1 */ fsquare(t0,z2_10_0);
    /* 2^12 - 2^2 */ fsquare(t1,t0);
    /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
    /* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0);
  
    /* 2^21 - 2^1 */ fsquare(t0,z2_20_0);
    /* 2^22 - 2^2 */ fsquare(t1,t0);
    /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
    /* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0);
  
    /* 2^41 - 2^1 */ fsquare(t1,t0);
    /* 2^42 - 2^2 */ fsquare(t0,t1);
    /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
    /* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0);
  
    /* 2^51 - 2^1 */ fsquare(t0,z2_50_0);
    /* 2^52 - 2^2 */ fsquare(t1,t0);
    /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
    /* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0);
  
    /* 2^101 - 2^1 */ fsquare(t1,z2_100_0);
    /* 2^102 - 2^2 */ fsquare(t0,t1);
    /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
    /* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0);
  
    /* 2^201 - 2^1 */ fsquare(t0,t1);
    /* 2^202 - 2^2 */ fsquare(t1,t0);
    /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
    /* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0);
  
    /* 2^251 - 2^1 */ fsquare(t1,t0);
    /* 2^252 - 2^2 */ fsquare(t0,t1);
    /* 2^253 - 2^3 */ fsquare(t1,t0);
    /* 2^254 - 2^4 */ fsquare(t0,t1);
    /* 2^255 - 2^5 */ fsquare(t1,t0);
    /* 2^255 - 21 */ fmul(out,t1,z11);
  }
  
  static inline int
  curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
    limb bp[10], x[10], z[11], zmone[10];
    uint8_t e[32];
    int i;
  
    for (i = 0; i < 32; ++i) e[i] = secret[i];
    e[0] &= 248;
    e[31] &= 127;
    e[31] |= 64;
  
    fexpand(bp, basepoint);
    cmult(x, z, e, bp);
    crecip(zmone, z);
    fmul(z, x, zmone);
    fcontract(mypublic, z);
    return 0;
  }
  
  #define X25519_BYTES (256/8)
  
  const u8 X25519_BASE_POINT[X25519_BYTES] = {9};
  
  static int
  x25519_exchange(u8 *output, const u8 *secret, const u8 *point)
  {
  	u8 bp[X25519_PUBKEY_LEN];
  	memcpy(bp, point, X25519_PUBKEY_LEN);
  	bp[31] &= 0x7f;
  	return curve25519_donna(output, secret, bp);
  }
  
  static int
  x25519_basepoint(u8 *output, const u8 *secret)
  {
  	static const u8 basepoint[32] = {9};
  	return x25519_exchange(output, secret, basepoint);
  }
  
  /*
   * X25519 is the Diffie-Hellman primitive built from curve25519. It is
   * sometimes referred to as “curve25519”, but “X25519” is a more precise name.
   * See http://cr.yp.to/ecdh.html and https://tools.ietf.org/html/rfc7748.
   *
   * To generate a private key, get 32 random bytes and:
   *
   *  sec[0] &= 248;
   *  sec[31] &= 127;
   *  sec[31] |= 64;
   */
  
  static void
  x25519_prepare_private_key(u8 *private_key, const u8* seed)
  {
  	/* To generate a private key, generate 32 random bytes and: */
  	memcpy(private_key, seed, 32);
  	private_key[0] &= 248;
  	private_key[31] &= 127;
  	private_key[31] |= 64;
  }
  
  /*
   * To generate the public key, just do:
   * static const uint8_t basepoint[32] = {9};
   * curve25519_donna(mypublic, mysecret, basepoint);
   */
  
  static void 
  x25519_public_from_private(const u8 *private_key, u8 *public_key)
  {
  	static const u8 bp[32] = {9};
  	curve25519_donna(public_key, private_key, bp);
  }
  
  static void
  x25519_derive(u8* secret, const u8 *key, const u8 *pub)
  {
  	curve25519_donna(secret, key, pub);
  }
  
  /*
   * To generate a shared key do:
   *
   * uint8_t shared_key[32];
   * curve25519_donna(shared_key, mysecret, theirpublic);
   */
  
  static void
  x25519_key_exchange(const u8* public, const u8 *secret, u8 *shared)
  {
  	curve25519_donna(shared, secret, public);
  }
  
  static void
  x25519_key_pair(u8 *public_key, u8 *private_key)
  {
  }
  
  #ifdef TEST_VECTORS
  
  #include <stdlib.h>
  #include <stdio.h>
  #include <strings.h>
  
  /* From RFC 7748, section 6.1 */
  /* Alice's private key, a: */
  const char
  a16[] = "77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a";
  /* Alice's public key, X25519(a, 9): */
  const char 
  a_pub16[] = "8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a";
  /* Bob's private key, b: */
  const char
  b16[] = "5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb";
  /* Bob's public key, X25519(b, 9): */
  const char 
  b_pub16[] = "de9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f";
  /* Their shared secret, K: */
  const char
  k16[] = "4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742";
  
  static inline void
  x25519_test_testvec(void)
  {
  	u8 a[32], b[32], a_pub[32], b_pub[32], k1[32], k2[32];
  	
  	x25519_basepoint(a_pub, a);
  	x25519_basepoint(b_pub, b);
  
  	x25519_exchange(k1, a, b_pub);
  	x25519_exchange(k2, b, a_pub);
  }
  
  int main()
  {
      printf("ecc-x25519 Test Vectors\n");
  /*
      test(pt1, ct1, sizeof(pt1), key1, sizeof(key1));
      printf("Test vector 1: OK\n");
  
      test(pt2, ct2, sizeof(pt2), key2, sizeof(key2));
      printf("Test vector 2: OK\n");
  
      test(pt3, ct3, sizeof(pt3), key3, sizeof(key3));
      printf("Test vector 3: OK\n");
  */
      printf("\nAll tests passed.\n");
  
      return 0;
  }
  
  void
  crypto_ecc_group_register(struct ecc_group *alg)
  {
  }
  
  #endif /* TEST_VECTORS */
  
  struct ecc_group x25519_ecc_group = {
  	.name = "x25519",
  	.id = ECC_GROUP_X25519,
  	.pub_key_len = X25519_PUBKEY_LEN,
  	.prv_key_len = X25519_PUBKEY_LEN,
  	.size = X25519_BYTES,
  	.key_pair = x25519_key_pair,
  	.key_exchange = x25519_key_exchange,
  	.public_from_private = x25519_public_from_private
  };
  
  void ecc_x25519_init(void)
  {
  	crypto_ecc_group_register(&x25519_ecc_group);
  }
  