tor

curve25519-donna.c (31995B)

---

/* Copyright 2008, Google Inc.
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are
* met:
*
*     * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*     * Redistributions in binary form must reproduce the above
* copyright notice, this list of conditions and the following disclaimer
* in the documentation and/or other materials provided with the
* distribution.
*     * Neither the name of Google Inc. nor the names of its
* contributors may be used to endorse or promote products derived from
* this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* curve25519-donna: 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 "orconfig.h"

#include <string.h>
#include "lib/cc/torint.h"

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

/* Field element representation:
*
* Field elements are written as an array of signed, 64-bit limbs, least
* significant first. The value of the field element is:
*   x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
*
* i.e. the limbs are 26, 25, 26, 25, ... bits wide. */

/* Sum two numbers: output += in */
static 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];
 }
}

/* Find the difference of two numbers: output = in - output
* (note the order of the arguments!). */
static void fdifference(limb *output, const limb *in) {
 unsigned i;
 for (i = 0; i < 10; ++i) {
   output[i] = in[i] - output[i];
 }
}

/* Multiply a number by a scalar: output = in * scalar */
static void fscalar_product(limb *output, const limb *in, const limb 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.
*
* output[x] <= 14 * the largest product of the input limbs. */
static 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]);
}

/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
*
* On entry: |output[i]| < 14*2^54
* On exit: |output[0..8]| < 280*2^54 */
static void freduce_degree(limb *output) {
 /* Each of these shifts and adds ends up multiplying the value by 19.
  *
  * For output[0..8], the absolute entry value is < 14*2^54 and we add, at
  * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
 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

/* return v / 2^26, using only shifts and adds.
*
* On entry: v can take any value. */
static 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;
}

/* return v / (2^25), using only shifts and adds.
*
* On entry: v can take any value. */
static 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;
}

#if 0
/* return v / (2^25), using only shifts and adds.
*
* On entry: v can take any value. */
static inline s32
div_s32_by_2_25(const s32 v)
{
  const s32 roundoff = ((uint32_t)(v >> 31)) >> 7;
  return (v + roundoff) >> 25;
}
#endif

/* Reduce all coefficients of the short form input so that |x| < 2^26.
*
* On entry: |output[i]| < 280*2^54 */
static 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. */
}

/* A helpful wrapper around fproduct: output = in * in2.
*
* On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
*
* output must be distinct to both inputs. The output is reduced degree
* (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
static 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);
}

/* Square a number: output = in**2
*
* output must be distinct from the input. The inputs are reduced coefficient
* form, the output is not.
*
* output[x] <= 14 * the largest product of the input limbs. */
static 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]);
}

/* fsquare sets output = in^2.
*
* On entry: The |in| argument is in reduced coefficients form and |in[i]| <
* 2^27.
*
* On exit: The |output| argument is in reduced coefficients form (indeed, one
* need only provide storage for 10 limbs) and |out[i]| < 2^26. */
static 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);
}

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

/* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
static 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;
}

/* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
* both non-negative. */
static s32 s32_gte(s32 a, s32 b) {
 a -= b;
 /* a >= 0 iff a >= b. */
 return ~(a >> 31);
}

/* Take a fully reduced polynomial form number and contract it into a
* little-endian, 32-byte array.
*
* On entry: |input_limbs[i]| < 2^26 */
static void
fcontract(u8 *output, limb *input_limbs) {
 int i;
 int j;
 s32 input[10];

 /* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
 for (i = 0; i < 10; i++) {
   input[i] = (s32) input_limbs[i];
 }

 for (j = 0; j < 2; ++j) {
   for (i = 0; i < 9; ++i) {
     if ((i & 1) == 1) {
       /* This calculation is a time-invariant way to make input[i]
        * non-negative by borrowing from the next-larger limb. */
       const s32 mask = input[i] >> 31;
       const s32 carry = -((input[i] & mask) >> 25);
       input[i] = input[i] + (carry << 25);
       input[i+1] = input[i+1] - carry;
     } else {
       const s32 mask = input[i] >> 31;
       const s32 carry = -((input[i] & mask) >> 26);
       input[i] = input[i] + (carry << 26);
       input[i+1] = input[i+1] - carry;
     }
   }

   /* There's no greater limb for input[9] to borrow from, but we can multiply
    * by 19 and borrow from input[0], which is valid mod 2^255-19. */
   {
     const s32 mask = input[9] >> 31;
     const s32 carry = -((input[9] & mask) >> 25);
     input[9] = input[9] + (carry << 25);
     input[0] = input[0] - (carry * 19);
   }

   /* After the first iteration, input[1..9] are non-negative and fit within
    * 25 or 26 bits, depending on position. However, input[0] may be
    * negative. */
 }

 /* The first borrow-propagation pass above ended with every limb
    except (possibly) input[0] non-negative.

    If input[0] was negative after the first pass, then it was because of a
    carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
    one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.

    In the second pass, each limb is decreased by at most one. Thus the second
    borrow-propagation pass could only have wrapped around to decrease
    input[0] again if the first pass left input[0] negative *and* input[1]
    through input[9] were all zero.  In that case, input[1] is now 2^25 - 1,
    and this last borrow-propagation step will leave input[1] non-negative. */
 {
   const s32 mask = input[0] >> 31;
   const s32 carry = -((input[0] & mask) >> 26);
   input[0] = input[0] + (carry << 26);
   input[1] = input[1] - carry;
 }

 /* All input[i] are now non-negative. However, there might be values between
  * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
 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;
   }
 }

 /* If the first carry-chain pass, just above, ended up with a carry from
  * input[9], and that caused input[0] to be out-of-bounds, then input[0] was
  * < 2^26 + 2*19, because the carry was, at most, two.
  *
  * If the second pass carried from input[9] again then input[0] is < 2*19 and
  * the input[9] -> input[0] carry didn't push input[0] out of bounds. */

 /* It still remains the case that input might be between 2^255-19 and 2^255.
  * In this case, input[1..9] must take their maximum value and input[0] must
  * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
 s32 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);
   }
 }

 /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
  * this conditionally subtracts 2^255-19. */
 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
}

/* 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
*
* On entry and exit, the absolute value of the limbs of all inputs and outputs
* are < 2^26. */
static 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 */
}

/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
* them unchanged if 'iswap' is 0.  Runs in data-invariant time to avoid
* side-channel attacks.
*
* NOTE that this function requires that 'iswap' be 1 or 0; other values give
* wrong results.  Also, the two limb arrays must be in reduced-coefficient,
* reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
* and all all values in a[0..9],b[0..9] must have magnitude less than
* INT32_MAX. */
static 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;
 }
}

/* 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(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);
}

// -----------------------------------------------------------------------------
// Shamelessly copied from djb's code
// -----------------------------------------------------------------------------
static 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);
}

int curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint);

int
curve25519_donna(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);
 return 0;
}