i2pd/core/crypto/ed25519/fe_tobytes.cpp

#include "fe.h"

/*
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_tobytes(unsigned char *s,const fe h)
{
  crypto_int32 h0 = h[0];
  crypto_int32 h1 = h[1];
  crypto_int32 h2 = h[2];
  crypto_int32 h3 = h[3];
  crypto_int32 h4 = h[4];
  crypto_int32 h5 = h[5];
  crypto_int32 h6 = h[6];
  crypto_int32 h7 = h[7];
  crypto_int32 h8 = h[8];
  crypto_int32 h9 = h[9];
  crypto_int32 q;
  crypto_int32 carry0;
  crypto_int32 carry1;
  crypto_int32 carry2;
  crypto_int32 carry3;
  crypto_int32 carry4;
  crypto_int32 carry5;
  crypto_int32 carry6;
  crypto_int32 carry7;
  crypto_int32 carry8;
  crypto_int32 carry9;

  q = (19 * h9 + (((crypto_int32) 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 << 26;
  carry1 = h1 >> 25; h2 += carry1; h1 -= carry1 << 25;
  carry2 = h2 >> 26; h3 += carry2; h2 -= carry2 << 26;
  carry3 = h3 >> 25; h4 += carry3; h3 -= carry3 << 25;
  carry4 = h4 >> 26; h5 += carry4; h4 -= carry4 << 26;
  carry5 = h5 >> 25; h6 += carry5; h5 -= carry5 << 25;
  carry6 = h6 >> 26; h7 += carry6; h6 -= carry6 << 26;
  carry7 = h7 >> 25; h8 += carry7; h7 -= carry7 << 25;
  carry8 = h8 >> 26; h9 += carry8; h8 -= carry8 << 26;
  carry9 = h9 >> 25;               h9 -= carry9 << 25;
                  /* 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;
}