doxygen/ed25519__fe_8cpp_source.html

/*

* 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)

   {

   const int32_t f0 = f[0];

   const int32_t f1 = f[1];

   const int32_t f2 = f[2];

   const int32_t f3 = f[3];

   const int32_t f4 = f[4];

   const int32_t f5 = f[5];

   const int32_t f6 = f[6];

   const int32_t f7 = f[7];

   const int32_t f8 = f[8];

   const int32_t f9 = f[9];


   const int32_t g0 = g[0];

   const int32_t g1 = g[1];

   const int32_t g2 = g[2];

   const int32_t g3 = g[3];

   const int32_t g4 = g[4];

   const int32_t g5 = g[5];

   const int32_t g6 = g[6];

   const int32_t g7 = g[7];

   const int32_t g8 = g[8];

   const int32_t g9 = g[9];


   const int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */

   const int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */

   const int32_t g3_19 = 19 * g3;

   const int32_t g4_19 = 19 * g4;

   const int32_t g5_19 = 19 * g5;

   const int32_t g6_19 = 19 * g6;

   const int32_t g7_19 = 19 * g7;

   const int32_t g8_19 = 19 * g8;

   const int32_t g9_19 = 19 * g9;

   const int32_t f1_2 = 2 * f1;

   const int32_t f3_2 = 2 * f3;

   const int32_t f5_2 = 2 * f5;

   const int32_t f7_2 = 2 * f7;

   const int32_t f9_2 = 2 * f9;


   const int64_t f0g0    = f0   * static_cast<int64_t>(g0);

   const int64_t f0g1    = f0   * static_cast<int64_t>(g1);

   const int64_t f0g2    = f0   * static_cast<int64_t>(g2);

   const int64_t f0g3    = f0   * static_cast<int64_t>(g3);

   const int64_t f0g4    = f0   * static_cast<int64_t>(g4);

   const int64_t f0g5    = f0   * static_cast<int64_t>(g5);

   const int64_t f0g6    = f0   * static_cast<int64_t>(g6);

   const int64_t f0g7    = f0   * static_cast<int64_t>(g7);

   const int64_t f0g8    = f0   * static_cast<int64_t>(g8);

   const int64_t f0g9    = f0   * static_cast<int64_t>(g9);

   const int64_t f1g0    = f1   * static_cast<int64_t>(g0);

   const int64_t f1g1_2  = f1_2 * static_cast<int64_t>(g1);

   const int64_t f1g2    = f1   * static_cast<int64_t>(g2);

   const int64_t f1g3_2  = f1_2 * static_cast<int64_t>(g3);

   const int64_t f1g4    = f1   * static_cast<int64_t>(g4);

   const int64_t f1g5_2  = f1_2 * static_cast<int64_t>(g5);

   const int64_t f1g6    = f1   * static_cast<int64_t>(g6);

   const int64_t f1g7_2  = f1_2 * static_cast<int64_t>(g7);

   const int64_t f1g8    = f1   * static_cast<int64_t>(g8);

   const int64_t f1g9_38 = f1_2 * static_cast<int64_t>(g9_19);

   const int64_t f2g0    = f2   * static_cast<int64_t>(g0);

   const int64_t f2g1    = f2   * static_cast<int64_t>(g1);

   const int64_t f2g2    = f2   * static_cast<int64_t>(g2);

   const int64_t f2g3    = f2   * static_cast<int64_t>(g3);

   const int64_t f2g4    = f2   * static_cast<int64_t>(g4);

   const int64_t f2g5    = f2   * static_cast<int64_t>(g5);

   const int64_t f2g6    = f2   * static_cast<int64_t>(g6);

   const int64_t f2g7    = f2   * static_cast<int64_t>(g7);

   const int64_t f2g8_19 = f2   * static_cast<int64_t>(g8_19);

   const int64_t f2g9_19 = f2   * static_cast<int64_t>(g9_19);

   const int64_t f3g0    = f3   * static_cast<int64_t>(g0);

   const int64_t f3g1_2  = f3_2 * static_cast<int64_t>(g1);

   const int64_t f3g2    = f3   * static_cast<int64_t>(g2);

   const int64_t f3g3_2  = f3_2 * static_cast<int64_t>(g3);

   const int64_t f3g4    = f3   * static_cast<int64_t>(g4);

   const int64_t f3g5_2  = f3_2 * static_cast<int64_t>(g5);

   const int64_t f3g6    = f3   * static_cast<int64_t>(g6);

   const int64_t f3g7_38 = f3_2 * static_cast<int64_t>(g7_19);

   const int64_t f3g8_19 = f3   * static_cast<int64_t>(g8_19);

   const int64_t f3g9_38 = f3_2 * static_cast<int64_t>(g9_19);

   const int64_t f4g0    = f4   * static_cast<int64_t>(g0);

   const int64_t f4g1    = f4   * static_cast<int64_t>(g1);

   const int64_t f4g2    = f4   * static_cast<int64_t>(g2);

   const int64_t f4g3    = f4   * static_cast<int64_t>(g3);

   const int64_t f4g4    = f4   * static_cast<int64_t>(g4);

   const int64_t f4g5    = f4   * static_cast<int64_t>(g5);

   const int64_t f4g6_19 = f4   * static_cast<int64_t>(g6_19);

   const int64_t f4g7_19 = f4   * static_cast<int64_t>(g7_19);

   const int64_t f4g8_19 = f4   * static_cast<int64_t>(g8_19);

   const int64_t f4g9_19 = f4   * static_cast<int64_t>(g9_19);

   const int64_t f5g0    = f5   * static_cast<int64_t>(g0);

   const int64_t f5g1_2  = f5_2 * static_cast<int64_t>(g1);

   const int64_t f5g2    = f5   * static_cast<int64_t>(g2);

   const int64_t f5g3_2  = f5_2 * static_cast<int64_t>(g3);

   const int64_t f5g4    = f5   * static_cast<int64_t>(g4);

   const int64_t f5g5_38 = f5_2 * static_cast<int64_t>(g5_19);

   const int64_t f5g6_19 = f5   * static_cast<int64_t>(g6_19);

   const int64_t f5g7_38 = f5_2 * static_cast<int64_t>(g7_19);

   const int64_t f5g8_19 = f5   * static_cast<int64_t>(g8_19);

   const int64_t f5g9_38 = f5_2 * static_cast<int64_t>(g9_19);

   const int64_t f6g0    = f6   * static_cast<int64_t>(g0);

   const int64_t f6g1    = f6   * static_cast<int64_t>(g1);

   const int64_t f6g2    = f6   * static_cast<int64_t>(g2);

   const int64_t f6g3    = f6   * static_cast<int64_t>(g3);

   const int64_t f6g4_19 = f6   * static_cast<int64_t>(g4_19);

   const int64_t f6g5_19 = f6   * static_cast<int64_t>(g5_19);

   const int64_t f6g6_19 = f6   * static_cast<int64_t>(g6_19);

   const int64_t f6g7_19 = f6   * static_cast<int64_t>(g7_19);

   const int64_t f6g8_19 = f6   * static_cast<int64_t>(g8_19);

   const int64_t f6g9_19 = f6   * static_cast<int64_t>(g9_19);

   const int64_t f7g0    = f7   * static_cast<int64_t>(g0);

   const int64_t f7g1_2  = f7_2 * static_cast<int64_t>(g1);

   const int64_t f7g2    = f7   * static_cast<int64_t>(g2);

   const int64_t f7g3_38 = f7_2 * static_cast<int64_t>(g3_19);

   const int64_t f7g4_19 = f7   * static_cast<int64_t>(g4_19);

   const int64_t f7g5_38 = f7_2 * static_cast<int64_t>(g5_19);

   const int64_t f7g6_19 = f7   * static_cast<int64_t>(g6_19);

   const int64_t f7g7_38 = f7_2 * static_cast<int64_t>(g7_19);

   const int64_t f7g8_19 = f7   * static_cast<int64_t>(g8_19);

   const int64_t f7g9_38 = f7_2 * static_cast<int64_t>(g9_19);

   const int64_t f8g0    = f8   * static_cast<int64_t>(g0);

   const int64_t f8g1    = f8   * static_cast<int64_t>(g1);

   const int64_t f8g2_19 = f8   * static_cast<int64_t>(g2_19);

   const int64_t f8g3_19 = f8   * static_cast<int64_t>(g3_19);

   const int64_t f8g4_19 = f8   * static_cast<int64_t>(g4_19);

   const int64_t f8g5_19 = f8   * static_cast<int64_t>(g5_19);

   const int64_t f8g6_19 = f8   * static_cast<int64_t>(g6_19);

   const int64_t f8g7_19 = f8   * static_cast<int64_t>(g7_19);

   const int64_t f8g8_19 = f8   * static_cast<int64_t>(g8_19);

   const int64_t f8g9_19 = f8   * static_cast<int64_t>(g9_19);

   const int64_t f9g0    = f9   * static_cast<int64_t>(g0);

   const int64_t f9g1_38 = f9_2 * static_cast<int64_t>(g1_19);

   const int64_t f9g2_19 = f9   * static_cast<int64_t>(g2_19);

   const int64_t f9g3_38 = f9_2 * static_cast<int64_t>(g3_19);

   const int64_t f9g4_19 = f9   * static_cast<int64_t>(g4_19);

   const int64_t f9g5_38 = f9_2 * static_cast<int64_t>(g5_19);

   const int64_t f9g6_19 = f9   * static_cast<int64_t>(g6_19);

   const int64_t f9g7_38 = f9_2 * static_cast<int64_t>(g7_19);

   const int64_t f9g8_19 = f9   * static_cast<int64_t>(g8_19);

   const 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   ;


   /*

   |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

   */

   carry<26>(h0, h1);

   carry<26>(h4, h5);


   /* |h0| <= 2^25 */

   /* |h4| <= 2^25 */

   /* |h1| <= 1.71*2^59 */

   /* |h5| <= 1.71*2^59 */


   carry<25>(h1, h2);

   carry<25>(h5, h6);


   /* |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 */


   carry<26>(h2, h3);

   carry<26>(h6, h7);

   /* |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 */


   carry<25>(h3, h4);

   carry<25>(h7, h8);

   /* |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 */


   carry<26>(h4, h5);

   carry<26>(h8, h9);

   /* |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 */


   carry<25, 19>(h9, h0);


   /* |h9| <= 2^24; from now on fits into int32 unchanged */

   /* |h0| <= 1.1*2^39 */


   carry<26>(h0, h1);

   /* |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)

   {

   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;


      carry<26>(h0, h1);

      carry<26>(h4, h5);

      carry<25>(h1, h2);

      carry<25>(h5, h6);

      carry<26>(h2, h3);

      carry<26>(h6, h7);


      carry<25>(h3, h4);

      carry<25>(h7, h8);


      carry<26>(h4, h5);

      carry<26>(h8, h9);

      carry<25,19>(h9, h0);

      carry<26>(h0, h1);


      f0 = static_cast<int32_t>(h0);

      f1 = static_cast<int32_t>(h1);

      f2 = static_cast<int32_t>(h2);

      f3 = static_cast<int32_t>(h3);

      f4 = static_cast<int32_t>(h4);

      f5 = static_cast<int32_t>(h5);

      f6 = static_cast<int32_t>(h6);

      f7 = static_cast<int32_t>(h7);

      f8 = static_cast<int32_t>(h8);

      f9 = static_cast<int32_t>(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 int32_t f0 = f[0];

   const int32_t f1 = f[1];

   const int32_t f2 = f[2];

   const int32_t f3 = f[3];

   const int32_t f4 = f[4];

   const int32_t f5 = f[5];

   const int32_t f6 = f[6];

   const int32_t f7 = f[7];

   const int32_t f8 = f[8];

   const int32_t f9 = f[9];

   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;


   h0 += h0;

   h1 += h1;

   h2 += h2;

   h3 += h3;

   h4 += h4;

   h5 += h5;

   h6 += h6;

   h7 += h7;

   h8 += h8;

   h9 += h9;


   carry<26>(h0, h1);

   carry<26>(h4, h5);


   carry<25>(h1, h2);

   carry<25>(h5, h6);


   carry<26>(h2, h3);

   carry<26>(h6, h7);


   carry<25>(h3, h4);

   carry<25>(h7, h8);

   carry<26>(h4, h5);

   carry<26>(h8, h9);

   carry<25,19>(h9, h0);

   carry<26>(h0, h1);


   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])

   {

   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;


   carry<25,19>(h9, h0);

   carry<25>(h1, h2);

   carry<25>(h3, h4);

   carry<25>(h5, h6);

   carry<25>(h7, h8);


   carry<26>(h0, h1);

   carry<26>(h2, h3);

   carry<26>(h4, h5);

   carry<26>(h6, h7);

   carry<26>(h8, h9);


   m_fe[0] = static_cast<int32_t>(h0);

   m_fe[1] = static_cast<int32_t>(h1);

   m_fe[2] = static_cast<int32_t>(h2);

   m_fe[3] = static_cast<int32_t>(h3);

   m_fe[4] = static_cast<int32_t>(h4);

   m_fe[5] = static_cast<int32_t>(h5);

   m_fe[6] = static_cast<int32_t>(h6);

   m_fe[7] = static_cast<int32_t>(h7);

   m_fe[8] = static_cast<int32_t>(h8);

   m_fe[9] = static_cast<int32_t>(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);


   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;


   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<26>(h0, h1);

   carry0<25>(h1, h2);

   carry0<26>(h2, h3);

   carry0<25>(h3, h4);

   carry0<26>(h4, h5);

   carry0<25>(h5, h6);

   carry0<26>(h6, h7);

   carry0<25>(h7, h8);

   carry0<26>(h8, h9);


   int32_t 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] = static_cast<uint8_t>(h0 >> 0);

   s[1] = static_cast<uint8_t>(h0 >> 8);

   s[2] = static_cast<uint8_t>(h0 >> 16);

   s[3] = static_cast<uint8_t>((h0 >> 24) | (h1 << 2));

   s[4] = static_cast<uint8_t>(h1 >> 6);

   s[5] = static_cast<uint8_t>(h1 >> 14);

   s[6] = static_cast<uint8_t>((h1 >> 22) | (h2 << 3));

   s[7] = static_cast<uint8_t>(h2 >> 5);

   s[8] = static_cast<uint8_t>(h2 >> 13);

   s[9] = static_cast<uint8_t>((h2 >> 21) | (h3 << 5));

   s[10] = static_cast<uint8_t>(h3 >> 3);

   s[11] = static_cast<uint8_t>(h3 >> 11);

   s[12] = static_cast<uint8_t>((h3 >> 19) | (h4 << 6));

   s[13] = static_cast<uint8_t>(h4 >> 2);

   s[14] = static_cast<uint8_t>(h4 >> 10);

   s[15] = static_cast<uint8_t>(h4 >> 18);

   s[16] = static_cast<uint8_t>(h5 >> 0);

   s[17] = static_cast<uint8_t>(h5 >> 8);

   s[18] = static_cast<uint8_t>(h5 >> 16);

   s[19] = static_cast<uint8_t>((h5 >> 24) | (h6 << 1));

   s[20] = static_cast<uint8_t>(h6 >> 7);

   s[21] = static_cast<uint8_t>(h6 >> 15);

   s[22] = static_cast<uint8_t>((h6 >> 23) | (h7 << 3));

   s[23] = static_cast<uint8_t>(h7 >> 5);

   s[24] = static_cast<uint8_t>(h7 >> 13);

   s[25] = static_cast<uint8_t>((h7 >> 21) | (h8 << 4));

   s[26] = static_cast<uint8_t>(h8 >> 4);

   s[27] = static_cast<uint8_t>(h8 >> 12);

   s[28] = static_cast<uint8_t>((h8 >> 20) | (h9 << 6));

   s[29] = static_cast<uint8_t>(h9 >> 2);

   s[30] = static_cast<uint8_t>(h9 >> 10);

   s[31] = static_cast<uint8_t>(h9 >> 18);

   }


}

Botan::FE_25519
Definition: ed25519_fe.h:23

Botan::FE_25519::to_bytes
void to_bytes(uint8_t b[32]) const
Definition: ed25519_fe.cpp:667

Botan::FE_25519::from_bytes
void from_bytes(const uint8_t b[32])
Definition: ed25519_fe.cpp:605

Botan::FE_25519::invert
static FE_25519 invert(const FE_25519 &a)
Definition: ed25519_fe.cpp:17

Botan::FE_25519::mul
static FE_25519 mul(const FE_25519 &a, const FE_25519 &b)
Definition: ed25519_fe.cpp:115

Botan::FE_25519::pow_22523
static FE_25519 pow_22523(const FE_25519 &a)
Definition: ed25519_fe.cpp:50

Botan::FE_25519::FE_25519
FE_25519(int init=0)
Definition: ed25519_fe.h:30

Botan::FE_25519::sqr_iter
static FE_25519 sqr_iter(const FE_25519 &a, size_t iter)
Definition: ed25519_fe.cpp:337

Botan::FE_25519::sqr2
static FE_25519 sqr2(const FE_25519 &a)
Definition: ed25519_fe.cpp:479

Botan
Definition: alg_id.cpp:13

Botan::load_4
uint64_t load_4(const uint8_t *in)
Definition: ed25519_internal.h:26

Botan::fe_mul
void fe_mul(fe &x, const fe &a, const fe &b)
Definition: ed25519_fe.h:195

Botan::fe_sq
void fe_sq(fe &x, const fe &z)
Definition: ed25519_fe.h:200

Botan::fe_sq_iter
void fe_sq_iter(fe &x, const fe &z, size_t iter)
Definition: ed25519_fe.h:205

Botan::load_3
uint64_t load_3(const uint8_t in[3])
Definition: ed25519_internal.h:19