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 *

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 *

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 *

 * curve25519: Curve25519 elliptic curve, public key function

 *

 * http://code.google.com/p/curve25519-donna/

 *

 * Adam Langley <agl@imperialviolet.org>

 *

 * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>

 *

 * More information about curve25519 can be found here

 *   http://cr.yp.to/ecdh.html

 *

 * djb's sample implementation of curve25519 is written in a special assembly

 * language called qhasm and uses the floating point registers.

 *

 * This is, almost, a clean room reimplementation from the curve25519 paper. It

 * uses many of the tricks described therein. Only the crecip function is taken

 * from the sample implementation.

 */

#include "os.h"

#include <libsec.h>

typedef vlong felem;

/* Sum two numbers: output += in */

static void fsum(felem *output, felem *in) {

  unsigned i;

  for (i = 0; i < 10; i += 2) {

    output[0+i] = (output[0+i] + in[0+i]);

    output[1+i] = (output[1+i] + in[1+i]);

  }

}

/* Find the difference of two numbers: output = in - output

 * (note the order of the arguments!)

 */

static void fdifference(felem *output, felem *in) {

  unsigned i;

  for (i = 0; i < 10; ++i) {

    output[i] = (in[i] - output[i]);

  }

}

/* Multiply a number my a scalar: output = in * scalar */

static void fscalar_product(felem *output, felem *in, felem scalar) {

  unsigned i;

  for (i = 0; i < 10; ++i) {

    output[i] = in[i] * scalar;

  }

}

/* Multiply two numbers: output = in2 * in

 *

 * output must be distinct to both inputs. The inputs are reduced coefficient

 * form, the output is not.

 */

static void fproduct(felem *output, felem *in2, felem *in) {

  output[0] =      in2[0] * in[0];

  output[1] =      in2[0] * in[1] +

                   in2[1] * in[0];

  output[2] =  2 * in2[1] * in[1] +

                   in2[0] * in[2] +

                   in2[2] * in[0];

  output[3] =      in2[1] * in[2] +

                   in2[2] * in[1] +

                   in2[0] * in[3] +

                   in2[3] * in[0];

  output[4] =      in2[2] * in[2] +

               2 * (in2[1] * in[3] +

                    in2[3] * in[1]) +

                   in2[0] * in[4] +

                   in2[4] * in[0];

  output[5] =      in2[2] * in[3] +

                   in2[3] * in[2] +

                   in2[1] * in[4] +

                   in2[4] * in[1] +

                   in2[0] * in[5] +

                   in2[5] * in[0];

  output[6] =  2 * (in2[3] * in[3] +

                    in2[1] * in[5] +

                    in2[5] * in[1]) +

                   in2[2] * in[4] +

                   in2[4] * in[2] +

                   in2[0] * in[6] +

                   in2[6] * in[0];

  output[7] =      in2[3] * in[4] +

                   in2[4] * in[3] +

                   in2[2] * in[5] +

                   in2[5] * in[2] +

                   in2[1] * in[6] +

                   in2[6] * in[1] +

                   in2[0] * in[7] +

                   in2[7] * in[0];

  output[8] =      in2[4] * in[4] +

               2 * (in2[3] * in[5] +

                    in2[5] * in[3] +

                    in2[1] * in[7] +

                    in2[7] * in[1]) +

                   in2[2] * in[6] +

                   in2[6] * in[2] +

                   in2[0] * in[8] +

                   in2[8] * in[0];

  output[9] =      in2[4] * in[5] +

                   in2[5] * in[4] +

                   in2[3] * in[6] +

                   in2[6] * in[3] +

                   in2[2] * in[7] +

                   in2[7] * in[2] +

                   in2[1] * in[8] +

                   in2[8] * in[1] +

                   in2[0] * in[9] +

                   in2[9] * in[0];

  output[10] = 2 * (in2[5] * in[5] +

                    in2[3] * in[7] +

                    in2[7] * in[3] +

                    in2[1] * in[9] +

                    in2[9] * in[1]) +

                   in2[4] * in[6] +

                   in2[6] * in[4] +

                   in2[2] * in[8] +

                   in2[8] * in[2];

  output[11] =     in2[5] * in[6] +

                   in2[6] * in[5] +

                   in2[4] * in[7] +

                   in2[7] * in[4] +

                   in2[3] * in[8] +

                   in2[8] * in[3] +

                   in2[2] * in[9] +

                   in2[9] * in[2];

  output[12] =     in2[6] * in[6] +

               2 * (in2[5] * in[7] +

                    in2[7] * in[5] +

                    in2[3] * in[9] +

                    in2[9] * in[3]) +

                   in2[4] * in[8] +

                   in2[8] * in[4];

  output[13] =     in2[6] * in[7] +

                   in2[7] * in[6] +

                   in2[5] * in[8] +

                   in2[8] * in[5] +

                   in2[4] * in[9] +

                   in2[9] * in[4];

  output[14] = 2 * (in2[7] * in[7] +

                    in2[5] * in[9] +

                    in2[9] * in[5]) +

                   in2[6] * in[8] +

                   in2[8] * in[6];

  output[15] =     in2[7] * in[8] +

                   in2[8] * in[7] +

                   in2[6] * in[9] +

                   in2[9] * in[6];

  output[16] =     in2[8] * in[8] +

               2 * (in2[7] * in[9] +

                    in2[9] * in[7]);

  output[17] =     in2[8] * in[9] +

                   in2[9] * in[8];

  output[18] = 2 * in2[9] * in[9];

}

/* Reduce a long form to a short form by taking the input mod 2^255 - 19. */

static void freduce_degree(felem *output) {

  output[8] += 19 * output[18];

  output[7] += 19 * output[17];

  output[6] += 19 * output[16];

  output[5] += 19 * output[15];

  output[4] += 19 * output[14];

  output[3] += 19 * output[13];

  output[2] += 19 * output[12];

  output[1] += 19 * output[11];

  output[0] += 19 * output[10];

}

/* Reduce all coefficients of the short form input to be -2**25 <= x <= 2**25

 */

static void freduce_coefficients(felem *output) {

  unsigned i;

  do {

    output[10] = 0;

    for (i = 0; i < 10; i += 2) {

      felem over = output[i] / 0x2000000l;

      felem over2 = (over + ((over >> 63) * 2) + 1) / 2;

      output[i+1] += over2;

      output[i] -= over2 * 0x4000000l;

      over = output[i+1] / 0x2000000;

      output[i+2] += over;

      output[i+1] -= over * 0x2000000;

    }

    output[0] += 19 * output[10];

  } while (output[10]);

}

/* A helpful wrapper around fproduct: output = in * in2.

 *

 * output must be distinct to both inputs. The output is reduced degree and

 * reduced coefficient.

 */

static void

fmul(felem *output, felem *in, felem *in2) {

  felem t[19];

  fproduct(t, in, in2);

  freduce_degree(t);

  freduce_coefficients(t);

  memcpy(output, t, sizeof(felem) * 10);

}

static void fsquare_inner(felem *output, felem *in) {

  felem tmp;

  output[0] =      in[0] * in[0];

  output[1] =  2 * in[0] * in[1];

  output[2] =  2 * (in[1] * in[1] +

                    in[0] * in[2]);

  output[3] =  2 * (in[1] * in[2] +

                    in[0] * in[3]);

  output[4] =      in[2] * in[2] +

               4 * in[1] * in[3] +

               2 * in[0] * in[4];

  output[5] =  2 * (in[2] * in[3] +

                    in[1] * in[4] +

                    in[0] * in[5]);

  output[6] =  2 * (in[3] * in[3] +

                    in[2] * in[4] +

                    in[0] * in[6] +

                2 * in[1] * in[5]);

  output[7] =  2 * (in[3] * in[4] +

                    in[2] * in[5] +

                    in[1] * in[6] +

                    in[0] * in[7]);

  tmp = in[1] * in[7] + in[3] * in[5];

  output[8] =      in[4] * in[4] +

               2 * (in[2] * in[6] +

                    in[0] * in[8] +

                        2 * tmp);

  output[9] =  2 * (in[4] * in[5] +

                    in[3] * in[6] +

                    in[2] * in[7] +

                    in[1] * in[8] +

                    in[0] * in[9]);

  tmp = in[3] * in[7] + in[1] * in[9];

  output[10] = 2 * (in[5] * in[5] +

                   in[4] * in[6] +

                   in[2] * in[8] +

                       2 * tmp);

  output[11] = 2 * (in[5] * in[6] +

                    in[4] * in[7] +

                    in[3] * in[8] +

                    in[2] * in[9]);

  output[12] =     in[6] * in[6] +

               2 * (in[4] * in[8] +

                2 * (in[5] * in[7] +

                     in[3] * in[9]));

  output[13] = 2 * (in[6] * in[7] +

                    in[5] * in[8] +

                    in[4] * in[9]);

  output[14] = 2 * (in[7] * in[7] +

                    in[6] * in[8] +

                2 * in[5] * in[9]);

  output[15] = 2 * (in[7] * in[8] +

                    in[6] * in[9]);

  output[16] =     in[8] * in[8] +

               4 * in[7] * in[9];

  output[17] = 2 * in[8] * in[9];

  output[18] = 2 * in[9] * in[9];

}

static void

fsquare(felem *output, felem *in) {

  felem t[19];

  fsquare_inner(t, in);

  freduce_degree(t);

  freduce_coefficients(t);

  memcpy(output, t, sizeof(felem) * 10);

}

/* Take a little-endian, 32-byte number and expand it into polynomial form */

static void

fexpand(felem *output, uchar *input) {

#define F(n,start,shift,mask) \

  output[n] = ((((felem) input[start + 0]) | \

                ((felem) input[start + 1]) << 8 | \

                ((felem) input[start + 2]) << 16 | \

                ((felem) 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

}

/* Take a fully reduced polynomial form number and contract it into a

 * little-endian, 32-byte array

 */

static void

fcontract(uchar *output, felem *input) {

  int i;

  do {

    for (i = 0; i < 9; ++i) {

      if ((i & 1) == 1) {

        while (input[i] < 0) {

          input[i] += 0x2000000;

          input[i + 1]--;

        }

      } else {

        while (input[i] < 0) {

          input[i] += 0x4000000;

          input[i + 1]--;

        }

      }

    }

    while (input[9] < 0) {

      input[9] += 0x2000000;

      input[0] -= 19;

    }

  } while (input[0] < 0);

  input[1] <<= 2;

  input[2] <<= 3;

  input[3] <<= 5;

  input[4] <<= 6;

  input[6] <<= 1;

  input[7] <<= 3;

  input[8] <<= 4;

  input[9] <<= 6;

#define F(i, s) \

  output[s+0] |=  input[i] & 0xff; \

  output[s+1]  = (input[i] >> 8) & 0xff; \

  output[s+2]  = (input[i] >> 16) & 0xff; \

  output[s+3]  = (input[i] >> 24) & 0xff;

  output[0] = 0;

  output[16] = 0;

  F(0,0);

  F(1,3);

  F(2,6);

  F(3,9);

  F(4,12);

  F(5,16);

  F(6,19);

  F(7,22);

  F(8,25);

  F(9,28);

#undef F

}

/* Input: Q, Q', Q-Q'

 * Output: 2Q, Q+Q'

 *

 *   x2 z3: long form

 *   x3 z3: long form

 *   x z: short form, destroyed

 *   xprime zprime: short form, destroyed

 *   qmqp: short form, preserved

 */

static void fmonty(felem *x2, felem *z2,  /* output 2Q */

                   felem *x3, felem *z3,  /* output Q + Q' */

                   felem *x, felem *z,    /* input Q */

                   felem *xprime, felem *zprime,  /* input Q' */

                   felem *qmqp /* input Q - Q' */) {

  felem origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],

        zzprime[19], zzzprime[19], xxxprime[19];

  memcpy(origx, x, 10 * sizeof(felem));

  fsum(x, z);

  fdifference(z, origx);  // does x - z

  memcpy(origxprime, xprime, sizeof(felem) * 10);

  fsum(xprime, zprime);

  fdifference(zprime, origxprime);

  fproduct(xxprime, xprime, z);

  fproduct(zzprime, x, zprime);

  freduce_degree(xxprime);

  freduce_coefficients(xxprime);

  freduce_degree(zzprime);

  freduce_coefficients(zzprime);

  memcpy(origxprime, xxprime, sizeof(felem) * 10);

  fsum(xxprime, zzprime);

  fdifference(zzprime, origxprime);

  fsquare(xxxprime, xxprime);

  fsquare(zzzprime, zzprime);

  fproduct(zzprime, zzzprime, qmqp);

  freduce_degree(zzprime);

  freduce_coefficients(zzprime);

  memcpy(x3, xxxprime, sizeof(felem) * 10);

  memcpy(z3, zzprime, sizeof(felem) * 10);

  fsquare(xx, x);

  fsquare(zz, z);

  fproduct(x2, xx, zz);

  freduce_degree(x2);

  freduce_coefficients(x2);

  fdifference(zz, xx);  // does zz = xx - zz

  memset(zzz + 10, 0, sizeof(felem) * 9);

  fscalar_product(zzz, zz, 121665);

  freduce_degree(zzz);

  freduce_coefficients(zzz);

  fsum(zzz, xx);

  fproduct(z2, zz, zzz);

  freduce_degree(z2);

  freduce_coefficients(z2);

}

/* Calculates nQ where Q is the x-coordinate of a point on the curve

 *

 *   resultx/resultz: the x coordinate of the resulting curve point (short form)

 *   n: a little endian, 32-byte number

 *   q: a point of the curve (short form)

 */

static void

cmult(felem *resultx, felem *resultz, uchar *n, felem *q) {

  felem a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};

  felem *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;

  felem e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};

  felem *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;

  unsigned i, j;

  memcpy(nqpqx, q, sizeof(felem) * 10);

  for (i = 0; i < 32; ++i) {

    uchar byte = n[31 - i];

    for (j = 0; j < 8; ++j) {

      if (byte & 0x80) {

        fmonty(nqpqx2, nqpqz2,

               nqx2, nqz2,

               nqpqx, nqpqz,

               nqx, nqz,

               q);

      } else {

        fmonty(nqx2, nqz2,

               nqpqx2, nqpqz2,

               nqx, nqz,

               nqpqx, nqpqz,

               q);

      }

      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(felem) * 10);

  memcpy(resultz, nqz, sizeof(felem) * 10);

}

// -----------------------------------------------------------------------------

// Shamelessly copied from djb's code

// -----------------------------------------------------------------------------

static void

crecip(felem *out, felem *z) {

  felem z2[10];

  felem z9[10];

  felem z11[10];

  felem z2_5_0[10];

  felem z2_10_0[10];

  felem z2_20_0[10];

  felem z2_50_0[10];

  felem z2_100_0[10];

  felem t0[10];

  felem 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);

}

void

curve25519(uchar mypublic[32], uchar secret[32], uchar basepoint[32]) {

  felem bp[10], x[10], z[10], zmone[10];

  fexpand(bp, basepoint);

  cmult(x, z, secret, bp);

  crecip(zmone, z);

  fmul(z, x, zmone);

  fcontract(mypublic, z);

}