path: root/src/lib/pubkey/ed25519/ed25519_fe.cpp

/*
* Ed25519 field element
* (C) 2017 Ribose Inc
*
* Based on the public domain code from SUPERCOP ref10 by
* Peter Schwabe, Daniel J. Bernstein, Niels Duif, Tanja Lange, Bo-Yin Yang
*
* Botan is released under the Simplified BSD License (see license.txt)
*/

#include <botan/internal/ed25519_fe.h>
#include <botan/internal/ed25519_internal.h>

namespace Botan {

//static
FE_25519 FE_25519::invert(const FE_25519& z)
   {
   fe t0;
   fe t1;
   fe t2;
   fe t3;

   fe_sq(t0, z);
   fe_sq_iter(t1, t0, 2);
   fe_mul(t1, z, t1);
   fe_mul(t0, t0, t1);
   fe_sq(t2, t0);
   fe_mul(t1, t1, t2);
   fe_sq_iter(t2, t1, 5);
   fe_mul(t1, t2, t1);
   fe_sq_iter(t2, t1, 10);
   fe_mul(t2, t2, t1);
   fe_sq_iter(t3, t2, 20);
   fe_mul(t2, t3, t2);
   fe_sq_iter(t2, t2, 10);
   fe_mul(t1, t2, t1);
   fe_sq_iter(t2, t1, 50);
   fe_mul(t2, t2, t1);
   fe_sq_iter(t3, t2, 100);
   fe_mul(t2, t3, t2);
   fe_sq_iter(t2, t2, 50);
   fe_mul(t1, t2, t1);
   fe_sq_iter(t1, t1, 5);

   fe_mul(t0, t1, t0);
   return t0;
   }

FE_25519 FE_25519::pow_22523(const fe& z)
   {
   fe t0;
   fe t1;
   fe t2;

   fe_sq(t0, z);
   fe_sq_iter(t1, t0, 2);
   fe_mul(t1, z, t1);
   fe_mul(t0, t0, t1);
   fe_sq(t0, t0);
   fe_mul(t0, t1, t0);
   fe_sq_iter(t1, t0, 5);
   fe_mul(t0, t1, t0);
   fe_sq_iter(t1, t0, 10);
   fe_mul(t1, t1, t0);
   fe_sq_iter(t2, t1, 20);
   fe_mul(t1, t2, t1);
   fe_sq_iter(t1, t1, 10);
   fe_mul(t0, t1, t0);
   fe_sq_iter(t1, t0, 50);
   fe_mul(t1, t1, t0);
   fe_sq_iter(t2, t1, 100);
   fe_mul(t1, t2, t1);
   fe_sq_iter(t1, t1, 50);
   fe_mul(t0, t1, t0);
   fe_sq_iter(t0, t0, 2);

   fe_mul(t0, t0, z);
   return t0;
   }

/*
h = f * g
Can overlap h with f or g.

Preconditions:
|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
|g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.

Postconditions:
|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
*/

/*
Notes on implementation strategy:

Using schoolbook multiplication.
Karatsuba would save a little in some cost models.

Most multiplications by 2 and 19 are 32-bit precomputations;
cheaper than 64-bit postcomputations.

There is one remaining multiplication by 19 in the carry chain;
one *19 precomputation can be merged into this,
but the resulting data flow is considerably less clean.

There are 12 carries below.
10 of them are 2-way parallelizable and vectorizable.
Can get away with 11 carries, but then data flow is much deeper.

With tighter constraints on inputs can squeeze carries into int32.
*/

//static
FE_25519 FE_25519::mul(const FE_25519& f, const FE_25519& g)
   {
   int32_t f0 = f[0];
   int32_t f1 = f[1];
   int32_t f2 = f[2];
   int32_t f3 = f[3];
   int32_t f4 = f[4];
   int32_t f5 = f[5];
   int32_t f6 = f[6];
   int32_t f7 = f[7];
   int32_t f8 = f[8];
   int32_t f9 = f[9];

   int32_t g0 = g[0];
   int32_t g1 = g[1];
   int32_t g2 = g[2];
   int32_t g3 = g[3];
   int32_t g4 = g[4];
   int32_t g5 = g[5];
   int32_t g6 = g[6];
   int32_t g7 = g[7];
   int32_t g8 = g[8];
   int32_t g9 = g[9];

   int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
   int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
   int32_t g3_19 = 19 * g3;
   int32_t g4_19 = 19 * g4;
   int32_t g5_19 = 19 * g5;
   int32_t g6_19 = 19 * g6;
   int32_t g7_19 = 19 * g7;
   int32_t g8_19 = 19 * g8;
   int32_t g9_19 = 19 * g9;
   int32_t f1_2 = 2 * f1;
   int32_t f3_2 = 2 * f3;
   int32_t f5_2 = 2 * f5;
   int32_t f7_2 = 2 * f7;
   int32_t f9_2 = 2 * f9;
   int64_t f0g0    = f0   * static_cast<int64_t>(g0);
   int64_t f0g1    = f0   * static_cast<int64_t>(g1);
   int64_t f0g2    = f0   * static_cast<int64_t>(g2);
   int64_t f0g3    = f0   * static_cast<int64_t>(g3);
   int64_t f0g4    = f0   * static_cast<int64_t>(g4);
   int64_t f0g5    = f0   * static_cast<int64_t>(g5);
   int64_t f0g6    = f0   * static_cast<int64_t>(g6);
   int64_t f0g7    = f0   * static_cast<int64_t>(g7);
   int64_t f0g8    = f0   * static_cast<int64_t>(g8);
   int64_t f0g9    = f0   * static_cast<int64_t>(g9);
   int64_t f1g0    = f1   * static_cast<int64_t>(g0);
   int64_t f1g1_2  = f1_2 * static_cast<int64_t>(g1);
   int64_t f1g2    = f1   * static_cast<int64_t>(g2);
   int64_t f1g3_2  = f1_2 * static_cast<int64_t>(g3);
   int64_t f1g4    = f1   * static_cast<int64_t>(g4);
   int64_t f1g5_2  = f1_2 * static_cast<int64_t>(g5);
   int64_t f1g6    = f1   * static_cast<int64_t>(g6);
   int64_t f1g7_2  = f1_2 * static_cast<int64_t>(g7);
   int64_t f1g8    = f1   * static_cast<int64_t>(g8);
   int64_t f1g9_38 = f1_2 * static_cast<int64_t>(g9_19);
   int64_t f2g0    = f2   * static_cast<int64_t>(g0);
   int64_t f2g1    = f2   * static_cast<int64_t>(g1);
   int64_t f2g2    = f2   * static_cast<int64_t>(g2);
   int64_t f2g3    = f2   * static_cast<int64_t>(g3);
   int64_t f2g4    = f2   * static_cast<int64_t>(g4);
   int64_t f2g5    = f2   * static_cast<int64_t>(g5);
   int64_t f2g6    = f2   * static_cast<int64_t>(g6);
   int64_t f2g7    = f2   * static_cast<int64_t>(g7);
   int64_t f2g8_19 = f2   * static_cast<int64_t>(g8_19);
   int64_t f2g9_19 = f2   * static_cast<int64_t>(g9_19);
   int64_t f3g0    = f3   * static_cast<int64_t>(g0);
   int64_t f3g1_2  = f3_2 * static_cast<int64_t>(g1);
   int64_t f3g2    = f3   * static_cast<int64_t>(g2);
   int64_t f3g3_2  = f3_2 * static_cast<int64_t>(g3);
   int64_t f3g4    = f3   * static_cast<int64_t>(g4);
   int64_t f3g5_2  = f3_2 * static_cast<int64_t>(g5);
   int64_t f3g6    = f3   * static_cast<int64_t>(g6);
   int64_t f3g7_38 = f3_2 * static_cast<int64_t>(g7_19);
   int64_t f3g8_19 = f3   * static_cast<int64_t>(g8_19);
   int64_t f3g9_38 = f3_2 * static_cast<int64_t>(g9_19);
   int64_t f4g0    = f4   * static_cast<int64_t>(g0);
   int64_t f4g1    = f4   * static_cast<int64_t>(g1);
   int64_t f4g2    = f4   * static_cast<int64_t>(g2);
   int64_t f4g3    = f4   * static_cast<int64_t>(g3);
   int64_t f4g4    = f4   * static_cast<int64_t>(g4);
   int64_t f4g5    = f4   * static_cast<int64_t>(g5);
   int64_t f4g6_19 = f4   * static_cast<int64_t>(g6_19);
   int64_t f4g7_19 = f4   * static_cast<int64_t>(g7_19);
   int64_t f4g8_19 = f4   * static_cast<int64_t>(g8_19);
   int64_t f4g9_19 = f4   * static_cast<int64_t>(g9_19);
   int64_t f5g0    = f5   * static_cast<int64_t>(g0);
   int64_t f5g1_2  = f5_2 * static_cast<int64_t>(g1);
   int64_t f5g2    = f5   * static_cast<int64_t>(g2);
   int64_t f5g3_2  = f5_2 * static_cast<int64_t>(g3);
   int64_t f5g4    = f5   * static_cast<int64_t>(g4);
   int64_t f5g5_38 = f5_2 * static_cast<int64_t>(g5_19);
   int64_t f5g6_19 = f5   * static_cast<int64_t>(g6_19);
   int64_t f5g7_38 = f5_2 * static_cast<int64_t>(g7_19);
   int64_t f5g8_19 = f5   * static_cast<int64_t>(g8_19);
   int64_t f5g9_38 = f5_2 * static_cast<int64_t>(g9_19);
   int64_t f6g0    = f6   * static_cast<int64_t>(g0);
   int64_t f6g1    = f6   * static_cast<int64_t>(g1);
   int64_t f6g2    = f6   * static_cast<int64_t>(g2);
   int64_t f6g3    = f6   * static_cast<int64_t>(g3);
   int64_t f6g4_19 = f6   * static_cast<int64_t>(g4_19);
   int64_t f6g5_19 = f6   * static_cast<int64_t>(g5_19);
   int64_t f6g6_19 = f6   * static_cast<int64_t>(g6_19);
   int64_t f6g7_19 = f6   * static_cast<int64_t>(g7_19);
   int64_t f6g8_19 = f6   * static_cast<int64_t>(g8_19);
   int64_t f6g9_19 = f6   * static_cast<int64_t>(g9_19);
   int64_t f7g0    = f7   * static_cast<int64_t>(g0);
   int64_t f7g1_2  = f7_2 * static_cast<int64_t>(g1);
   int64_t f7g2    = f7   * static_cast<int64_t>(g2);
   int64_t f7g3_38 = f7_2 * static_cast<int64_t>(g3_19);
   int64_t f7g4_19 = f7   * static_cast<int64_t>(g4_19);
   int64_t f7g5_38 = f7_2 * static_cast<int64_t>(g5_19);
   int64_t f7g6_19 = f7   * static_cast<int64_t>(g6_19);
   int64_t f7g7_38 = f7_2 * static_cast<int64_t>(g7_19);
   int64_t f7g8_19 = f7   * static_cast<int64_t>(g8_19);
   int64_t f7g9_38 = f7_2 * static_cast<int64_t>(g9_19);
   int64_t f8g0    = f8   * static_cast<int64_t>(g0);
   int64_t f8g1    = f8   * static_cast<int64_t>(g1);
   int64_t f8g2_19 = f8   * static_cast<int64_t>(g2_19);
   int64_t f8g3_19 = f8   * static_cast<int64_t>(g3_19);
   int64_t f8g4_19 = f8   * static_cast<int64_t>(g4_19);
   int64_t f8g5_19 = f8   * static_cast<int64_t>(g5_19);
   int64_t f8g6_19 = f8   * static_cast<int64_t>(g6_19);
   int64_t f8g7_19 = f8   * static_cast<int64_t>(g7_19);
   int64_t f8g8_19 = f8   * static_cast<int64_t>(g8_19);
   int64_t f8g9_19 = f8   * static_cast<int64_t>(g9_19);
   int64_t f9g0    = f9   * static_cast<int64_t>(g0);
   int64_t f9g1_38 = f9_2 * static_cast<int64_t>(g1_19);
   int64_t f9g2_19 = f9   * static_cast<int64_t>(g2_19);
   int64_t f9g3_38 = f9_2 * static_cast<int64_t>(g3_19);
   int64_t f9g4_19 = f9   * static_cast<int64_t>(g4_19);
   int64_t f9g5_38 = f9_2 * static_cast<int64_t>(g5_19);
   int64_t f9g6_19 = f9   * static_cast<int64_t>(g6_19);
   int64_t f9g7_38 = f9_2 * static_cast<int64_t>(g7_19);
   int64_t f9g8_19 = f9   * static_cast<int64_t>(g8_19);
   int64_t f9g9_38 = f9_2 * static_cast<int64_t>(g9_19);
   int64_t h0 = f0g0+f1g9_38+f2g8_19+f3g7_38+f4g6_19+f5g5_38+f6g4_19+f7g3_38+f8g2_19+f9g1_38;
   int64_t h1 = f0g1+f1g0   +f2g9_19+f3g8_19+f4g7_19+f5g6_19+f6g5_19+f7g4_19+f8g3_19+f9g2_19;
   int64_t h2 = f0g2+f1g1_2 +f2g0   +f3g9_38+f4g8_19+f5g7_38+f6g6_19+f7g5_38+f8g4_19+f9g3_38;
   int64_t h3 = f0g3+f1g2   +f2g1   +f3g0   +f4g9_19+f5g8_19+f6g7_19+f7g6_19+f8g5_19+f9g4_19;
   int64_t h4 = f0g4+f1g3_2 +f2g2   +f3g1_2 +f4g0   +f5g9_38+f6g8_19+f7g7_38+f8g6_19+f9g5_38;
   int64_t h5 = f0g5+f1g4   +f2g3   +f3g2   +f4g1   +f5g0   +f6g9_19+f7g8_19+f8g7_19+f9g6_19;
   int64_t h6 = f0g6+f1g5_2 +f2g4   +f3g3_2 +f4g2   +f5g1_2 +f6g0   +f7g9_38+f8g8_19+f9g7_38;
   int64_t h7 = f0g7+f1g6   +f2g5   +f3g4   +f4g3   +f5g2   +f6g1   +f7g0   +f8g9_19+f9g8_19;
   int64_t h8 = f0g8+f1g7_2 +f2g6   +f3g5_2 +f4g4   +f5g3_2 +f6g2   +f7g1_2 +f8g0   +f9g9_38;
   int64_t h9 = f0g9+f1g8   +f2g7   +f3g6   +f4g5   +f5g4   +f6g3   +f7g2   +f8g1   +f9g0   ;
   int64_t carry0;
   int64_t carry1;
   int64_t carry2;
   int64_t carry3;
   int64_t carry4;
   int64_t carry5;
   int64_t carry6;
   int64_t carry7;
   int64_t carry8;
   int64_t carry9;

   /*
   |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
   |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9
   */
   const int64_t X24 = (1 << 24);
   const int64_t X25 = (1 << 25);
   const int64_t X26 = (1 << 26);

   carry0 = (h0 + X25) >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;
   carry4 = (h4 + X25) >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;
   /* |h0| <= 2^25 */
   /* |h4| <= 2^25 */
   /* |h1| <= 1.71*2^59 */
   /* |h5| <= 1.71*2^59 */

   carry1 = (h1 + X24) >> 25;
   h2 += carry1;
   h1 -= carry1 * X25;
   carry5 = (h5 + X24) >> 25;
   h6 += carry5;
   h5 -= carry5 * X25;
   /* |h1| <= 2^24; from now on fits into int32 */
   /* |h5| <= 2^24; from now on fits into int32 */
   /* |h2| <= 1.41*2^60 */
   /* |h6| <= 1.41*2^60 */

   carry2 = (h2 + X25) >> 26;
   h3 += carry2;
   h2 -= carry2 * X26;
   carry6 = (h6 + X25) >> 26;
   h7 += carry6;
   h6 -= carry6 * X26;
   /* |h2| <= 2^25; from now on fits into int32 unchanged */
   /* |h6| <= 2^25; from now on fits into int32 unchanged */
   /* |h3| <= 1.71*2^59 */
   /* |h7| <= 1.71*2^59 */

   carry3 = (h3 + X24) >> 25;
   h4 += carry3;
   h3 -= carry3 * X25;
   carry7 = (h7 + X24) >> 25;
   h8 += carry7;
   h7 -= carry7 * X25;
   /* |h3| <= 2^24; from now on fits into int32 unchanged */
   /* |h7| <= 2^24; from now on fits into int32 unchanged */
   /* |h4| <= 1.72*2^34 */
   /* |h8| <= 1.41*2^60 */

   carry4 = (h4 + X25) >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;
   carry8 = (h8 + X25) >> 26;
   h9 += carry8;
   h8 -= carry8 * X26;
   /* |h4| <= 2^25; from now on fits into int32 unchanged */
   /* |h8| <= 2^25; from now on fits into int32 unchanged */
   /* |h5| <= 1.01*2^24 */
   /* |h9| <= 1.71*2^59 */

   carry9 = (h9 + X24) >> 25;
   h0 += carry9 * 19;
   h9 -= carry9 * X25;
   /* |h9| <= 2^24; from now on fits into int32 unchanged */
   /* |h0| <= 1.1*2^39 */

   carry0 = (h0 + X25) >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;
   /* |h0| <= 2^25; from now on fits into int32 unchanged */
   /* |h1| <= 1.01*2^24 */

   return FE_25519(h0, h1, h2, h3, h4, h5, h6, h7, h8, h9);
   }

/*
h = f * f
Can overlap h with f.

Preconditions:
|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.

Postconditions:
|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
*/

/*
See fe_mul.c for discussion of implementation strategy.
*/

//static
FE_25519 FE_25519::sqr_iter(const FE_25519& f, size_t iter)
   {
   const int64_t X24 = (1 << 24);
   const int64_t X25 = (1 << 25);
   const int64_t X26 = (1 << 26);

   int32_t f0 = f[0];
   int32_t f1 = f[1];
   int32_t f2 = f[2];
   int32_t f3 = f[3];
   int32_t f4 = f[4];
   int32_t f5 = f[5];
   int32_t f6 = f[6];
   int32_t f7 = f[7];
   int32_t f8 = f[8];
   int32_t f9 = f[9];

   for(size_t i = 0; i != iter; ++i)
      {
      const int32_t f0_2 = 2 * f0;
      const int32_t f1_2 = 2 * f1;
      const int32_t f2_2 = 2 * f2;
      const int32_t f3_2 = 2 * f3;
      const int32_t f4_2 = 2 * f4;
      const int32_t f5_2 = 2 * f5;
      const int32_t f6_2 = 2 * f6;
      const int32_t f7_2 = 2 * f7;
      const int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
      const int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
      const int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
      const int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
      const int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */

      const int64_t f0f0    = f0   * static_cast<int64_t>(f0);
      const int64_t f0f1_2  = f0_2 * static_cast<int64_t>(f1);
      const int64_t f0f2_2  = f0_2 * static_cast<int64_t>(f2);
      const int64_t f0f3_2  = f0_2 * static_cast<int64_t>(f3);
      const int64_t f0f4_2  = f0_2 * static_cast<int64_t>(f4);
      const int64_t f0f5_2  = f0_2 * static_cast<int64_t>(f5);
      const int64_t f0f6_2  = f0_2 * static_cast<int64_t>(f6);
      const int64_t f0f7_2  = f0_2 * static_cast<int64_t>(f7);
      const int64_t f0f8_2  = f0_2 * static_cast<int64_t>(f8);
      const int64_t f0f9_2  = f0_2 * static_cast<int64_t>(f9);
      const int64_t f1f1_2  = f1_2 * static_cast<int64_t>(f1);
      const int64_t f1f2_2  = f1_2 * static_cast<int64_t>(f2);
      const int64_t f1f3_4  = f1_2 * static_cast<int64_t>(f3_2);
      const int64_t f1f4_2  = f1_2 * static_cast<int64_t>(f4);
      const int64_t f1f5_4  = f1_2 * static_cast<int64_t>(f5_2);
      const int64_t f1f6_2  = f1_2 * static_cast<int64_t>(f6);
      const int64_t f1f7_4  = f1_2 * static_cast<int64_t>(f7_2);
      const int64_t f1f8_2  = f1_2 * static_cast<int64_t>(f8);
      const int64_t f1f9_76 = f1_2 * static_cast<int64_t>(f9_38);
      const int64_t f2f2    = f2   * static_cast<int64_t>(f2);
      const int64_t f2f3_2  = f2_2 * static_cast<int64_t>(f3);
      const int64_t f2f4_2  = f2_2 * static_cast<int64_t>(f4);
      const int64_t f2f5_2  = f2_2 * static_cast<int64_t>(f5);
      const int64_t f2f6_2  = f2_2 * static_cast<int64_t>(f6);
      const int64_t f2f7_2  = f2_2 * static_cast<int64_t>(f7);
      const int64_t f2f8_38 = f2_2 * static_cast<int64_t>(f8_19);
      const int64_t f2f9_38 = f2   * static_cast<int64_t>(f9_38);
      const int64_t f3f3_2  = f3_2 * static_cast<int64_t>(f3);
      const int64_t f3f4_2  = f3_2 * static_cast<int64_t>(f4);
      const int64_t f3f5_4  = f3_2 * static_cast<int64_t>(f5_2);
      const int64_t f3f6_2  = f3_2 * static_cast<int64_t>(f6);
      const int64_t f3f7_76 = f3_2 * static_cast<int64_t>(f7_38);
      const int64_t f3f8_38 = f3_2 * static_cast<int64_t>(f8_19);
      const int64_t f3f9_76 = f3_2 * static_cast<int64_t>(f9_38);
      const int64_t f4f4    = f4   * static_cast<int64_t>(f4);
      const int64_t f4f5_2  = f4_2 * static_cast<int64_t>(f5);
      const int64_t f4f6_38 = f4_2 * static_cast<int64_t>(f6_19);
      const int64_t f4f7_38 = f4   * static_cast<int64_t>(f7_38);
      const int64_t f4f8_38 = f4_2 * static_cast<int64_t>(f8_19);
      const int64_t f4f9_38 = f4   * static_cast<int64_t>(f9_38);
      const int64_t f5f5_38 = f5   * static_cast<int64_t>(f5_38);
      const int64_t f5f6_38 = f5_2 * static_cast<int64_t>(f6_19);
      const int64_t f5f7_76 = f5_2 * static_cast<int64_t>(f7_38);
      const int64_t f5f8_38 = f5_2 * static_cast<int64_t>(f8_19);
      const int64_t f5f9_76 = f5_2 * static_cast<int64_t>(f9_38);
      const int64_t f6f6_19 = f6   * static_cast<int64_t>(f6_19);
      const int64_t f6f7_38 = f6   * static_cast<int64_t>(f7_38);
      const int64_t f6f8_38 = f6_2 * static_cast<int64_t>(f8_19);
      const int64_t f6f9_38 = f6   * static_cast<int64_t>(f9_38);
      const int64_t f7f7_38 = f7   * static_cast<int64_t>(f7_38);
      const int64_t f7f8_38 = f7_2 * static_cast<int64_t>(f8_19);
      const int64_t f7f9_76 = f7_2 * static_cast<int64_t>(f9_38);
      const int64_t f8f8_19 = f8   * static_cast<int64_t>(f8_19);
      const int64_t f8f9_38 = f8   * static_cast<int64_t>(f9_38);
      const int64_t f9f9_38 = f9   * static_cast<int64_t>(f9_38);

      int64_t h0 = f0f0  +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
      int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
      int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
      int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
      int64_t h4 = f0f4_2+f1f3_4 +f2f2   +f5f9_76+f6f8_38+f7f7_38;
      int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
      int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
      int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
      int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4   +f9f9_38;
      int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;

      int64_t carry0;
      int64_t carry1;
      int64_t carry2;
      int64_t carry3;
      int64_t carry4;
      int64_t carry5;
      int64_t carry6;
      int64_t carry7;
      int64_t carry8;
      int64_t carry9;

      carry0 = (h0 + X25) >> 26;
      h1 += carry0;
      h0 -= carry0 * X26;
      carry4 = (h4 + X25) >> 26;
      h5 += carry4;
      h4 -= carry4 * X26;

      carry1 = (h1 + X24) >> 25;
      h2 += carry1;
      h1 -= carry1 * X25;
      carry5 = (h5 + X24) >> 25;
      h6 += carry5;
      h5 -= carry5 * X25;

      carry2 = (h2 + X25) >> 26;
      h3 += carry2;
      h2 -= carry2 * X26;
      carry6 = (h6 + X25) >> 26;
      h7 += carry6;
      h6 -= carry6 * X26;

      carry3 = (h3 + X24) >> 25;
      h4 += carry3;
      h3 -= carry3 * X25;
      carry7 = (h7 + X24) >> 25;
      h8 += carry7;
      h7 -= carry7 * X25;

      carry4 = (h4 + X25) >> 26;
      h5 += carry4;
      h4 -= carry4 * X26;
      carry8 = (h8 + X25) >> 26;
      h9 += carry8;
      h8 -= carry8 * X26;

      carry9 = (h9 + X24) >> 25;
      h0 += carry9 * 19;
      h9 -= carry9 * X25;
      carry0 = (h0 + X25) >> 26;
      h1 += carry0;
      h0 -= carry0 * X26;

      f0 = h0;
      f1 = h1;
      f2 = h2;
      f3 = h3;
      f4 = h4;
      f5 = h5;
      f6 = h6;
      f7 = h7;
      f8 = h8;
      f9 = h9;
      }

   return FE_25519(f0, f1, f2, f3, f4, f5, f6, f7, f8, f9);
   }

/*
h = 2 * f * f
Can overlap h with f.

Preconditions:
|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.

Postconditions:
|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
*/

/*
See fe_mul.c for discussion of implementation strategy.
*/

//static
FE_25519 FE_25519::sqr2(const FE_25519& f)
   {
   const int64_t X24 = (1 << 24);
   const int64_t X25 = (1 << 25);
   const int64_t X26 = (1 << 26);

   int32_t f0 = f[0];
   int32_t f1 = f[1];
   int32_t f2 = f[2];
   int32_t f3 = f[3];
   int32_t f4 = f[4];
   int32_t f5 = f[5];
   int32_t f6 = f[6];
   int32_t f7 = f[7];
   int32_t f8 = f[8];
   int32_t f9 = f[9];
   int32_t f0_2 = 2 * f0;
   int32_t f1_2 = 2 * f1;
   int32_t f2_2 = 2 * f2;
   int32_t f3_2 = 2 * f3;
   int32_t f4_2 = 2 * f4;
   int32_t f5_2 = 2 * f5;
   int32_t f6_2 = 2 * f6;
   int32_t f7_2 = 2 * f7;
   int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
   int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
   int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
   int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
   int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
   int64_t f0f0    = f0   * static_cast<int64_t>(f0);
   int64_t f0f1_2  = f0_2 * static_cast<int64_t>(f1);
   int64_t f0f2_2  = f0_2 * static_cast<int64_t>(f2);
   int64_t f0f3_2  = f0_2 * static_cast<int64_t>(f3);
   int64_t f0f4_2  = f0_2 * static_cast<int64_t>(f4);
   int64_t f0f5_2  = f0_2 * static_cast<int64_t>(f5);
   int64_t f0f6_2  = f0_2 * static_cast<int64_t>(f6);
   int64_t f0f7_2  = f0_2 * static_cast<int64_t>(f7);
   int64_t f0f8_2  = f0_2 * static_cast<int64_t>(f8);
   int64_t f0f9_2  = f0_2 * static_cast<int64_t>(f9);
   int64_t f1f1_2  = f1_2 * static_cast<int64_t>(f1);
   int64_t f1f2_2  = f1_2 * static_cast<int64_t>(f2);
   int64_t f1f3_4  = f1_2 * static_cast<int64_t>(f3_2);
   int64_t f1f4_2  = f1_2 * static_cast<int64_t>(f4);
   int64_t f1f5_4  = f1_2 * static_cast<int64_t>(f5_2);
   int64_t f1f6_2  = f1_2 * static_cast<int64_t>(f6);
   int64_t f1f7_4  = f1_2 * static_cast<int64_t>(f7_2);
   int64_t f1f8_2  = f1_2 * static_cast<int64_t>(f8);
   int64_t f1f9_76 = f1_2 * static_cast<int64_t>(f9_38);
   int64_t f2f2    = f2   * static_cast<int64_t>(f2);
   int64_t f2f3_2  = f2_2 * static_cast<int64_t>(f3);
   int64_t f2f4_2  = f2_2 * static_cast<int64_t>(f4);
   int64_t f2f5_2  = f2_2 * static_cast<int64_t>(f5);
   int64_t f2f6_2  = f2_2 * static_cast<int64_t>(f6);
   int64_t f2f7_2  = f2_2 * static_cast<int64_t>(f7);
   int64_t f2f8_38 = f2_2 * static_cast<int64_t>(f8_19);
   int64_t f2f9_38 = f2   * static_cast<int64_t>(f9_38);
   int64_t f3f3_2  = f3_2 * static_cast<int64_t>(f3);
   int64_t f3f4_2  = f3_2 * static_cast<int64_t>(f4);
   int64_t f3f5_4  = f3_2 * static_cast<int64_t>(f5_2);
   int64_t f3f6_2  = f3_2 * static_cast<int64_t>(f6);
   int64_t f3f7_76 = f3_2 * static_cast<int64_t>(f7_38);
   int64_t f3f8_38 = f3_2 * static_cast<int64_t>(f8_19);
   int64_t f3f9_76 = f3_2 * static_cast<int64_t>(f9_38);
   int64_t f4f4    = f4   * static_cast<int64_t>(f4);
   int64_t f4f5_2  = f4_2 * static_cast<int64_t>(f5);
   int64_t f4f6_38 = f4_2 * static_cast<int64_t>(f6_19);
   int64_t f4f7_38 = f4   * static_cast<int64_t>(f7_38);
   int64_t f4f8_38 = f4_2 * static_cast<int64_t>(f8_19);
   int64_t f4f9_38 = f4   * static_cast<int64_t>(f9_38);
   int64_t f5f5_38 = f5   * static_cast<int64_t>(f5_38);
   int64_t f5f6_38 = f5_2 * static_cast<int64_t>(f6_19);
   int64_t f5f7_76 = f5_2 * static_cast<int64_t>(f7_38);
   int64_t f5f8_38 = f5_2 * static_cast<int64_t>(f8_19);
   int64_t f5f9_76 = f5_2 * static_cast<int64_t>(f9_38);
   int64_t f6f6_19 = f6   * static_cast<int64_t>(f6_19);
   int64_t f6f7_38 = f6   * static_cast<int64_t>(f7_38);
   int64_t f6f8_38 = f6_2 * static_cast<int64_t>(f8_19);
   int64_t f6f9_38 = f6   * static_cast<int64_t>(f9_38);
   int64_t f7f7_38 = f7   * static_cast<int64_t>(f7_38);
   int64_t f7f8_38 = f7_2 * static_cast<int64_t>(f8_19);
   int64_t f7f9_76 = f7_2 * static_cast<int64_t>(f9_38);
   int64_t f8f8_19 = f8   * static_cast<int64_t>(f8_19);
   int64_t f8f9_38 = f8   * static_cast<int64_t>(f9_38);
   int64_t f9f9_38 = f9   * static_cast<int64_t>(f9_38);
   int64_t h0 = f0f0  +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
   int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
   int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
   int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
   int64_t h4 = f0f4_2+f1f3_4 +f2f2   +f5f9_76+f6f8_38+f7f7_38;
   int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
   int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
   int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
   int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4   +f9f9_38;
   int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;
   int64_t carry0;
   int64_t carry1;
   int64_t carry2;
   int64_t carry3;
   int64_t carry4;
   int64_t carry5;
   int64_t carry6;
   int64_t carry7;
   int64_t carry8;
   int64_t carry9;

   h0 += h0;
   h1 += h1;
   h2 += h2;
   h3 += h3;
   h4 += h4;
   h5 += h5;
   h6 += h6;
   h7 += h7;
   h8 += h8;
   h9 += h9;

   carry0 = (h0 + X25) >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;
   carry4 = (h4 + X25) >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;

   carry1 = (h1 + X24) >> 25;
   h2 += carry1;
   h1 -= carry1 * X25;
   carry5 = (h5 + X24) >> 25;
   h6 += carry5;
   h5 -= carry5 * X25;

   carry2 = (h2 + X25) >> 26;
   h3 += carry2;
   h2 -= carry2 * X26;
   carry6 = (h6 + X25) >> 26;
   h7 += carry6;
   h6 -= carry6 * X26;

   carry3 = (h3 + X24) >> 25;
   h4 += carry3;
   h3 -= carry3 * X25;
   carry7 = (h7 + X24) >> 25;
   h8 += carry7;
   h7 -= carry7 * X25;

   carry4 = (h4 + X25) >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;
   carry8 = (h8 + X25) >> 26;
   h9 += carry8;
   h8 -= carry8 * X26;

   carry9 = (h9 + X24) >> 25;
   h0 += carry9 * 19;
   h9 -= carry9 * X25;

   carry0 = (h0 + X25) >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;

   return FE_25519(h0, h1, h2, h3, h4, h5, h6, h7, h8, h9);
   }

/*
Ignores top bit of h.
*/

void FE_25519::from_bytes(const uint8_t s[32])
   {
   const int64_t X24 = (1 << 24);
   const int64_t X25 = (1 << 25);
   const int64_t X26 = (1 << 26);

   int64_t h0 = load_4(s);
   int64_t h1 = load_3(s + 4) << 6;
   int64_t h2 = load_3(s + 7) << 5;
   int64_t h3 = load_3(s + 10) << 3;
   int64_t h4 = load_3(s + 13) << 2;
   int64_t h5 = load_4(s + 16);
   int64_t h6 = load_3(s + 20) << 7;
   int64_t h7 = load_3(s + 23) << 5;
   int64_t h8 = load_3(s + 26) << 4;
   int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;

   const int64_t carry9 = (h9 + X24) >> 25;
   h0 += carry9 * 19;
   h9 -= carry9 * X25;
   const int64_t carry1 = (h1 + X24) >> 25;
   h2 += carry1;
   h1 -= carry1 * X25;
   const int64_t carry3 = (h3 + X24) >> 25;
   h4 += carry3;
   h3 -= carry3 * X25;
   const int64_t carry5 = (h5 + X24) >> 25;
   h6 += carry5;
   h5 -= carry5 * X25;
   const int64_t carry7 = (h7 + X24) >> 25;
   h8 += carry7;
   h7 -= carry7 * X25;

   const int64_t carry0 = (h0 + X25) >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;
   const int64_t carry2 = (h2 + X25) >> 26;
   h3 += carry2;
   h2 -= carry2 * X26;
   const int64_t carry4 = (h4 + X25) >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;
   const int64_t carry6 = (h6 + X25) >> 26;
   h7 += carry6;
   h6 -= carry6 * X26;
   const int64_t carry8 = (h8 + X25) >> 26;
   h9 += carry8;
   h8 -= carry8 * X26;

   m_fe[0] = h0;
   m_fe[1] = h1;
   m_fe[2] = h2;
   m_fe[3] = h3;
   m_fe[4] = h4;
   m_fe[5] = h5;
   m_fe[6] = h6;
   m_fe[7] = h7;
   m_fe[8] = h8;
   m_fe[9] = h9;
   }

/*
Preconditions:
|h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.

Write p=2^255-19; q=floor(h/p).
Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).

Proof:
Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.

Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
Then 0<y<1.

Write r=h-pq.
Have 0<=r<=p-1=2^255-20.
Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.

Write x=r+19(2^-255)r+y.
Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.

Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
*/

void FE_25519::to_bytes(uint8_t s[32]) const
   {
   const int64_t X25 = (1 << 25);
   const int64_t X26 = (1 << 26);

   int32_t h0 = m_fe[0];
   int32_t h1 = m_fe[1];
   int32_t h2 = m_fe[2];
   int32_t h3 = m_fe[3];
   int32_t h4 = m_fe[4];
   int32_t h5 = m_fe[5];
   int32_t h6 = m_fe[6];
   int32_t h7 = m_fe[7];
   int32_t h8 = m_fe[8];
   int32_t h9 = m_fe[9];
   int32_t q;
   int32_t carry0;
   int32_t carry1;
   int32_t carry2;
   int32_t carry3;
   int32_t carry4;
   int32_t carry5;
   int32_t carry6;
   int32_t carry7;
   int32_t carry8;
   int32_t carry9;

   q = (19 * h9 + ((static_cast<int32_t>(1) << 24))) >> 25;
   q = (h0 + q) >> 26;
   q = (h1 + q) >> 25;
   q = (h2 + q) >> 26;
   q = (h3 + q) >> 25;
   q = (h4 + q) >> 26;
   q = (h5 + q) >> 25;
   q = (h6 + q) >> 26;
   q = (h7 + q) >> 25;
   q = (h8 + q) >> 26;
   q = (h9 + q) >> 25;

   /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
   h0 += 19 * q;
   /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */

   carry0 = h0 >> 26;
   h1 += carry0;
   h0 -= carry0 * X26;
   carry1 = h1 >> 25;
   h2 += carry1;
   h1 -= carry1 * X25;
   carry2 = h2 >> 26;
   h3 += carry2;
   h2 -= carry2 * X26;
   carry3 = h3 >> 25;
   h4 += carry3;
   h3 -= carry3 * X25;
   carry4 = h4 >> 26;
   h5 += carry4;
   h4 -= carry4 * X26;
   carry5 = h5 >> 25;
   h6 += carry5;
   h5 -= carry5 * X25;
   carry6 = h6 >> 26;
   h7 += carry6;
   h6 -= carry6 * X26;
   carry7 = h7 >> 25;
   h8 += carry7;
   h7 -= carry7 * X25;
   carry8 = h8 >> 26;
   h9 += carry8;
   h8 -= carry8 * X26;
   carry9 = h9 >> 25;
   h9 -= carry9 * X25;
   /* h10 = carry9 */

   /*
   Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
   Have h0+...+2^230 h9 between 0 and 2^255-1;
   evidently 2^255 h10-2^255 q = 0.
   Goal: Output h0+...+2^230 h9.
   */

   s[0] = h0 >> 0;
   s[1] = h0 >> 8;
   s[2] = h0 >> 16;
   s[3] = (h0 >> 24) | (h1 << 2);
   s[4] = h1 >> 6;
   s[5] = h1 >> 14;
   s[6] = (h1 >> 22) | (h2 << 3);
   s[7] = h2 >> 5;
   s[8] = h2 >> 13;
   s[9] = (h2 >> 21) | (h3 << 5);
   s[10] = h3 >> 3;
   s[11] = h3 >> 11;
   s[12] = (h3 >> 19) | (h4 << 6);
   s[13] = h4 >> 2;
   s[14] = h4 >> 10;
   s[15] = h4 >> 18;
   s[16] = h5 >> 0;
   s[17] = h5 >> 8;
   s[18] = h5 >> 16;
   s[19] = (h5 >> 24) | (h6 << 1);
   s[20] = h6 >> 7;
   s[21] = h6 >> 15;
   s[22] = (h6 >> 23) | (h7 << 3);
   s[23] = h7 >> 5;
   s[24] = h7 >> 13;
   s[25] = (h7 >> 21) | (h8 << 4);
   s[26] = h8 >> 4;
   s[27] = h8 >> 12;
   s[28] = (h8 >> 20) | (h9 << 6);
   s[29] = h9 >> 2;
   s[30] = h9 >> 10;
   s[31] = h9 >> 18;
   }

}