path: root/src/libcryptobox/curve25519/curve25519-donna.c

/* 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 <string.h>
#include <stdint.h>
#include "curve25519.h"

#ifdef _MSC_VER
#define inline __inline
#endif

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;
}

/* 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];
	s32 mask;

	/* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
	for (i = 0; i < 10; i++) {
		input[i] = 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. */
	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 scalarmult_donna (u8 *mypublic, const u8 *secret, const u8 *basepoint)
{
	limb bp[10], x[10], z[11], zmone[10];
	unsigned char e[32];

	memcpy (e, secret, 32);
	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;
}

int
scalarmult_base_donna (u8 *mypublic, const u8 *secret)
{
	return scalarmult_donna (mypublic, secret,
			curve25519_basepoint);
}