ZeroTierOne/node/C25519.cpp

/*
Matthew Dempsky
Public domain.
Derived from public domain code by D. J. Bernstein.
*/

// Modified very slightly for ZeroTier One by Adam Ierymenko
// This code remains in the public domain.

#include <stdint.h>
#include <stdlib.h>
#include <string.h>

#include "Constants.hpp"
#include "C25519.hpp"
#include "SHA512.hpp"
#include "Buffer.hpp"
#include "Hashtable.hpp"
#include "Mutex.hpp"

#ifdef __WINDOWS__
#pragma warning(disable: 4146)
#endif

#ifdef __GNUC__
#pragma GCC diagnostic ignored "-Wunused-function"
#endif

namespace {

#define crypto_int32 int32_t
#define crypto_uint32 uint32_t
#define crypto_int64 int64_t
#define crypto_uint64 uint64_t
#define crypto_hash_sha512_BYTES 64

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

typedef uint8_t u8;
typedef int32_t s32;
typedef int64_t limb;

static ZT_ALWAYS_INLINE 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];
  }
}

static ZT_ALWAYS_INLINE void fdifference(limb *output, const limb *in) {
  unsigned i;
  for (i = 0; i < 10; ++i) {
    output[i] = in[i] - output[i];
  }
}

static ZT_ALWAYS_INLINE void fscalar_product(limb *output, const limb *in, const limb scalar) {
  unsigned i;
  for (i = 0; i < 10; ++i) {
    output[i] = in[i] * scalar;
  }
}

static ZT_ALWAYS_INLINE 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]);
}

static ZT_ALWAYS_INLINE void freduce_degree(limb *output) {
  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

static ZT_ALWAYS_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;
}

static ZT_ALWAYS_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;
}

static ZT_ALWAYS_INLINE 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. */
}

static inline 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);
}

static ZT_ALWAYS_INLINE 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]);
}

static inline 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);
}

static ZT_ALWAYS_INLINE 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

static ZT_ALWAYS_INLINE 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;
}

static ZT_ALWAYS_INLINE s32 s32_gte(s32 a, s32 b) {
  a -= b;
  /* a >= 0 iff a >= b. */
  return ~(a >> 31);
}

static inline void fcontract(u8 *output, limb *input_limbs) {
  int i;
  int j;
  s32 input[10];
  s32 mask;

  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) {
        const s32 mm = input[i] >> 31;
        const s32 carry = -((input[i] & mm) >> 25);
        input[i] = input[i] + (carry << 25);
        input[i+1] = input[i+1] - carry;
      } else {
        const s32 mm = input[i] >> 31;
        const s32 carry = -((input[i] & mm) >> 26);
        input[i] = input[i] + (carry << 26);
        input[i+1] = input[i+1] - carry;
      }
    }
    {
      const s32 mm = input[9] >> 31;
      const s32 carry = -((input[9] & mm) >> 25);
      input[9] = input[9] + (carry << 25);
      input[0] = input[0] - (carry * 19);
    }
  }
  {
    const s32 mm = input[0] >> 31;
    const s32 carry = -((input[0] & mm) >> 26);
    input[0] = input[0] + (carry << 26);
    input[1] = input[1] - carry;
  }
  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;
    }
  }
  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);
    }
  }
  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
}

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 */
}

static ZT_ALWAYS_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;
  }
}

static inline 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);
}

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 void crypto_scalarmult(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);
}

static const unsigned char base[32] = {9};
static ZT_ALWAYS_INLINE void crypto_scalarmult_base(unsigned char *q,const unsigned char *n) { crypto_scalarmult(q,n,base); }

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

// Ed25519 ref from: http://bench.cr.yp.to/supercop.html

typedef struct
{
	crypto_uint32 v[32];
}
fe25519;

typedef struct
{
	crypto_uint32 v[32];
}
sc25519;

typedef struct
{
	crypto_uint32 v[16];
}
shortsc25519;

typedef struct
{
	fe25519 x;
	fe25519 y;
	fe25519 z;
	fe25519 t;
} ge25519;

#define ge25519_p3 ge25519

typedef struct
{
	fe25519 x;
	fe25519 z;
	fe25519 y;
	fe25519 t;
} ge25519_p1p1;

typedef struct
{
	fe25519 x;
	fe25519 y;
	fe25519 z;
} ge25519_p2;

typedef struct
{
	fe25519 x;
	fe25519 y;
} ge25519_aff;

static inline void fe25519_sub(fe25519 *r, const fe25519 *x, const fe25519 *y);

static ZT_ALWAYS_INLINE crypto_uint32 equal(crypto_uint32 a,crypto_uint32 b) /* 16-bit inputs */
{
	crypto_uint32 x = a ^ b; /* 0: yes; 1..65535: no */
	x -= 1; /* 4294967295: yes; 0..65534: no */
	x >>= 31; /* 1: yes; 0: no */
	return x;
}

static ZT_ALWAYS_INLINE crypto_uint32 ge(crypto_uint32 a,crypto_uint32 b) /* 16-bit inputs */
{
	unsigned int x = a;
	x -= (unsigned int) b; /* 0..65535: yes; 4294901761..4294967295: no */
	x >>= 31; /* 0: yes; 1: no */
	x ^= 1; /* 1: yes; 0: no */
	return x;
}

static ZT_ALWAYS_INLINE crypto_uint32 times19(crypto_uint32 a) { return (a << 4) + (a << 1) + a; }
static ZT_ALWAYS_INLINE crypto_uint32 times38(crypto_uint32 a) { return (a << 5) + (a << 2) + (a << 1); }

static ZT_ALWAYS_INLINE void reduce_add_sub(fe25519 *r)
{
	int i,rep;
	for(rep=0;rep<4;rep++)
	{
		crypto_uint32 t = r->v[31] >> 7;
		r->v[31] &= 127;
		t = times19(t);
		r->v[0] += t;
		for(i=0;i<31;i++)
		{
			t = r->v[i] >> 8;
			r->v[i+1] += t;
			r->v[i] &= 255;
		}
	}
}

static ZT_ALWAYS_INLINE void reduce_mul(fe25519 *r)
{
	int i,rep;
	for(rep=0;rep<2;rep++)
	{
		crypto_uint32 t = r->v[31] >> 7;
		r->v[31] &= 127;
		t = times19(t);
		r->v[0] += t;
		for(i=0;i<31;i++)
		{
			t = r->v[i] >> 8;
			r->v[i+1] += t;
			r->v[i] &= 255;
		}
	}
}

static ZT_ALWAYS_INLINE void fe25519_freeze(fe25519 *r)
{
	int i;
	crypto_uint32 mm = equal(r->v[31],127);
	for(i=30;i>0;i--)
		mm &= equal(r->v[i],255);
	mm &= ge(r->v[0],237);

	mm = -mm;

	r->v[31] -= mm&127;
	for(i=30;i>0;i--)
		r->v[i] -= mm&255;
	r->v[0] -= mm&237;
}

static ZT_ALWAYS_INLINE void fe25519_unpack(fe25519 *r, const unsigned char x[32])
{
	int i;
	for(i=0;i<32;i++) r->v[i] = x[i];
	r->v[31] &= 127;
}

static ZT_ALWAYS_INLINE void fe25519_pack(unsigned char r[32], const fe25519 *x)
{
	int i;
	fe25519 y = *x;
	fe25519_freeze(&y);
	for(i=0;i<32;i++)
		r[i] = y.v[i];
}

static inline int fe25519_iseq_vartime(const fe25519 *x, const fe25519 *y)
{
	int i;
	fe25519 t1 = *x;
	fe25519 t2 = *y;
	fe25519_freeze(&t1);
	fe25519_freeze(&t2);
	for(i=0;i<32;i++)
		if(t1.v[i] != t2.v[i]) return 0;
	return 1;
}

static ZT_ALWAYS_INLINE void fe25519_cmov(fe25519 *r, const fe25519 *x, unsigned char b)
{
	int i;
	crypto_uint32 mask = b;
	mask = -mask;
	for(i=0;i<32;i++) r->v[i] ^= mask & (x->v[i] ^ r->v[i]);
}

static ZT_ALWAYS_INLINE unsigned char fe25519_getparity(const fe25519 *x)
{
	fe25519 t = *x;
	fe25519_freeze(&t);
	return t.v[0] & 1;
}

static ZT_ALWAYS_INLINE void fe25519_setone(fe25519 *r)
{
	int i;
	r->v[0] = 1;
	for(i=1;i<32;i++) r->v[i]=0;
}

static ZT_ALWAYS_INLINE void fe25519_setzero(fe25519 *r)
{
	int i;
	for(i=0;i<32;i++) r->v[i]=0;
}

static ZT_ALWAYS_INLINE void fe25519_neg(fe25519 *r, const fe25519 *x)
{
	fe25519 t;
	int i;
	for(i=0;i<32;i++) t.v[i]=x->v[i];
	fe25519_setzero(r);
	fe25519_sub(r, r, &t);
}

static ZT_ALWAYS_INLINE void fe25519_add(fe25519 *r, const fe25519 *x, const fe25519 *y)
{
	int i;
	for(i=0;i<32;i++) r->v[i] = x->v[i] + y->v[i];
	reduce_add_sub(r);
}

static ZT_ALWAYS_INLINE void fe25519_sub(fe25519 *r, const fe25519 *x, const fe25519 *y)
{
	int i;
	crypto_uint32 t[32];
	t[0] = x->v[0] + 0x1da;
	t[31] = x->v[31] + 0xfe;
	for(i=1;i<31;i++) t[i] = x->v[i] + 0x1fe;
	for(i=0;i<32;i++) r->v[i] = t[i] - y->v[i];
	reduce_add_sub(r);
}

static ZT_ALWAYS_INLINE void fe25519_mul(fe25519 *r, const fe25519 *x, const fe25519 *y)
{
	int i,j;
	crypto_uint32 t[63];
	for(i=0;i<63;i++)t[i] = 0;

	for(i=0;i<32;i++)
		for(j=0;j<32;j++)
			t[i+j] += x->v[i] * y->v[j];

	for(i=32;i<63;i++)
		r->v[i-32] = t[i-32] + times38(t[i]);
	r->v[31] = t[31]; /* result now in r[0]...r[31] */

	reduce_mul(r);
}

static ZT_ALWAYS_INLINE void fe25519_square(fe25519 *r, const fe25519 *x) { fe25519_mul(r, x, x); }

static inline void fe25519_invert(fe25519 *r, const fe25519 *x)
{
	fe25519 z2;
	fe25519 z9;
	fe25519 z11;
	fe25519 z2_5_0;
	fe25519 z2_10_0;
	fe25519 z2_20_0;
	fe25519 z2_50_0;
	fe25519 z2_100_0;
	fe25519 t0;
	fe25519 t1;
	int i;

	/* 2 */ fe25519_square(&z2,x);
	/* 4 */ fe25519_square(&t1,&z2);
	/* 8 */ fe25519_square(&t0,&t1);
	/* 9 */ fe25519_mul(&z9,&t0,x);
	/* 11 */ fe25519_mul(&z11,&z9,&z2);
	/* 22 */ fe25519_square(&t0,&z11);
	/* 2^5 - 2^0 = 31 */ fe25519_mul(&z2_5_0,&t0,&z9);

	/* 2^6 - 2^1 */ fe25519_square(&t0,&z2_5_0);
	/* 2^7 - 2^2 */ fe25519_square(&t1,&t0);
	/* 2^8 - 2^3 */ fe25519_square(&t0,&t1);
	/* 2^9 - 2^4 */ fe25519_square(&t1,&t0);
	/* 2^10 - 2^5 */ fe25519_square(&t0,&t1);
	/* 2^10 - 2^0 */ fe25519_mul(&z2_10_0,&t0,&z2_5_0);

	/* 2^11 - 2^1 */ fe25519_square(&t0,&z2_10_0);
	/* 2^12 - 2^2 */ fe25519_square(&t1,&t0);
	/* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fe25519_square(&t0,&t1); fe25519_square(&t1,&t0); }
	/* 2^20 - 2^0 */ fe25519_mul(&z2_20_0,&t1,&z2_10_0);

	/* 2^21 - 2^1 */ fe25519_square(&t0,&z2_20_0);
	/* 2^22 - 2^2 */ fe25519_square(&t1,&t0);
	/* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fe25519_square(&t0,&t1); fe25519_square(&t1,&t0); }
	/* 2^40 - 2^0 */ fe25519_mul(&t0,&t1,&z2_20_0);

	/* 2^41 - 2^1 */ fe25519_square(&t1,&t0);
	/* 2^42 - 2^2 */ fe25519_square(&t0,&t1);
	/* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fe25519_square(&t1,&t0); fe25519_square(&t0,&t1); }
	/* 2^50 - 2^0 */ fe25519_mul(&z2_50_0,&t0,&z2_10_0);

	/* 2^51 - 2^1 */ fe25519_square(&t0,&z2_50_0);
	/* 2^52 - 2^2 */ fe25519_square(&t1,&t0);
	/* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fe25519_square(&t0,&t1); fe25519_square(&t1,&t0); }
	/* 2^100 - 2^0 */ fe25519_mul(&z2_100_0,&t1,&z2_50_0);

	/* 2^101 - 2^1 */ fe25519_square(&t1,&z2_100_0);
	/* 2^102 - 2^2 */ fe25519_square(&t0,&t1);
	/* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fe25519_square(&t1,&t0); fe25519_square(&t0,&t1); }
	/* 2^200 - 2^0 */ fe25519_mul(&t1,&t0,&z2_100_0);

	/* 2^201 - 2^1 */ fe25519_square(&t0,&t1);
	/* 2^202 - 2^2 */ fe25519_square(&t1,&t0);
	/* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fe25519_square(&t0,&t1); fe25519_square(&t1,&t0); }
	/* 2^250 - 2^0 */ fe25519_mul(&t0,&t1,&z2_50_0);

	/* 2^251 - 2^1 */ fe25519_square(&t1,&t0);
	/* 2^252 - 2^2 */ fe25519_square(&t0,&t1);
	/* 2^253 - 2^3 */ fe25519_square(&t1,&t0);
	/* 2^254 - 2^4 */ fe25519_square(&t0,&t1);
	/* 2^255 - 2^5 */ fe25519_square(&t1,&t0);
	/* 2^255 - 21 */ fe25519_mul(r,&t1,&z11);
}

static inline void fe25519_pow2523(fe25519 *r, const fe25519 *x)
{
	fe25519 z2;
	fe25519 z9;
	fe25519 z11;
	fe25519 z2_5_0;
	fe25519 z2_10_0;
	fe25519 z2_20_0;
	fe25519 z2_50_0;
	fe25519 z2_100_0;
	fe25519 t;
	int i;

	/* 2 */ fe25519_square(&z2,x);
	/* 4 */ fe25519_square(&t,&z2);
	/* 8 */ fe25519_square(&t,&t);
	/* 9 */ fe25519_mul(&z9,&t,x);
	/* 11 */ fe25519_mul(&z11,&z9,&z2);
	/* 22 */ fe25519_square(&t,&z11);
	/* 2^5 - 2^0 = 31 */ fe25519_mul(&z2_5_0,&t,&z9);

	/* 2^6 - 2^1 */ fe25519_square(&t,&z2_5_0);
	/* 2^10 - 2^5 */ for (i = 1;i < 5;i++) { fe25519_square(&t,&t); }
	/* 2^10 - 2^0 */ fe25519_mul(&z2_10_0,&t,&z2_5_0);

	/* 2^11 - 2^1 */ fe25519_square(&t,&z2_10_0);
	/* 2^20 - 2^10 */ for (i = 1;i < 10;i++) { fe25519_square(&t,&t); }
	/* 2^20 - 2^0 */ fe25519_mul(&z2_20_0,&t,&z2_10_0);

	/* 2^21 - 2^1 */ fe25519_square(&t,&z2_20_0);
	/* 2^40 - 2^20 */ for (i = 1;i < 20;i++) { fe25519_square(&t,&t); }
	/* 2^40 - 2^0 */ fe25519_mul(&t,&t,&z2_20_0);

	/* 2^41 - 2^1 */ fe25519_square(&t,&t);
	/* 2^50 - 2^10 */ for (i = 1;i < 10;i++) { fe25519_square(&t,&t); }
	/* 2^50 - 2^0 */ fe25519_mul(&z2_50_0,&t,&z2_10_0);

	/* 2^51 - 2^1 */ fe25519_square(&t,&z2_50_0);
	/* 2^100 - 2^50 */ for (i = 1;i < 50;i++) { fe25519_square(&t,&t); }
	/* 2^100 - 2^0 */ fe25519_mul(&z2_100_0,&t,&z2_50_0);

	/* 2^101 - 2^1 */ fe25519_square(&t,&z2_100_0);
	/* 2^200 - 2^100 */ for (i = 1;i < 100;i++) { fe25519_square(&t,&t); }
	/* 2^200 - 2^0 */ fe25519_mul(&t,&t,&z2_100_0);

	/* 2^201 - 2^1 */ fe25519_square(&t,&t);
	/* 2^250 - 2^50 */ for (i = 1;i < 50;i++) { fe25519_square(&t,&t); }
	/* 2^250 - 2^0 */ fe25519_mul(&t,&t,&z2_50_0);

	/* 2^251 - 2^1 */ fe25519_square(&t,&t);
	/* 2^252 - 2^2 */ fe25519_square(&t,&t);
	/* 2^252 - 3 */ fe25519_mul(r,&t,x);
}

static const crypto_uint32 m[32] = {0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2, 0xDE, 0xF9, 0xDE, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10};
static const crypto_uint32 mu[33] = {0x1B, 0x13, 0x2C, 0x0A, 0xA3, 0xE5, 0x9C, 0xED, 0xA7, 0x29, 0x63, 0x08, 0x5D, 0x21, 0x06, 0x21, 0xEB, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x0F};

static ZT_ALWAYS_INLINE crypto_uint32 lt(crypto_uint32 a,crypto_uint32 b) /* 16-bit inputs */
{
	unsigned int x = a;
	x -= (unsigned int) b; /* 0..65535: no; 4294901761..4294967295: yes */
	x >>= 31; /* 0: no; 1: yes */
	return x;
}

static ZT_ALWAYS_INLINE void reduce_add_sub(sc25519 *r)
{
	crypto_uint32 pb = 0;
	crypto_uint32 b;
	crypto_uint32 mask;
	int i;
	unsigned char t[32];

	for(i=0;i<32;i++)
	{
		pb += m[i];
		b = lt(r->v[i],pb);
		t[i] = r->v[i]-pb+(b<<8);
		pb = b;
	}
	mask = b - 1;
	for(i=0;i<32;i++)
		r->v[i] ^= mask & (r->v[i] ^ t[i]);
}

static inline void barrett_reduce(sc25519 *r, const crypto_uint32 x[64])
{
	/* See HAC, Alg. 14.42 */
	int i,j;
	crypto_uint32 q2[66];
	crypto_uint32 *q3 = q2 + 33;
	crypto_uint32 r1[33];
	crypto_uint32 r2[33];
	crypto_uint32 pb = 0;

	for (i = 0;i < 66;++i) q2[i] = 0;
	for (i = 0;i < 33;++i) r2[i] = 0;

	for(i=0;i<33;i++)
		for(j=0;j<33;j++)
			if(i+j >= 31) q2[i+j] += mu[i]*x[j+31];
	q2[32] += (q2[31] >> 8);
	q2[33] += (q2[32] >> 8);

	for(i=0;i<33;i++)r1[i] = x[i];
	for(i=0;i<32;i++)
		for(j=0;j<33;j++)
			if(i+j < 33) r2[i+j] += m[i]*q3[j];

	for(i=0;i<32;i++)
	{
		r2[i+1] += (r2[i] >> 8);
		r2[i] &= 0xff;
	}

	for(i=0;i<32;i++)
	{
		pb += r2[i];
		crypto_uint32 b = lt(r1[i],pb);
		r->v[i] = r1[i]-pb+(b<<8);
		pb = b;
	}

	reduce_add_sub(r);
	reduce_add_sub(r);
}

static inline void sc25519_from32bytes(sc25519 *r, const unsigned char x[32])
{
	int i;
	crypto_uint32 t[64];
	for(i=0;i<32;i++) t[i] = x[i];
	for(i=32;i<64;++i) t[i] = 0;
	barrett_reduce(r, t);
}

static inline void sc25519_from64bytes(sc25519 *r, const unsigned char x[64])
{
	int i;
	crypto_uint32 t[64];
	for(i=0;i<64;i++) t[i] = x[i];
	barrett_reduce(r, t);
}

static ZT_ALWAYS_INLINE void sc25519_to32bytes(unsigned char r[32], const sc25519 *x)
{
	int i;
	for(i=0;i<32;i++) r[i] = x->v[i];
}

static inline void sc25519_add(sc25519 *r, const sc25519 *x, const sc25519 *y)
{
	int i;
	for(i=0;i<32;i++) r->v[i] = x->v[i] + y->v[i];
	for(i=0;i<31;i++)
	{
		r->v[i+1] += (r->v[i] >> 8);
		r->v[i] &= 0xff;
	}
	reduce_add_sub(r);
}

static inline void sc25519_mul(sc25519 *r, const sc25519 *x, const sc25519 *y)
{
	int i,j;
	crypto_uint32 t[64];
	for(i=0;i<64;i++)t[i] = 0;

	for(i=0;i<32;i++)
		for(j=0;j<32;j++)
			t[i+j] += x->v[i] * y->v[j];

	for(i=0;i<63;i++)
	{
		t[i+1] += (t[i] >> 8);
		t[i] &= 0xff;
	}

	barrett_reduce(r, t);
}

static ZT_ALWAYS_INLINE void sc25519_window3(signed char r[85], const sc25519 *s)
{
	char carry;
	int i;
	for(i=0;i<10;i++)
	{
		r[8*i+0]  =  s->v[3*i+0]       & 7;
		r[8*i+1]  = (s->v[3*i+0] >> 3) & 7;
		r[8*i+2]  = (s->v[3*i+0] >> 6) & 7;
		r[8*i+2] ^= (s->v[3*i+1] << 2) & 7;
		r[8*i+3]  = (s->v[3*i+1] >> 1) & 7;
		r[8*i+4]  = (s->v[3*i+1] >> 4) & 7;
		r[8*i+5]  = (s->v[3*i+1] >> 7) & 7;
		r[8*i+5] ^= (s->v[3*i+2] << 1) & 7;
		r[8*i+6]  = (s->v[3*i+2] >> 2) & 7;
		r[8*i+7]  = (s->v[3*i+2] >> 5) & 7;
	}
	r[8*i+0]  =  s->v[3*i+0]       & 7;
	r[8*i+1]  = (s->v[3*i+0] >> 3) & 7;
	r[8*i+2]  = (s->v[3*i+0] >> 6) & 7;
	r[8*i+2] ^= (s->v[3*i+1] << 2) & 7;
	r[8*i+3]  = (s->v[3*i+1] >> 1) & 7;
	r[8*i+4]  = (s->v[3*i+1] >> 4) & 7;

	/* Making it signed */
	carry = 0;
	for(i=0;i<84;i++)
	{
		r[i] += carry;
		r[i+1] += r[i] >> 3;
		r[i] &= 7;
		carry = r[i] >> 2;
		r[i] -= carry<<3;
	}
	r[84] += carry;
}

static ZT_ALWAYS_INLINE void sc25519_2interleave2(unsigned char r[127], const sc25519 *s1, const sc25519 *s2)
{
	int i;
	for(i=0;i<31;i++)
	{
		r[4*i]   = ( s1->v[i]       & 3) ^ (( s2->v[i]       & 3) << 2);
		r[4*i+1] = ((s1->v[i] >> 2) & 3) ^ (((s2->v[i] >> 2) & 3) << 2);
		r[4*i+2] = ((s1->v[i] >> 4) & 3) ^ (((s2->v[i] >> 4) & 3) << 2);
		r[4*i+3] = ((s1->v[i] >> 6) & 3) ^ (((s2->v[i] >> 6) & 3) << 2);
	}
	r[124] = ( s1->v[31]       & 3) ^ (( s2->v[31]       & 3) << 2);
	r[125] = ((s1->v[31] >> 2) & 3) ^ (((s2->v[31] >> 2) & 3) << 2);
	r[126] = ((s1->v[31] >> 4) & 3) ^ (((s2->v[31] >> 4) & 3) << 2);
}

/* d */
static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
											0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
/* 2*d */
static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
											 0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
/* sqrt(-1) */
static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
												 0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};

/* Packed coordinates of the base point */
static const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
																0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
															{{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
																0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
															{{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
																0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
															{{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
																0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};

/* Multiples of the base point in affine representation */
static const ge25519_aff ge25519_base_multiples_affine[425] = {
{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
{{{0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21}} ,
 {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}}},
{{{0x0e, 0xce, 0x43, 0x28, 0x4e, 0xa1, 0xc5, 0x83, 0x5f, 0xa4, 0xd7, 0x15, 0x45, 0x8e, 0x0d, 0x08, 0xac, 0xe7, 0x33, 0x18, 0x7d, 0x3b, 0x04, 0x3d, 0x6c, 0x04, 0x5a, 0x9f, 0x4c, 0x38, 0xab, 0x36}} ,
 {{0xc9, 0xa3, 0xf8, 0x6a, 0xae, 0x46, 0x5f, 0x0e, 0x56, 0x51, 0x38, 0x64, 0x51, 0x0f, 0x39, 0x97, 0x56, 0x1f, 0xa2, 0xc9, 0xe8, 0x5e, 0xa2, 0x1d, 0xc2, 0x29, 0x23, 0x09, 0xf3, 0xcd, 0x60, 0x22}}},
{{{0x5c, 0xe2, 0xf8, 0xd3, 0x5f, 0x48, 0x62, 0xac, 0x86, 0x48, 0x62, 0x81, 0x19, 0x98, 0x43, 0x63, 0x3a, 0xc8, 0xda, 0x3e, 0x74, 0xae, 0xf4, 0x1f, 0x49, 0x8f, 0x92, 0x22, 0x4a, 0x9c, 0xae, 0x67}} ,
 {{0xd4, 0xb4, 0xf5, 0x78, 0x48, 0x68, 0xc3, 0x02, 0x04, 0x03, 0x24, 0x67, 0x17, 0xec, 0x16, 0x9f, 0xf7, 0x9e, 0x26, 0x60, 0x8e, 0xa1, 0x26, 0xa1, 0xab, 0x69, 0xee, 0x77, 0xd1, 0xb1, 0x67, 0x12}}},
{{{0x70, 0xf8, 0xc9, 0xc4, 0x57, 0xa6, 0x3a, 0x49, 0x47, 0x15, 0xce, 0x93, 0xc1, 0x9e, 0x73, 0x1a, 0xf9, 0x20, 0x35, 0x7a, 0xb8, 0xd4, 0x25, 0x83, 0x46, 0xf1, 0xcf, 0x56, 0xdb, 0xa8, 0x3d, 0x20}} ,
 {{0x2f, 0x11, 0x32, 0xca, 0x61, 0xab, 0x38, 0xdf, 0xf0, 0x0f, 0x2f, 0xea, 0x32, 0x28, 0xf2, 0x4c, 0x6c, 0x71, 0xd5, 0x80, 0x85, 0xb8, 0x0e, 0x47, 0xe1, 0x95, 0x15, 0xcb, 0x27, 0xe8, 0xd0, 0x47}}},
{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
{{{0xc8, 0x84, 0xa5, 0x08, 0xbc, 0xfd, 0x87, 0x3b, 0x99, 0x8b, 0x69, 0x80, 0x7b, 0xc6, 0x3a, 0xeb, 0x93, 0xcf, 0x4e, 0xf8, 0x5c, 0x2d, 0x86, 0x42, 0xb6, 0x71, 0xd7, 0x97, 0x5f, 0xe1, 0x42, 0x67}} ,
 {{0xb4, 0xb9, 0x37, 0xfc, 0xa9, 0x5b, 0x2f, 0x1e, 0x93, 0xe4, 0x1e, 0x62, 0xfc, 0x3c, 0x78, 0x81, 0x8f, 0xf3, 0x8a, 0x66, 0x09, 0x6f, 0xad, 0x6e, 0x79, 0x73, 0xe5, 0xc9, 0x00, 0x06, 0xd3, 0x21}}},
{{{0xf8, 0xf9, 0x28, 0x6c, 0x6d, 0x59, 0xb2, 0x59, 0x74, 0x23, 0xbf, 0xe7, 0x33, 0x8d, 0x57, 0x09, 0x91, 0x9c, 0x24, 0x08, 0x15, 0x2b, 0xe2, 0xb8, 0xee, 0x3a, 0xe5, 0x27, 0x06, 0x86, 0xa4, 0x23}} ,
 {{0xeb, 0x27, 0x67, 0xc1, 0x37, 0xab, 0x7a, 0xd8, 0x27, 0x9c, 0x07, 0x8e, 0xff, 0x11, 0x6a, 0xb0, 0x78, 0x6e, 0xad, 0x3a, 0x2e, 0x0f, 0x98, 0x9f, 0x72, 0xc3, 0x7f, 0x82, 0xf2, 0x96, 0x96, 0x70}}},
{{{0x81, 0x6b, 0x88, 0xe8, 0x1e, 0xc7, 0x77, 0x96, 0x0e, 0xa1, 0xa9, 0x52, 0xe0, 0xd8, 0x0e, 0x61, 0x9e, 0x79, 0x2d, 0x95, 0x9c, 0x8d, 0x96, 0xe0, 0x06, 0x40, 0x5d, 0x87, 0x28, 0x5f, 0x98, 0x70}} ,
 {{0xf1, 0x79, 0x7b, 0xed, 0x4f, 0x44, 0xb2, 0xe7, 0x08, 0x0d, 0xc2, 0x08, 0x12, 0xd2, 0x9f, 0xdf, 0xcd, 0x93, 0x20, 0x8a, 0xcf, 0x33, 0xca, 0x6d, 0x89, 0xb9, 0x77, 0xc8, 0x93, 0x1b, 0x4e, 0x60}}},
{{{0x26, 0x4f, 0x7e, 0x97, 0xf6, 0x40, 0xdd, 0x4f, 0xfc, 0x52, 0x78, 0xf9, 0x90, 0x31, 0x03, 0xe6, 0x7d, 0x56, 0x39, 0x0b, 0x1d, 0x56, 0x82, 0x85, 0xf9, 0x1a, 0x42, 0x17, 0x69, 0x6c, 0xcf, 0x39}} ,
 {{0x69, 0xd2, 0x06, 0x3a, 0x4f, 0x39, 0x2d, 0xf9, 0x38, 0x40, 0x8c, 0x4c, 0xe7, 0x05, 0x12, 0xb4, 0x78, 0x8b, 0xf8, 0xc0, 0xec, 0x93, 0xde, 0x7a, 0x6b, 0xce, 0x2c, 0xe1, 0x0e, 0xa9, 0x34, 0x44}}},
{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
{{{0x0b, 0xa4, 0x3c, 0xb0, 0x0f, 0x7a, 0x51, 0xf1, 0x78, 0xd6, 0xd9, 0x6a, 0xfd, 0x46, 0xe8, 0xb8, 0xa8, 0x79, 0x1d, 0x87, 0xf9, 0x90, 0xf2, 0x9c, 0x13, 0x29, 0xf8, 0x0b, 0x20, 0x64, 0xfa, 0x05}} ,
 {{0x26, 0x09, 0xda, 0x17, 0xaf, 0x95, 0xd6, 0xfb, 0x6a, 0x19, 0x0d, 0x6e, 0x5e, 0x12, 0xf1, 0x99, 0x4c, 0xaa, 0xa8, 0x6f, 0x79, 0x86, 0xf4, 0x72, 0x28, 0x00, 0x26, 0xf9, 0xea, 0x9e, 0x19, 0x3d}}},
{{{0x87, 0xdd, 0xcf, 0xf0, 0x5b, 0x49, 0xa2, 0x5d, 0x40, 0x7a, 0x23, 0x26, 0xa4, 0x7a, 0x83, 0x8a, 0xb7, 0x8b, 0xd2, 0x1a, 0xbf, 0xea, 0x02, 0x24, 0x08, 0x5f, 0x7b, 0xa9, 0xb1, 0xbe, 0x9d, 0x37}} ,
 {{0xfc, 0x86, 0x4b, 0x08, 0xee, 0xe7, 0xa0, 0xfd, 0x21, 0x45, 0x09, 0x34, 0xc1, 0x61, 0x32, 0x23, 0xfc, 0x9b, 0x55, 0x48, 0x53, 0x99, 0xf7, 0x63, 0xd0, 0x99, 0xce, 0x01, 0xe0, 0x9f, 0xeb, 0x28}}},
{{{0x47, 0xfc, 0xab, 0x5a, 0x17, 0xf0, 0x85, 0x56, 0x3a, 0x30, 0x86, 0x20, 0x28, 0x4b, 0x8e, 0x44, 0x74, 0x3a, 0x6e, 0x02, 0xf1, 0x32, 0x8f, 0x9f, 0x3f, 0x08, 0x35, 0xe9, 0xca, 0x16, 0x5f, 0x6e}} ,
 {{0x1c, 0x59, 0x1c, 0x65, 0x5d, 0x34, 0xa4, 0x09, 0xcd, 0x13, 0x9c, 0x70, 0x7d, 0xb1, 0x2a, 0xc5, 0x88, 0xaf, 0x0b, 0x60, 0xc7, 0x9f, 0x34, 0x8d, 0xd6, 0xb7, 0x7f, 0xea, 0x78, 0x65, 0x8d, 0x77}}},
{{{0x56, 0xa5, 0xc2, 0x0c, 0xdd, 0xbc, 0xb8, 0x20, 0x6d, 0x57, 0x61, 0xb5, 0xfb, 0x78, 0xb5, 0xd4, 0x49, 0x54, 0x90, 0x26, 0xc1, 0xcb, 0xe9, 0xe6, 0xbf, 0xec, 0x1d, 0x4e, 0xed, 0x07, 0x7e, 0x5e}} ,
 {{0xc7, 0xf6, 0x6c, 0x56, 0x31, 0x20, 0x14, 0x0e, 0xa8, 0xd9, 0x27, 0xc1, 0x9a, 0x3d, 0x1b, 0x7d, 0x0e, 0x26, 0xd3, 0x81, 0xaa, 0xeb, 0xf5, 0x6b, 0x79, 0x02, 0xf1, 0x51, 0x5c, 0x75, 0x55, 0x0f}}},
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{{{0xdb, 0xa7, 0x9d, 0x9d, 0xbf, 0xa0, 0xcc, 0xed, 0x53, 0xa2, 0xa2, 0x19, 0x39, 0x48, 0x83, 0x19, 0x37, 0x58, 0xd1, 0x04, 0x28, 0x40, 0xf7, 0x8a, 0xc2, 0x08, 0xb7, 0xa5, 0x42, 0xcf, 0x53, 0x4c}} ,
 {{0xa7, 0xbb, 0xf6, 0x8e, 0xad, 0xdd, 0xf7, 0x90, 0xdd, 0x5f, 0x93, 0x89, 0xae, 0x04, 0x37, 0xe6, 0x9a, 0xb7, 0xe8, 0xc0, 0xdf, 0x16, 0x2a, 0xbf, 0xc4, 0x3a, 0x3c, 0x41, 0xd5, 0x89, 0x72, 0x5a}}},
{{{0x1f, 0x96, 0xff, 0x34, 0x2c, 0x13, 0x21, 0xcb, 0x0a, 0x89, 0x85, 0xbe, 0xb3, 0x70, 0x9e, 0x1e, 0xde, 0x97, 0xaf, 0x96, 0x30, 0xf7, 0x48, 0x89, 0x40, 0x8d, 0x07, 0xf1, 0x25, 0xf0, 0x30, 0x58}} ,
 {{0x1e, 0xd4, 0x93, 0x57, 0xe2, 0x17, 0xe7, 0x9d, 0xab, 0x3c, 0x55, 0x03, 0x82, 0x2f, 0x2b, 0xdb, 0x56, 0x1e, 0x30, 0x2e, 0x24, 0x47, 0x6e, 0xe6, 0xff, 0x33, 0x24, 0x2c, 0x75, 0x51, 0xd4, 0x67}}},
{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
{{{0x2b, 0x06, 0xd9, 0xa1, 0x5d, 0xe1, 0xf4, 0xd1, 0x1e, 0x3c, 0x9a, 0xc6, 0x29, 0x2b, 0x13, 0x13, 0x78, 0xc0, 0xd8, 0x16, 0x17, 0x2d, 0x9e, 0xa9, 0xc9, 0x79, 0x57, 0xab, 0x24, 0x91, 0x92, 0x19}} ,
 {{0x69, 0xfb, 0xa1, 0x9c, 0xa6, 0x75, 0x49, 0x7d, 0x60, 0x73, 0x40, 0x42, 0xc4, 0x13, 0x0a, 0x95, 0x79, 0x1e, 0x04, 0x83, 0x94, 0x99, 0x9b, 0x1e, 0x0c, 0xe8, 0x1f, 0x54, 0xef, 0xcb, 0xc0, 0x52}}},
{{{0x14, 0x89, 0x73, 0xa1, 0x37, 0x87, 0x6a, 0x7a, 0xcf, 0x1d, 0xd9, 0x2e, 0x1a, 0x67, 0xed, 0x74, 0xc0, 0xf0, 0x9c, 0x33, 0xdd, 0xdf, 0x08, 0xbf, 0x7b, 0xd1, 0x66, 0xda, 0xe6, 0xc9, 0x49, 0x08}} ,
 {{0xe9, 0xdd, 0x5e, 0x55, 0xb0, 0x0a, 0xde, 0x21, 0x4c, 0x5a, 0x2e, 0xd4, 0x80, 0x3a, 0x57, 0x92, 0x7a, 0xf1, 0xc4, 0x2c, 0x40, 0xaf, 0x2f, 0xc9, 0x92, 0x03, 0xe5, 0x5a, 0xbc, 0xdc, 0xf4, 0x09}}},
{{{0xf3, 0xe1, 0x2b, 0x7c, 0x05, 0x86, 0x80, 0x93, 0x4a, 0xad, 0xb4, 0x8f, 0x7e, 0x99, 0x0c, 0xfd, 0xcd, 0xef, 0xd1, 0xff, 0x2c, 0x69, 0x34, 0x13, 0x41, 0x64, 0xcf, 0x3b, 0xd0, 0x90, 0x09, 0x1e}} ,
 {{0x9d, 0x45, 0xd6, 0x80, 0xe6, 0x45, 0xaa, 0xf4, 0x15, 0xaa, 0x5c, 0x34, 0x87, 0x99, 0xa2, 0x8c, 0x26, 0x84, 0x62, 0x7d, 0xb6, 0x29, 0xc0, 0x52, 0xea, 0xf5, 0x81, 0x18, 0x0f, 0x35, 0xa9, 0x0e}}},
{{{0xe7, 0x20, 0x72, 0x7c, 0x6d, 0x94, 0x5f, 0x52, 0x44, 0x54, 0xe3, 0xf1, 0xb2, 0xb0, 0x36, 0x46, 0x0f, 0xae, 0x92, 0xe8, 0x70, 0x9d, 0x6e, 0x79, 0xb1, 0xad, 0x37, 0xa9, 0x5f, 0xc0, 0xde, 0x03}} ,
 {{0x15, 0x55, 0x37, 0xc6, 0x1c, 0x27, 0x1c, 0x6d, 0x14, 0x4f, 0xca, 0xa4, 0xc4, 0x88, 0x25, 0x46, 0x39, 0xfc, 0x5a, 0xe5, 0xfe, 0x29, 0x11, 0x69, 0xf5, 0x72, 0x84, 0x4d, 0x78, 0x9f, 0x94, 0x15}}},
{{{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}},
{{{0xec, 0xd3, 0xff, 0x57, 0x0b, 0xb0, 0xb2, 0xdc, 0xf8, 0x4f, 0xe2, 0x12, 0xd5, 0x36, 0xbe, 0x6b, 0x09, 0x43, 0x6d, 0xa3, 0x4d, 0x90, 0x2d, 0xb8, 0x74, 0xe8, 0x71, 0x45, 0x19, 0x8b, 0x0c, 0x6a}} ,
 {{0xb8, 0x42, 0x1c, 0x03, 0xad, 0x2c, 0x03, 0x8e, 0xac, 0xd7, 0x98, 0x29, 0x13, 0xc6, 0x02, 0x29, 0xb5, 0xd4, 0xe7, 0xcf, 0xcc, 0x8b, 0x83, 0xec, 0x35, 0xc7, 0x9c, 0x74, 0xb7, 0xad, 0x85, 0x5f}}},
{{{0x78, 0x84, 0xe1, 0x56, 0x45, 0x69, 0x68, 0x5a, 0x4f, 0xb8, 0xb1, 0x29, 0xff, 0x33, 0x03, 0x31, 0xb7, 0xcb, 0x96, 0x25, 0xe6, 0xe6, 0x41, 0x98, 0x1a, 0xbb, 0x03, 0x56, 0xf2, 0xb2, 0x91, 0x34}} ,
 {{0x2c, 0x6c, 0xf7, 0x66, 0xa4, 0x62, 0x6b, 0x39, 0xb3, 0xba, 0x65, 0xd3, 0x1c, 0xf8, 0x11, 0xaa, 0xbe, 0xdc, 0x80, 0x59, 0x87, 0xf5, 0x7b, 0xe5, 0xe3, 0xb3, 0x3e, 0x39, 0xda, 0xbe, 0x88, 0x09}}},
{{{0x8b, 0xf1, 0xa0, 0xf5, 0xdc, 0x29, 0xb4, 0xe2, 0x07, 0xc6, 0x7a, 0x00, 0xd0, 0x89, 0x17, 0x51, 0xd4, 0xbb, 0xd4, 0x22, 0xea, 0x7e, 0x7d, 0x7c, 0x24, 0xea, 0xf2, 0xe8, 0x22, 0x12, 0x95, 0x06}} ,
 {{0xda, 0x7c, 0xa4, 0x0c, 0xf4, 0xba, 0x6e, 0xe1, 0x89, 0xb5, 0x59, 0xca, 0xf1, 0xc0, 0x29, 0x36, 0x09, 0x44, 0xe2, 0x7f, 0xd1, 0x63, 0x15, 0x99, 0xea, 0x25, 0xcf, 0x0c, 0x9d, 0xc0, 0x44, 0x6f}}},
{{{0x1d, 0x86, 0x4e, 0xcf, 0xf7, 0x37, 0x10, 0x25, 0x8f, 0x12, 0xfb, 0x19, 0xfb, 0xe0, 0xed, 0x10, 0xc8, 0xe2, 0xf5, 0x75, 0xb1, 0x33, 0xc0, 0x96, 0x0d, 0xfb, 0x15, 0x6c, 0x0d, 0x07, 0x5f, 0x05}} ,
 {{0x69, 0x3e, 0x47, 0x97, 0x2c, 0xaf, 0x52, 0x7c, 0x78, 0x83, 0xad, 0x1b, 0x39, 0x82, 0x2f, 0x02, 0x6f, 0x47, 0xdb, 0x2a, 0xb0, 0xe1, 0x91, 0x99, 0x55, 0xb8, 0x99, 0x3a, 0xa0, 0x44, 0x11, 0x51}}}
};

static ZT_ALWAYS_INLINE void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p)
{
	fe25519_mul(&r->x, &p->x, &p->t);
	fe25519_mul(&r->y, &p->y, &p->z);
	fe25519_mul(&r->z, &p->z, &p->t);
}

static ZT_ALWAYS_INLINE void p1p1_to_p2_2(ge25519_p3 *r, const ge25519_p1p1 *p)
{
	fe25519_mul(&r->x, &p->x, &p->t);
	fe25519_mul(&r->y, &p->y, &p->z);
	fe25519_mul(&r->z, &p->z, &p->t);
}

static ZT_ALWAYS_INLINE void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p)
{
	p1p1_to_p2_2(r, p);
	fe25519_mul(&r->t, &p->x, &p->y);
}

static inline void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q)
{
	fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
	fe25519_mul(&qt, &q->x, &q->y);
	fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
	fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
	fe25519_sub(&t1, &q->y, &q->x);
	fe25519_add(&t2, &q->y, &q->x);
	fe25519_mul(&a, &a, &t1);
	fe25519_mul(&b, &b, &t2);
	fe25519_sub(&e, &b, &a); /* E = B-A */
	fe25519_add(&h, &b, &a); /* H = B+A */
	fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
	fe25519_sub(&f, &d, &c); /* F = D-C */
	fe25519_add(&g, &d, &c); /* G = D+C */
	fe25519_mul(&r->x, &e, &f);
	fe25519_mul(&r->y, &h, &g);
	fe25519_mul(&r->z, &g, &f);
	fe25519_mul(&r->t, &e, &h);
}

static inline void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q)
{
	fe25519 a, b, c, d, t;

	fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
	fe25519_sub(&t, &q->y, &q->x);
	fe25519_mul(&a, &a, &t);
	fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
	fe25519_add(&t, &q->x, &q->y);
	fe25519_mul(&b, &b, &t);
	fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
	fe25519_add(&d, &d, &d);
	fe25519_sub(&r->x, &b, &a); /* E = B-A */
	fe25519_sub(&r->t, &d, &c); /* F = D-C */
	fe25519_add(&r->z, &d, &c); /* G = D+C */
	fe25519_add(&r->y, &b, &a); /* H = B+A */
}

/* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
static inline void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p)
{
	fe25519 a,b,c,d;
	fe25519_square(&a, &p->x);
	fe25519_square(&b, &p->y);
	fe25519_square(&c, &p->z);
	fe25519_add(&c, &c, &c);
	fe25519_neg(&d, &a);

	fe25519_add(&r->x, &p->x, &p->y);
	fe25519_square(&r->x, &r->x);
	fe25519_sub(&r->x, &r->x, &a);
	fe25519_sub(&r->x, &r->x, &b);
	fe25519_add(&r->z, &d, &b);
	fe25519_sub(&r->t, &r->z, &c);
	fe25519_sub(&r->y, &d, &b);
}

/* Constant-time version of: if(b) r = p */
static ZT_ALWAYS_INLINE void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b)
{
	fe25519_cmov(&r->x, &p->x, b);
	fe25519_cmov(&r->y, &p->y, b);
}

static ZT_ALWAYS_INLINE unsigned char equal(signed char b,signed char c)
{
	unsigned char ub = b;
	unsigned char uc = c;
	unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
	crypto_uint32 y = x; /* 0: yes; 1..255: no */
	y -= 1; /* 4294967295: yes; 0..254: no */
	y >>= 31; /* 1: yes; 0: no */
	return (unsigned char)y;
}

static ZT_ALWAYS_INLINE unsigned char negative(signed char b)
{
	unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
	x >>= 63; /* 1: yes; 0: no */
	return (unsigned char)x;
}

static inline void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
{
	/* constant time */
	fe25519 v;
	*t = ge25519_base_multiples_affine[5*pos+0];
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
	fe25519_neg(&v, &t->x);
	fe25519_cmov(&t->x, &v, negative(b));
}

static inline void setneutral(ge25519 *r)
{
	fe25519_setzero(&r->x);
	fe25519_setone(&r->y);
	fe25519_setone(&r->z);
	fe25519_setzero(&r->t);
}

/* return 0 on success, -1 otherwise */
static inline int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
{
	unsigned char par;
	fe25519 t, chk, num, den, den2, den4, den6;
	fe25519_setone(&r->z);
	par = p[31] >> 7;
	fe25519_unpack(&r->y, p);
	fe25519_square(&num, &r->y); /* x = y^2 */
	fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
	fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
	fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */

	/* Computation of sqrt(num/den) */
	/* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
	fe25519_square(&den2, &den);
	fe25519_square(&den4, &den2);
	fe25519_mul(&den6, &den4, &den2);
	fe25519_mul(&t, &den6, &num);
	fe25519_mul(&t, &t, &den);

	fe25519_pow2523(&t, &t);
	/* 2. computation of r->x = t * num * den^3 */
	fe25519_mul(&t, &t, &num);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&r->x, &t, &den);

	/* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (!fe25519_iseq_vartime(&chk, &num))
		fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);

	/* 4. Now we have one of the two square roots, except if input was not a square */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (!fe25519_iseq_vartime(&chk, &num))
		return -1;

	/* 5. Choose the desired square root according to parity: */
	if(fe25519_getparity(&r->x) != (1-par))
		fe25519_neg(&r->x, &r->x);

	fe25519_mul(&r->t, &r->x, &r->y);
	return 0;
}

static inline void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
{
	fe25519 tx, ty, zi;
	fe25519_invert(&zi, &p->z);
	fe25519_mul(&tx, &p->x, &zi);
	fe25519_mul(&ty, &p->y, &zi);
	fe25519_pack(r, &ty);
	r[31] ^= fe25519_getparity(&tx) << 7;
}

/* computes [s1]p1 + [s2]p2 */
static inline void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2)
{
	ge25519_p1p1 tp1p1;
	ge25519_p3 pre[16];
	char *pre5 = (char *)(&(pre[5])); // eliminate type punning warning
	unsigned char b[127];
	int i;

	/* precomputation                                                        s2 s1 */
	setneutral(pre);                                                      /* 00 00 */
	pre[1] = *p1;                                                         /* 00 01 */
	dbl_p1p1(&tp1p1,(ge25519_p2 *)p1);      p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */
	add_p1p1(&tp1p1,&pre[1], &pre[2]);      p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
	pre[4] = *p2;                                                         /* 01 00 */
	add_p1p1(&tp1p1,&pre[1], &pre[4]);      p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
	add_p1p1(&tp1p1,&pre[2], &pre[4]);      p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
	add_p1p1(&tp1p1,&pre[3], &pre[4]);      p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
	dbl_p1p1(&tp1p1,(ge25519_p2 *)p2);      p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */
	add_p1p1(&tp1p1,&pre[1], &pre[8]);      p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
	dbl_p1p1(&tp1p1,(ge25519_p2 *)pre5);    p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
	add_p1p1(&tp1p1,&pre[3], &pre[8]);      p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
	add_p1p1(&tp1p1,&pre[4], &pre[8]);      p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
	add_p1p1(&tp1p1,&pre[1],&pre[12]);      p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
	add_p1p1(&tp1p1,&pre[2],&pre[12]);      p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
	add_p1p1(&tp1p1,&pre[3],&pre[12]);      p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */

	sc25519_2interleave2(b,s1,s2);

	/* scalar multiplication */
	*r = pre[b[126]];
	for(i=125;i>=0;i--)
	{
		dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
		p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
		dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
		if(b[i]!=0)
		{
			p1p1_to_p3(r, &tp1p1);
			add_p1p1(&tp1p1, r, &pre[b[i]]);
		}
		if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
		else p1p1_to_p3(r, &tp1p1);
	}
}

static inline void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
{
	signed char b[85];
	int i;
	ge25519_aff t;
	sc25519_window3(b,s);

	choose_t((ge25519_aff *)r, 0, b[0]);
	fe25519_setone(&r->z);
	fe25519_mul(&r->t, &r->x, &r->y);
	for(i=1;i<85;i++)
	{
		choose_t(&t, (unsigned long long) i, b[i]);
		ge25519_mixadd2(r, &t);
	}
}

static ZT_ALWAYS_INLINE void get_hram(unsigned char *hram, const unsigned char *sm, const unsigned char *pk, unsigned char *playground, unsigned long long smlen)
{
	unsigned long long i;

	for (i =  0;i < 32;++i)    playground[i] = sm[i];
	for (i = 32;i < 64;++i)    playground[i] = pk[i-32];
	for (i = 64;i < smlen;++i) playground[i] = sm[i];

	ZeroTier::SHA512(hram,playground,(unsigned int)smlen);
}

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

} // anonymous namespace

namespace ZeroTier {

void C25519::agree(const uint8_t mine[ZT_C25519_PRIVATE_KEY_LEN],const uint8_t their[ZT_C25519_PUBLIC_KEY_LEN],uint8_t rawkey[32])
{
	crypto_scalarmult(rawkey,mine,their);
}

void C25519::sign(const uint8_t myPrivate[ZT_C25519_PRIVATE_KEY_LEN],const uint8_t myPublic[ZT_C25519_PUBLIC_KEY_LEN],const void *msg,unsigned int len,void *signature)
{
	unsigned char digest[64]; // we sign the first 32 bytes of SHA-512(msg)
	SHA512(digest,msg,len);

	sc25519 sck, scs, scsk;
	ge25519 ger;
	unsigned char r[32];
	unsigned char s[32];
	unsigned char extsk[64];
	unsigned char hmg[crypto_hash_sha512_BYTES];
	unsigned char hram[crypto_hash_sha512_BYTES];
	unsigned char *sig = (unsigned char *)signature;

	SHA512(extsk,myPrivate + 32,32);
	extsk[0] &= 248;
	extsk[31] &= 127;
	extsk[31] |= 64;

	for(unsigned int i=0;i<32;i++)
		sig[32 + i] = extsk[32 + i];
	for(unsigned int i=0;i<32;i++)
		sig[64 + i] = digest[i];

	SHA512(hmg,sig + 32,64);

	/* Computation of R */
	sc25519_from64bytes(&sck, hmg);
	ge25519_scalarmult_base(&ger, &sck);
	ge25519_pack(r, &ger);

	/* Computation of s */
	for(unsigned int i=0;i<32;i++)
		sig[i] = r[i];

	get_hram(hram,sig,myPublic + 32,sig,96);

	sc25519_from64bytes(&scs, hram);
	sc25519_from32bytes(&scsk, extsk);
	sc25519_mul(&scs, &scs, &scsk);

	sc25519_add(&scs, &scs, &sck);

	sc25519_to32bytes(s,&scs); /* cat s */
	for(unsigned int i=0;i<32;i++)
		sig[32 + i] = s[i];
}

bool C25519::verify(const uint8_t their[ZT_C25519_PUBLIC_KEY_LEN],const void *msg,unsigned int len,const void *signature,const unsigned int siglen)
{
	if (siglen < 64) return false;

	const unsigned char *sig = (const unsigned char *)signature;
	unsigned char digest[64]; // we sign the first 32 bytes of SHA-512(msg)
	unsigned char sigtmp[96];
	SHA512(digest,msg,len);

	if ((siglen == 96)&&(!Utils::secureEq(sig+64,digest,32))) {
		return false;
	} else if (siglen == 64) {
		memcpy(sigtmp,sig,64);
		memcpy(sigtmp+64,digest,32);
		sig = sigtmp;
	}

	unsigned char t2[32];
	ge25519 get1, get2;
	sc25519 schram, scs;
	unsigned char hram[crypto_hash_sha512_BYTES];
	unsigned char m[96];

	if (ge25519_unpackneg_vartime(&get1,their + 32))
		return false;

	get_hram(hram,sig,their + 32,m,96);

	sc25519_from64bytes(&schram, hram);

	sc25519_from32bytes(&scs, sig+32);

	ge25519_double_scalarmult_vartime(&get2, &get1, &schram, &ge25519_base, &scs);
	ge25519_pack(t2, &get2);

	return Utils::secureEq(sig,t2,32);
}

void C25519::_calcPubDH(uint8_t pub[ZT_C25519_PUBLIC_KEY_LEN],const uint8_t priv[ZT_C25519_PRIVATE_KEY_LEN])
{
	// First 32 bytes of pub and priv are the keys for ECDH key
	// agreement. This generates the public portion from the private.
	crypto_scalarmult_base(pub,priv);
}

void C25519::_calcPubED(uint8_t pub[ZT_C25519_PUBLIC_KEY_LEN],const uint8_t priv[ZT_C25519_PRIVATE_KEY_LEN])
{
	unsigned char extsk[64];
	sc25519 scsk;
	ge25519 gepk;

	// Second 32 bytes of pub and priv are the keys for ed25519
	// signing and verification.
	SHA512(extsk,priv + 32,32);
	extsk[0] &= 248;
	extsk[31] &= 127;
	extsk[31] |= 64;
	sc25519_from32bytes(&scsk,extsk);
	ge25519_scalarmult_base(&gepk,&scsk);
	ge25519_pack(pub + 32,&gepk);

	// In NaCl, the public key is crammed into the next 32 bytes
	// of the private key for signing since both keys are required
	// to sign. In this version we just get it from kp.pub, so we
	// leave that out of private.
}

} // namespace ZeroTier