ZeroTierOne/node/ECC.cpp

/*
Matthew Dempsky
Public domain.
Derived from public domain code by D. J. Bernstein.
*/

// Modified very slightly for ZeroTier One by Adam Ierymenko
// This code remains in the public domain.

#include "ECC.hpp"

#include "SHA512.hpp"

#include <stdint.h>
#include <string.h>

#ifdef __WINDOWS__
#pragma warning(disable : 4146)
#endif

#ifdef __GNUC__
#pragma GCC diagnostic ignored "-Wunused-function"
#endif

namespace {

#define crypto_int32			 int32_t
#define crypto_uint32			 uint32_t
#define crypto_int64			 int64_t
#define crypto_uint64			 uint64_t
#define crypto_hash_sha512_BYTES 64

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

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

static inline 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];
	}
}

static inline void fdifference(limb* output, const limb* in)
{
	unsigned i;
	for (i = 0; i < 10; ++i) {
		output[i] = in[i] - output[i];
	}
}

static inline void fscalar_product(limb* output, const limb* in, const limb scalar)
{
	unsigned i;
	for (i = 0; i < 10; ++i) {
		output[i] = in[i] * scalar;
	}
}

static inline 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]);
}

static inline void freduce_degree(limb* output)
{
	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

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;
}

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;
}

static inline 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. */
}

static inline 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);
}

static inline 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]);
}

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);
}

static inline 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

static inline 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;
}

static inline s32 s32_gte(s32 a, s32 b)
{
	a -= b;
	/* a >= 0 iff a >= b. */
	return ~(a >> 31);
}

static inline 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
}

static inline 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 */
}

static inline 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;
	}
}

static inline 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);
}

static inline 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);
}

static void crypto_scalarmult(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);
}

static const unsigned char base[32] = { 9 };
static inline void crypto_scalarmult_base(unsigned char* q, const unsigned char* n)
{
	crypto_scalarmult(q, n, base);
}

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

// Ed25519 ref from: http://bench.cr.yp.to/supercop.html

typedef struct {
	crypto_uint32 v[32];
} fe25519;

typedef struct {
	crypto_uint32 v[32];
} sc25519;

typedef struct {
	crypto_uint32 v[16];
} shortsc25519;

typedef struct {
	fe25519 x;
	fe25519 y;
	fe25519 z;
	fe25519 t;
} ge25519;

#define ge25519_p3 ge25519

typedef struct {
	fe25519 x;
	fe25519 z;
	fe25519 y;
	fe25519 t;
} ge25519_p1p1;

typedef struct {
	fe25519 x;
	fe25519 y;
	fe25519 z;
} ge25519_p2;

typedef struct {
	fe25519 x;
	fe25519 y;
} ge25519_aff;

static inline void fe25519_sub(fe25519* r, const fe25519* x, const fe25519* y);

static inline crypto_uint32 equal(crypto_uint32 a, crypto_uint32 b) /* 16-bit inputs */
{
	crypto_uint32 x = a ^ b; /* 0: yes; 1..65535: no */
	x -= 1;					 /* 4294967295: yes; 0..65534: no */
	x >>= 31;				 /* 1: yes; 0: no */
	return x;
}

static inline crypto_uint32 ge(crypto_uint32 a, crypto_uint32 b) /* 16-bit inputs */
{
	unsigned int x = a;
	x -= (unsigned int)b; /* 0..65535: yes; 4294901761..4294967295: no */
	x >>= 31;			  /* 0: yes; 1: no */
	x ^= 1;				  /* 1: yes; 0: no */
	return x;
}

static inline crypto_uint32 times19(crypto_uint32 a)
{
	return (a << 4) + (a << 1) + a;
}

static inline crypto_uint32 times38(crypto_uint32 a)
{
	return (a << 5) + (a << 2) + (a << 1);
}

static inline void reduce_add_sub(fe25519* r)
{
	crypto_uint32 t;
	int i, rep;

	for (rep = 0; rep < 4; rep++) {
		t = r->v[31] >> 7;
		r->v[31] &= 127;
		t = times19(t);
		r->v[0] += t;
		for (i = 0; i < 31; i++) {
			t = r->v[i] >> 8;
			r->v[i + 1] += t;
			r->v[i] &= 255;
		}
	}
}

static inline void reduce_mul(fe25519* r)
{
	crypto_uint32 t;
	int i, rep;

	for (rep = 0; rep < 2; rep++) {
		t = r->v[31] >> 7;
		r->v[31] &= 127;
		t = times19(t);
		r->v[0] += t;
		for (i = 0; i < 31; i++) {
			t = r->v[i] >> 8;
			r->v[i + 1] += t;
			r->v[i] &= 255;
		}
	}
}

/* reduction modulo 2^255-19 */
static inline void fe25519_freeze(fe25519* r)
{
	int i;
	crypto_uint32 m = equal(r->v[31], 127);
	for (i = 30; i > 0; i--) {
		m &= equal(r->v[i], 255);
	}
	m &= ge(r->v[0], 237);

	m = -m;

	r->v[31] -= m & 127;
	for (i = 30; i > 0; i--) {
		r->v[i] -= m & 255;
	}
	r->v[0] -= m & 237;
}

static inline void fe25519_unpack(fe25519* r, const unsigned char x[32])
{
	int i;
	for (i = 0; i < 32; i++) {
		r->v[i] = x[i];
	}
	r->v[31] &= 127;
}

/* Assumes input x being reduced below 2^255 */
static inline void fe25519_pack(unsigned char r[32], const fe25519* x)
{
	int i;
	fe25519 y = *x;
	fe25519_freeze(&y);
	for (i = 0; i < 32; i++) {
		r[i] = y.v[i];
	}
}

static inline int fe25519_iseq_vartime(const fe25519* x, const fe25519* y)
{
	int i;
	fe25519 t1 = *x;
	fe25519 t2 = *y;
	fe25519_freeze(&t1);
	fe25519_freeze(&t2);
	for (i = 0; i < 32; i++) {
		if (t1.v[i] != t2.v[i]) {
			return 0;
		}
	}
	return 1;
}

static inline void fe25519_cmov(fe25519* r, const fe25519* x, unsigned char b)
{
	int i;
	crypto_uint32 mask = b;
	mask = -mask;
	for (i = 0; i < 32; i++) {
		r->v[i] ^= mask & (x->v[i] ^ r->v[i]);
	}
}

static inline unsigned char fe25519_getparity(const fe25519* x)
{
	fe25519 t = *x;
	fe25519_freeze(&t);
	return t.v[0] & 1;
}

static inline void fe25519_setone(fe25519* r)
{
	int i;
	r->v[0] = 1;
	for (i = 1; i < 32; i++) {
		r->v[i] = 0;
	}
}

static inline void fe25519_setzero(fe25519* r)
{
	int i;
	for (i = 0; i < 32; i++) {
		r->v[i] = 0;
	}
}

static inline void fe25519_neg(fe25519* r, const fe25519* x)
{
	fe25519 t;
	int i;
	for (i = 0; i < 32; i++) {
		t.v[i] = x->v[i];
	}
	fe25519_setzero(r);
	fe25519_sub(r, r, &t);
}

static inline void fe25519_add(fe25519* r, const fe25519* x, const fe25519* y)
{
	int i;
	for (i = 0; i < 32; i++) {
		r->v[i] = x->v[i] + y->v[i];
	}
	reduce_add_sub(r);
}

static inline void fe25519_sub(fe25519* r, const fe25519* x, const fe25519* y)
{
	int i;
	crypto_uint32 t[32];
	t[0] = x->v[0] + 0x1da;
	t[31] = x->v[31] + 0xfe;
	for (i = 1; i < 31; i++) {
		t[i] = x->v[i] + 0x1fe;
	}
	for (i = 0; i < 32; i++) {
		r->v[i] = t[i] - y->v[i];
	}
	reduce_add_sub(r);
}

static inline void fe25519_mul(fe25519* r, const fe25519* x, const fe25519* y)
{
	int i, j;
	crypto_uint32 t[63];
	for (i = 0; i < 63; i++) {
		t[i] = 0;
	}

	for (i = 0; i < 32; i++) {
		for (j = 0; j < 32; j++) {
			t[i + j] += x->v[i] * y->v[j];
		}
	}

	for (i = 32; i < 63; i++) {
		r->v[i - 32] = t[i - 32] + times38(t[i]);
	}
	r->v[31] = t[31]; /* result now in r[0]...r[31] */

	reduce_mul(r);
}

static inline void fe25519_square(fe25519* r, const fe25519* x)
{
	fe25519_mul(r, x, x);
}

static inline void fe25519_invert(fe25519* r, const fe25519* x)
{
	fe25519 z2;
	fe25519 z9;
	fe25519 z11;
	fe25519 z2_5_0;
	fe25519 z2_10_0;
	fe25519 z2_20_0;
	fe25519 z2_50_0;
	fe25519 z2_100_0;
	fe25519 t0;
	fe25519 t1;
	int i;

	/* 2 */ fe25519_square(&z2, x);
	/* 4 */ fe25519_square(&t1, &z2);
	/* 8 */ fe25519_square(&t0, &t1);
	/* 9 */ fe25519_mul(&z9, &t0, x);
	/* 11 */ fe25519_mul(&z11, &z9, &z2);
	/* 22 */ fe25519_square(&t0, &z11);
	/* 2^5 - 2^0 = 31 */ fe25519_mul(&z2_5_0, &t0, &z9);

	/* 2^6 - 2^1 */ fe25519_square(&t0, &z2_5_0);
	/* 2^7 - 2^2 */ fe25519_square(&t1, &t0);
	/* 2^8 - 2^3 */ fe25519_square(&t0, &t1);
	/* 2^9 - 2^4 */ fe25519_square(&t1, &t0);
	/* 2^10 - 2^5 */ fe25519_square(&t0, &t1);
	/* 2^10 - 2^0 */ fe25519_mul(&z2_10_0, &t0, &z2_5_0);

	/* 2^11 - 2^1 */ fe25519_square(&t0, &z2_10_0);
	/* 2^12 - 2^2 */ fe25519_square(&t1, &t0);
	/* 2^20 - 2^10 */ for (i = 2; i < 10; i += 2) {
		fe25519_square(&t0, &t1);
		fe25519_square(&t1, &t0);
	}
	/* 2^20 - 2^0 */ fe25519_mul(&z2_20_0, &t1, &z2_10_0);

	/* 2^21 - 2^1 */ fe25519_square(&t0, &z2_20_0);
	/* 2^22 - 2^2 */ fe25519_square(&t1, &t0);
	/* 2^40 - 2^20 */ for (i = 2; i < 20; i += 2) {
		fe25519_square(&t0, &t1);
		fe25519_square(&t1, &t0);
	}
	/* 2^40 - 2^0 */ fe25519_mul(&t0, &t1, &z2_20_0);

	/* 2^41 - 2^1 */ fe25519_square(&t1, &t0);
	/* 2^42 - 2^2 */ fe25519_square(&t0, &t1);
	/* 2^50 - 2^10 */ for (i = 2; i < 10; i += 2) {
		fe25519_square(&t1, &t0);
		fe25519_square(&t0, &t1);
	}
	/* 2^50 - 2^0 */ fe25519_mul(&z2_50_0, &t0, &z2_10_0);

	/* 2^51 - 2^1 */ fe25519_square(&t0, &z2_50_0);
	/* 2^52 - 2^2 */ fe25519_square(&t1, &t0);
	/* 2^100 - 2^50 */ for (i = 2; i < 50; i += 2) {
		fe25519_square(&t0, &t1);
		fe25519_square(&t1, &t0);
	}
	/* 2^100 - 2^0 */ fe25519_mul(&z2_100_0, &t1, &z2_50_0);

	/* 2^101 - 2^1 */ fe25519_square(&t1, &z2_100_0);
	/* 2^102 - 2^2 */ fe25519_square(&t0, &t1);
	/* 2^200 - 2^100 */ for (i = 2; i < 100; i += 2) {
		fe25519_square(&t1, &t0);
		fe25519_square(&t0, &t1);
	}
	/* 2^200 - 2^0 */ fe25519_mul(&t1, &t0, &z2_100_0);

	/* 2^201 - 2^1 */ fe25519_square(&t0, &t1);
	/* 2^202 - 2^2 */ fe25519_square(&t1, &t0);
	/* 2^250 - 2^50 */ for (i = 2; i < 50; i += 2) {
		fe25519_square(&t0, &t1);
		fe25519_square(&t1, &t0);
	}
	/* 2^250 - 2^0 */ fe25519_mul(&t0, &t1, &z2_50_0);

	/* 2^251 - 2^1 */ fe25519_square(&t1, &t0);
	/* 2^252 - 2^2 */ fe25519_square(&t0, &t1);
	/* 2^253 - 2^3 */ fe25519_square(&t1, &t0);
	/* 2^254 - 2^4 */ fe25519_square(&t0, &t1);
	/* 2^255 - 2^5 */ fe25519_square(&t1, &t0);
	/* 2^255 - 21 */ fe25519_mul(r, &t1, &z11);
}

static inline void fe25519_pow2523(fe25519* r, const fe25519* x)
{
	fe25519 z2;
	fe25519 z9;
	fe25519 z11;
	fe25519 z2_5_0;
	fe25519 z2_10_0;
	fe25519 z2_20_0;
	fe25519 z2_50_0;
	fe25519 z2_100_0;
	fe25519 t;
	int i;

	/* 2 */ fe25519_square(&z2, x);
	/* 4 */ fe25519_square(&t, &z2);
	/* 8 */ fe25519_square(&t, &t);
	/* 9 */ fe25519_mul(&z9, &t, x);
	/* 11 */ fe25519_mul(&z11, &z9, &z2);
	/* 22 */ fe25519_square(&t, &z11);
	/* 2^5 - 2^0 = 31 */ fe25519_mul(&z2_5_0, &t, &z9);

	/* 2^6 - 2^1 */ fe25519_square(&t, &z2_5_0);
	/* 2^10 - 2^5 */ for (i = 1; i < 5; i++) { fe25519_square(&t, &t); }
	/* 2^10 - 2^0 */ fe25519_mul(&z2_10_0, &t, &z2_5_0);

	/* 2^11 - 2^1 */ fe25519_square(&t, &z2_10_0);
	/* 2^20 - 2^10 */ for (i = 1; i < 10; i++) { fe25519_square(&t, &t); }
	/* 2^20 - 2^0 */ fe25519_mul(&z2_20_0, &t, &z2_10_0);

	/* 2^21 - 2^1 */ fe25519_square(&t, &z2_20_0);
	/* 2^40 - 2^20 */ for (i = 1; i < 20; i++) { fe25519_square(&t, &t); }
	/* 2^40 - 2^0 */ fe25519_mul(&t, &t, &z2_20_0);

	/* 2^41 - 2^1 */ fe25519_square(&t, &t);
	/* 2^50 - 2^10 */ for (i = 1; i < 10; i++) { fe25519_square(&t, &t); }
	/* 2^50 - 2^0 */ fe25519_mul(&z2_50_0, &t, &z2_10_0);

	/* 2^51 - 2^1 */ fe25519_square(&t, &z2_50_0);
	/* 2^100 - 2^50 */ for (i = 1; i < 50; i++) { fe25519_square(&t, &t); }
	/* 2^100 - 2^0 */ fe25519_mul(&z2_100_0, &t, &z2_50_0);

	/* 2^101 - 2^1 */ fe25519_square(&t, &z2_100_0);
	/* 2^200 - 2^100 */ for (i = 1; i < 100; i++) { fe25519_square(&t, &t); }
	/* 2^200 - 2^0 */ fe25519_mul(&t, &t, &z2_100_0);

	/* 2^201 - 2^1 */ fe25519_square(&t, &t);
	/* 2^250 - 2^50 */ for (i = 1; i < 50; i++) { fe25519_square(&t, &t); }
	/* 2^250 - 2^0 */ fe25519_mul(&t, &t, &z2_50_0);

	/* 2^251 - 2^1 */ fe25519_square(&t, &t);
	/* 2^252 - 2^2 */ fe25519_square(&t, &t);
	/* 2^252 - 3 */ fe25519_mul(r, &t, x);
}

static const crypto_uint32 m[32] = { 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2, 0xDE, 0xF9, 0xDE, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10 };
static const crypto_uint32 mu[33] = { 0x1B, 0x13, 0x2C, 0x0A, 0xA3, 0xE5, 0x9C, 0xED, 0xA7, 0x29, 0x63, 0x08, 0x5D, 0x21, 0x06, 0x21, 0xEB, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x0F };

static inline crypto_uint32 lt(crypto_uint32 a, crypto_uint32 b) /* 16-bit inputs */
{
	unsigned int x = a;
	x -= (unsigned int)b; /* 0..65535: no; 4294901761..4294967295: yes */
	x >>= 31;			  /* 0: no; 1: yes */
	return x;
}

/* Reduce coefficients of r before calling reduce_add_sub */
static inline void reduce_add_sub(sc25519* r)
{
	crypto_uint32 pb = 0;
	crypto_uint32 b;
	crypto_uint32 mask;
	int i;
	unsigned char t[32];

	for (i = 0; i < 32; i++) {
		pb += m[i];
		b = lt(r->v[i], pb);
		t[i] = r->v[i] - pb + (b << 8);
		pb = b;
	}
	mask = b - 1;
	for (i = 0; i < 32; i++) {
		r->v[i] ^= mask & (r->v[i] ^ t[i]);
	}
}

/* Reduce coefficients of x before calling barrett_reduce */
static inline void barrett_reduce(sc25519* r, const crypto_uint32 x[64])
{
	/* See HAC, Alg. 14.42 */
	int i, j;
	crypto_uint32 q2[66];
	crypto_uint32* q3 = q2 + 33;
	crypto_uint32 r1[33];
	crypto_uint32 r2[33];
	crypto_uint32 carry;
	crypto_uint32 pb = 0;
	crypto_uint32 b;

	for (i = 0; i < 66; ++i) {
		q2[i] = 0;
	}
	for (i = 0; i < 33; ++i) {
		r2[i] = 0;
	}

	for (i = 0; i < 33; i++) {
		for (j = 0; j < 33; j++) {
			if (i + j >= 31) {
				q2[i + j] += mu[i] * x[j + 31];
			}
		}
	}
	carry = q2[31] >> 8;
	q2[32] += carry;
	carry = q2[32] >> 8;
	q2[33] += carry;

	for (i = 0; i < 33; i++) {
		r1[i] = x[i];
	}
	for (i = 0; i < 32; i++) {
		for (j = 0; j < 33; j++) {
			if (i + j < 33) {
				r2[i + j] += m[i] * q3[j];
			}
		}
	}

	for (i = 0; i < 32; i++) {
		carry = r2[i] >> 8;
		r2[i + 1] += carry;
		r2[i] &= 0xff;
	}

	for (i = 0; i < 32; i++) {
		pb += r2[i];
		b = lt(r1[i], pb);
		r->v[i] = r1[i] - pb + (b << 8);
		pb = b;
	}

	/* XXX: Can it really happen that r<0?, See HAC, Alg 14.42, Step 3
	 * If so: Handle  it here!
	 */

	reduce_add_sub(r);
	reduce_add_sub(r);
}

static inline void sc25519_from32bytes(sc25519* r, const unsigned char x[32])
{
	int i;
	crypto_uint32 t[64];
	for (i = 0; i < 32; i++) {
		t[i] = x[i];
	}
	for (i = 32; i < 64; ++i) {
		t[i] = 0;
	}
	barrett_reduce(r, t);
}

static inline void sc25519_from64bytes(sc25519* r, const unsigned char x[64])
{
	int i;
	crypto_uint32 t[64];
	for (i = 0; i < 64; i++) {
		t[i] = x[i];
	}
	barrett_reduce(r, t);
}

static inline void sc25519_to32bytes(unsigned char r[32], const sc25519* x)
{
	int i;
	for (i = 0; i < 32; i++) {
		r[i] = x->v[i];
	}
}

static inline void sc25519_add(sc25519* r, const sc25519* x, const sc25519* y)
{
	int i, carry;
	for (i = 0; i < 32; i++) {
		r->v[i] = x->v[i] + y->v[i];
	}
	for (i = 0; i < 31; i++) {
		carry = r->v[i] >> 8;
		r->v[i + 1] += carry;
		r->v[i] &= 0xff;
	}
	reduce_add_sub(r);
}

static inline void sc25519_mul(sc25519* r, const sc25519* x, const sc25519* y)
{
	int i, j, carry;
	crypto_uint32 t[64];
	for (i = 0; i < 64; i++) {
		t[i] = 0;
	}

	for (i = 0; i < 32; i++) {
		for (j = 0; j < 32; j++) {
			t[i + j] += x->v[i] * y->v[j];
		}
	}

	for (i = 0; i < 63; i++) {
		carry = t[i] >> 8;
		t[i + 1] += carry;
		t[i] &= 0xff;
	}

	barrett_reduce(r, t);
}

static inline void sc25519_window3(signed char r[85], const sc25519* s)
{
	char carry;
	int i;
	for (i = 0; i < 10; i++) {
		r[8 * i + 0] = s->v[3 * i + 0] & 7;
		r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
		r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
		r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
		r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
		r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;
		r[8 * i + 5] = (s->v[3 * i + 1] >> 7) & 7;
		r[8 * i + 5] ^= (s->v[3 * i + 2] << 1) & 7;
		r[8 * i + 6] = (s->v[3 * i + 2] >> 2) & 7;
		r[8 * i + 7] = (s->v[3 * i + 2] >> 5) & 7;
	}
	r[8 * i + 0] = s->v[3 * i + 0] & 7;
	r[8 * i + 1] = (s->v[3 * i + 0] >> 3) & 7;
	r[8 * i + 2] = (s->v[3 * i + 0] >> 6) & 7;
	r[8 * i + 2] ^= (s->v[3 * i + 1] << 2) & 7;
	r[8 * i + 3] = (s->v[3 * i + 1] >> 1) & 7;
	r[8 * i + 4] = (s->v[3 * i + 1] >> 4) & 7;

	/* Making it signed */
	carry = 0;
	for (i = 0; i < 84; i++) {
		r[i] += carry;
		r[i + 1] += r[i] >> 3;
		r[i] &= 7;
		carry = r[i] >> 2;
		r[i] -= carry << 3;
	}
	r[84] += carry;
}

static inline void sc25519_2interleave2(unsigned char r[127], const sc25519* s1, const sc25519* s2)
{
	int i;
	for (i = 0; i < 31; i++) {
		r[4 * i] = (s1->v[i] & 3) ^ ((s2->v[i] & 3) << 2);
		r[4 * i + 1] = ((s1->v[i] >> 2) & 3) ^ (((s2->v[i] >> 2) & 3) << 2);
		r[4 * i + 2] = ((s1->v[i] >> 4) & 3) ^ (((s2->v[i] >> 4) & 3) << 2);
		r[4 * i + 3] = ((s1->v[i] >> 6) & 3) ^ (((s2->v[i] >> 6) & 3) << 2);
	}
	r[124] = (s1->v[31] & 3) ^ ((s2->v[31] & 3) << 2);
	r[125] = ((s1->v[31] >> 2) & 3) ^ (((s2->v[31] >> 2) & 3) << 2);
	r[126] = ((s1->v[31] >> 4) & 3) ^ (((s2->v[31] >> 4) & 3) << 2);
}

/* d */
static const fe25519 ge25519_ecd = { { 0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00, 0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52 } };
/* 2*d */
static const fe25519 ge25519_ec2d = { { 0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00, 0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24 } };
/* sqrt(-1) */
static const fe25519 ge25519_sqrtm1 = { { 0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F, 0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B } };

/* Packed coordinates of the base point */
static const ge25519 ge25519_base = { { { 0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69, 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21 } },
									  { { 0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66 } },
									  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
									  { { 0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20, 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67 } } };

/* Multiples of the base point in affine representation */
static const ge25519_aff ge25519_base_multiples_affine[425] = {
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21 } },
	  { { 0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66 } } },
	{ { { 0x0e, 0xce, 0x43, 0x28, 0x4e, 0xa1, 0xc5, 0x83, 0x5f, 0xa4, 0xd7, 0x15, 0x45, 0x8e, 0x0d, 0x08, 0xac, 0xe7, 0x33, 0x18, 0x7d, 0x3b, 0x04, 0x3d, 0x6c, 0x04, 0x5a, 0x9f, 0x4c, 0x38, 0xab, 0x36 } },
	  { { 0xc9, 0xa3, 0xf8, 0x6a, 0xae, 0x46, 0x5f, 0x0e, 0x56, 0x51, 0x38, 0x64, 0x51, 0x0f, 0x39, 0x97, 0x56, 0x1f, 0xa2, 0xc9, 0xe8, 0x5e, 0xa2, 0x1d, 0xc2, 0x29, 0x23, 0x09, 0xf3, 0xcd, 0x60, 0x22 } } },
	{ { { 0x5c, 0xe2, 0xf8, 0xd3, 0x5f, 0x48, 0x62, 0xac, 0x86, 0x48, 0x62, 0x81, 0x19, 0x98, 0x43, 0x63, 0x3a, 0xc8, 0xda, 0x3e, 0x74, 0xae, 0xf4, 0x1f, 0x49, 0x8f, 0x92, 0x22, 0x4a, 0x9c, 0xae, 0x67 } },
	  { { 0xd4, 0xb4, 0xf5, 0x78, 0x48, 0x68, 0xc3, 0x02, 0x04, 0x03, 0x24, 0x67, 0x17, 0xec, 0x16, 0x9f, 0xf7, 0x9e, 0x26, 0x60, 0x8e, 0xa1, 0x26, 0xa1, 0xab, 0x69, 0xee, 0x77, 0xd1, 0xb1, 0x67, 0x12 } } },
	{ { { 0x70, 0xf8, 0xc9, 0xc4, 0x57, 0xa6, 0x3a, 0x49, 0x47, 0x15, 0xce, 0x93, 0xc1, 0x9e, 0x73, 0x1a, 0xf9, 0x20, 0x35, 0x7a, 0xb8, 0xd4, 0x25, 0x83, 0x46, 0xf1, 0xcf, 0x56, 0xdb, 0xa8, 0x3d, 0x20 } },
	  { { 0x2f, 0x11, 0x32, 0xca, 0x61, 0xab, 0x38, 0xdf, 0xf0, 0x0f, 0x2f, 0xea, 0x32, 0x28, 0xf2, 0x4c, 0x6c, 0x71, 0xd5, 0x80, 0x85, 0xb8, 0x0e, 0x47, 0xe1, 0x95, 0x15, 0xcb, 0x27, 0xe8, 0xd0, 0x47 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0xc8, 0x84, 0xa5, 0x08, 0xbc, 0xfd, 0x87, 0x3b, 0x99, 0x8b, 0x69, 0x80, 0x7b, 0xc6, 0x3a, 0xeb, 0x93, 0xcf, 0x4e, 0xf8, 0x5c, 0x2d, 0x86, 0x42, 0xb6, 0x71, 0xd7, 0x97, 0x5f, 0xe1, 0x42, 0x67 } },
	  { { 0xb4, 0xb9, 0x37, 0xfc, 0xa9, 0x5b, 0x2f, 0x1e, 0x93, 0xe4, 0x1e, 0x62, 0xfc, 0x3c, 0x78, 0x81, 0x8f, 0xf3, 0x8a, 0x66, 0x09, 0x6f, 0xad, 0x6e, 0x79, 0x73, 0xe5, 0xc9, 0x00, 0x06, 0xd3, 0x21 } } },
	{ { { 0xf8, 0xf9, 0x28, 0x6c, 0x6d, 0x59, 0xb2, 0x59, 0x74, 0x23, 0xbf, 0xe7, 0x33, 0x8d, 0x57, 0x09, 0x91, 0x9c, 0x24, 0x08, 0x15, 0x2b, 0xe2, 0xb8, 0xee, 0x3a, 0xe5, 0x27, 0x06, 0x86, 0xa4, 0x23 } },
	  { { 0xeb, 0x27, 0x67, 0xc1, 0x37, 0xab, 0x7a, 0xd8, 0x27, 0x9c, 0x07, 0x8e, 0xff, 0x11, 0x6a, 0xb0, 0x78, 0x6e, 0xad, 0x3a, 0x2e, 0x0f, 0x98, 0x9f, 0x72, 0xc3, 0x7f, 0x82, 0xf2, 0x96, 0x96, 0x70 } } },
	{ { { 0x81, 0x6b, 0x88, 0xe8, 0x1e, 0xc7, 0x77, 0x96, 0x0e, 0xa1, 0xa9, 0x52, 0xe0, 0xd8, 0x0e, 0x61, 0x9e, 0x79, 0x2d, 0x95, 0x9c, 0x8d, 0x96, 0xe0, 0x06, 0x40, 0x5d, 0x87, 0x28, 0x5f, 0x98, 0x70 } },
	  { { 0xf1, 0x79, 0x7b, 0xed, 0x4f, 0x44, 0xb2, 0xe7, 0x08, 0x0d, 0xc2, 0x08, 0x12, 0xd2, 0x9f, 0xdf, 0xcd, 0x93, 0x20, 0x8a, 0xcf, 0x33, 0xca, 0x6d, 0x89, 0xb9, 0x77, 0xc8, 0x93, 0x1b, 0x4e, 0x60 } } },
	{ { { 0x26, 0x4f, 0x7e, 0x97, 0xf6, 0x40, 0xdd, 0x4f, 0xfc, 0x52, 0x78, 0xf9, 0x90, 0x31, 0x03, 0xe6, 0x7d, 0x56, 0x39, 0x0b, 0x1d, 0x56, 0x82, 0x85, 0xf9, 0x1a, 0x42, 0x17, 0x69, 0x6c, 0xcf, 0x39 } },
	  { { 0x69, 0xd2, 0x06, 0x3a, 0x4f, 0x39, 0x2d, 0xf9, 0x38, 0x40, 0x8c, 0x4c, 0xe7, 0x05, 0x12, 0xb4, 0x78, 0x8b, 0xf8, 0xc0, 0xec, 0x93, 0xde, 0x7a, 0x6b, 0xce, 0x2c, 0xe1, 0x0e, 0xa9, 0x34, 0x44 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0x0b, 0xa4, 0x3c, 0xb0, 0x0f, 0x7a, 0x51, 0xf1, 0x78, 0xd6, 0xd9, 0x6a, 0xfd, 0x46, 0xe8, 0xb8, 0xa8, 0x79, 0x1d, 0x87, 0xf9, 0x90, 0xf2, 0x9c, 0x13, 0x29, 0xf8, 0x0b, 0x20, 0x64, 0xfa, 0x05 } },
	  { { 0x26, 0x09, 0xda, 0x17, 0xaf, 0x95, 0xd6, 0xfb, 0x6a, 0x19, 0x0d, 0x6e, 0x5e, 0x12, 0xf1, 0x99, 0x4c, 0xaa, 0xa8, 0x6f, 0x79, 0x86, 0xf4, 0x72, 0x28, 0x00, 0x26, 0xf9, 0xea, 0x9e, 0x19, 0x3d } } },
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	  { { 0x65, 0x9c, 0x5d, 0xf0, 0x37, 0x04, 0xf0, 0x34, 0x69, 0x2a, 0xf0, 0xa5, 0x64, 0xca, 0xde, 0x2b, 0x5b, 0x15, 0x10, 0xd2, 0xab, 0x06, 0xdd, 0xc4, 0xb0, 0xb6, 0x5b, 0xc1, 0x17, 0xdf, 0x8f, 0x02 } } },
	{ { { 0xbd, 0x59, 0x3d, 0xbf, 0x5c, 0x31, 0x44, 0x2c, 0x32, 0x94, 0x04, 0x60, 0x84, 0x0f, 0xad, 0x00, 0xb6, 0x8f, 0xc9, 0x1d, 0xcc, 0x5c, 0xa2, 0x49, 0x0e, 0x50, 0x91, 0x08, 0x9a, 0x43, 0x55, 0x05 } },
	  { { 0x5d, 0x93, 0x55, 0xdf, 0x9b, 0x12, 0x19, 0xec, 0x93, 0x85, 0x42, 0x9e, 0x66, 0x0f, 0x9d, 0xaf, 0x99, 0xaf, 0x26, 0x89, 0xbc, 0x61, 0xfd, 0xff, 0xce, 0x4b, 0xf4, 0x33, 0x95, 0xc9, 0x35, 0x58 } } },
	{ { { 0x12, 0x55, 0xf9, 0xda, 0xcb, 0x44, 0xa7, 0xdc, 0x57, 0xe2, 0xf9, 0x9a, 0xe6, 0x07, 0x23, 0x60, 0x54, 0xa7, 0x39, 0xa5, 0x9b, 0x84, 0x56, 0x6e, 0xaa, 0x8b, 0x8f, 0xb0, 0x2c, 0x87, 0xaf, 0x67 } },
	  { { 0x00, 0xa9, 0x4c, 0xb2, 0x12, 0xf8, 0x32, 0xa8, 0x7a, 0x00, 0x4b, 0x49, 0x32, 0xba, 0x1f, 0x5d, 0x44, 0x8e, 0x44, 0x7a, 0xdc, 0x11, 0xfb, 0x39, 0x08, 0x57, 0x87, 0xa5, 0x12, 0x42, 0x93, 0x0e } } },
	{ { { 0x17, 0xb4, 0xae, 0x72, 0x59, 0xd0, 0xaa, 0xa8, 0x16, 0x8b, 0x63, 0x11, 0xb3, 0x43, 0x04, 0xda, 0x0c, 0xa8, 0xb7, 0x68, 0xdd, 0x4e, 0x54, 0xe7, 0xaf, 0x5d, 0x5d, 0x05, 0x76, 0x36, 0xec, 0x0d } },
	  { { 0x6d, 0x7c, 0x82, 0x32, 0x38, 0x55, 0x57, 0x74, 0x5b, 0x7d, 0xc3, 0xc4, 0xfb, 0x06, 0x29, 0xf0, 0x13, 0x55, 0x54, 0xc6, 0xa7, 0xdc, 0x4c, 0x9f, 0x98, 0x49, 0x20, 0xa8, 0xc3, 0x8d, 0xfa, 0x48 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0x87, 0x47, 0x9d, 0xe9, 0x25, 0xd5, 0xe3, 0x47, 0x78, 0xdf, 0x85, 0xa7, 0x85, 0x5e, 0x7a, 0x4c, 0x5f, 0x79, 0x1a, 0xf3, 0xa2, 0xb2, 0x28, 0xa0, 0x9c, 0xdd, 0x30, 0x40, 0xd4, 0x38, 0xbd, 0x28 } },
	  { { 0xfc, 0xbb, 0xd5, 0x78, 0x6d, 0x1d, 0xd4, 0x99, 0xb4, 0xaa, 0x44, 0x44, 0x7a, 0x1b, 0xd8, 0xfe, 0xb4, 0x99, 0xb9, 0xcc, 0xe7, 0xc4, 0xd3, 0x3a, 0x73, 0x83, 0x41, 0x5c, 0x40, 0xd7, 0x2d, 0x55 } } },
	{ { { 0x26, 0xe1, 0x7b, 0x5f, 0xe5, 0xdc, 0x3f, 0x7d, 0xa1, 0xa7, 0x26, 0x44, 0x22, 0x23, 0xc0, 0x8f, 0x7d, 0xf1, 0xb5, 0x11, 0x47, 0x7b, 0x19, 0xd4, 0x75, 0x6f, 0x1e, 0xa5, 0x27, 0xfe, 0xc8, 0x0e } },
	  { { 0xd3, 0x11, 0x3d, 0xab, 0xef, 0x2c, 0xed, 0xb1, 0x3d, 0x7c, 0x32, 0x81, 0x6b, 0xfe, 0xf8, 0x1c, 0x3c, 0x7b, 0xc0, 0x61, 0xdf, 0xb8, 0x75, 0x76, 0x7f, 0xaa, 0xd8, 0x93, 0xaf, 0x3d, 0xe8, 0x3d } } },
	{ { { 0xfd, 0x5b, 0x4e, 0x8d, 0xb6, 0x7e, 0x82, 0x9b, 0xef, 0xce, 0x04, 0x69, 0x51, 0x52, 0xff, 0xef, 0xa0, 0x52, 0xb5, 0x79, 0x17, 0x5e, 0x2f, 0xde, 0xd6, 0x3c, 0x2d, 0xa0, 0x43, 0xb4, 0x0b, 0x19 } },
	  { { 0xc0, 0x61, 0x48, 0x48, 0x17, 0xf4, 0x9e, 0x18, 0x51, 0x2d, 0xea, 0x2f, 0xf2, 0xf2, 0xe0, 0xa3, 0x14, 0xb7, 0x8b, 0x3a, 0x30, 0xf5, 0x81, 0xc1, 0x5d, 0x71, 0x39, 0x62, 0x55, 0x1f, 0x60, 0x5a } } },
	{ { { 0xe5, 0x89, 0x8a, 0x76, 0x6c, 0xdb, 0x4d, 0x0a, 0x5b, 0x72, 0x9d, 0x59, 0x6e, 0x63, 0x63, 0x18, 0x7c, 0xe3, 0xfa, 0xe2, 0xdb, 0xa1, 0x8d, 0xf4, 0xa5, 0xd7, 0x16, 0xb2, 0xd0, 0xb3, 0x3f, 0x39 } },
	  { { 0xce, 0x60, 0x09, 0x6c, 0xf5, 0x76, 0x17, 0x24, 0x80, 0x3a, 0x96, 0xc7, 0x94, 0x2e, 0xf7, 0x6b, 0xef, 0xb5, 0x05, 0x96, 0xef, 0xd3, 0x7b, 0x51, 0xda, 0x05, 0x44, 0x67, 0xbc, 0x07, 0x21, 0x4e } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0xe9, 0x73, 0x6f, 0x21, 0xb9, 0xde, 0x22, 0x7d, 0xeb, 0x97, 0x31, 0x10, 0xa3, 0xea, 0xe1, 0xc6, 0x37, 0xeb, 0x8f, 0x43, 0x58, 0xde, 0x41, 0x64, 0x0e, 0x3e, 0x07, 0x99, 0x3d, 0xf1, 0xdf, 0x1e } },
	  { { 0xf8, 0xad, 0x43, 0xc2, 0x17, 0x06, 0xe2, 0xe4, 0xa9, 0x86, 0xcd, 0x18, 0xd7, 0x78, 0xc8, 0x74, 0x66, 0xd2, 0x09, 0x18, 0xa5, 0xf1, 0xca, 0xa6, 0x62, 0x92, 0xc1, 0xcb, 0x00, 0xeb, 0x42, 0x2e } } },
	{ { { 0x7b, 0x34, 0x24, 0x4c, 0xcf, 0x38, 0xe5, 0x6c, 0x0a, 0x01, 0x2c, 0x22, 0x0b, 0x24, 0x38, 0xad, 0x24, 0x7e, 0x19, 0xf0, 0x6c, 0xf9, 0x31, 0xf4, 0x35, 0x11, 0xf6, 0x46, 0x33, 0x3a, 0x23, 0x59 } },
	  { { 0x20, 0x0b, 0xa1, 0x08, 0x19, 0xad, 0x39, 0x54, 0xea, 0x3e, 0x23, 0x09, 0xb6, 0xe2, 0xd2, 0xbc, 0x4d, 0xfc, 0x9c, 0xf0, 0x13, 0x16, 0x22, 0x3f, 0xb9, 0xd2, 0x11, 0x86, 0x90, 0x55, 0xce, 0x3c } } },
	{ { { 0xc4, 0x0b, 0x4b, 0x62, 0x99, 0x37, 0x84, 0x3f, 0x74, 0xa2, 0xf9, 0xce, 0xe2, 0x0b, 0x0f, 0x2a, 0x3d, 0xa3, 0xe3, 0xdb, 0x5a, 0x9d, 0x93, 0xcc, 0xa5, 0xef, 0x82, 0x91, 0x1d, 0xe6, 0x6c, 0x68 } },
	  { { 0xa3, 0x64, 0x17, 0x9b, 0x8b, 0xc8, 0x3a, 0x61, 0xe6, 0x9d, 0xc6, 0xed, 0x7b, 0x03, 0x52, 0x26, 0x9d, 0x3a, 0xb3, 0x13, 0xcc, 0x8a, 0xfd, 0x2c, 0x1a, 0x1d, 0xed, 0x13, 0xd0, 0x55, 0x57, 0x0e } } },
	{ { { 0x1a, 0xea, 0xbf, 0xfd, 0x4a, 0x3c, 0x8e, 0xec, 0x29, 0x7e, 0x77, 0x77, 0x12, 0x99, 0xd7, 0x84, 0xf9, 0x55, 0x7f, 0xf1, 0x8b, 0xb4, 0xd2, 0x95, 0xa3, 0x8d, 0xf0, 0x8a, 0xa7, 0xeb, 0x82, 0x4b } },
	  { { 0x2c, 0x28, 0xf4, 0x3a, 0xf6, 0xde, 0x0a, 0xe0, 0x41, 0x44, 0x23, 0xf8, 0x3f, 0x03, 0x64, 0x9f, 0xc3, 0x55, 0x4c, 0xc6, 0xc1, 0x94, 0x1c, 0x24, 0x5d, 0x5f, 0x92, 0x45, 0x96, 0x57, 0x37, 0x14 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0xc1, 0xcd, 0x90, 0x66, 0xb9, 0x76, 0xa0, 0x5b, 0xa5, 0x85, 0x75, 0x23, 0xf9, 0x89, 0xa5, 0x82, 0xb2, 0x6f, 0xb1, 0xeb, 0xc4, 0x69, 0x6f, 0x18, 0x5a, 0xed, 0x94, 0x3d, 0x9d, 0xd9, 0x2c, 0x1a } },
	  { { 0x35, 0xb0, 0xe6, 0x73, 0x06, 0xb7, 0x37, 0xe0, 0xf8, 0xb0, 0x22, 0xe8, 0xd2, 0xed, 0x0b, 0xef, 0xe6, 0xc6, 0x5a, 0x99, 0x9e, 0x1a, 0x9f, 0x04, 0x97, 0xe4, 0x4d, 0x0b, 0xbe, 0xba, 0x44, 0x40 } } },
	{ { { 0xc1, 0x56, 0x96, 0x91, 0x5f, 0x1f, 0xbb, 0x54, 0x6f, 0x88, 0x89, 0x0a, 0xb2, 0xd6, 0x41, 0x42, 0x6a, 0x82, 0xee, 0x14, 0xaa, 0x76, 0x30, 0x65, 0x0f, 0x67, 0x39, 0xa6, 0x51, 0x7c, 0x49, 0x24 } },
	  { { 0x35, 0xa3, 0x78, 0xd1, 0x11, 0x0f, 0x75, 0xd3, 0x70, 0x46, 0xdb, 0x20, 0x51, 0xcb, 0x92, 0x80, 0x54, 0x10, 0x74, 0x36, 0x86, 0xa9, 0xd7, 0xa3, 0x08, 0x78, 0xf1, 0x01, 0x29, 0xf8, 0x80, 0x3b } } },
	{ { { 0xdb, 0xa7, 0x9d, 0x9d, 0xbf, 0xa0, 0xcc, 0xed, 0x53, 0xa2, 0xa2, 0x19, 0x39, 0x48, 0x83, 0x19, 0x37, 0x58, 0xd1, 0x04, 0x28, 0x40, 0xf7, 0x8a, 0xc2, 0x08, 0xb7, 0xa5, 0x42, 0xcf, 0x53, 0x4c } },
	  { { 0xa7, 0xbb, 0xf6, 0x8e, 0xad, 0xdd, 0xf7, 0x90, 0xdd, 0x5f, 0x93, 0x89, 0xae, 0x04, 0x37, 0xe6, 0x9a, 0xb7, 0xe8, 0xc0, 0xdf, 0x16, 0x2a, 0xbf, 0xc4, 0x3a, 0x3c, 0x41, 0xd5, 0x89, 0x72, 0x5a } } },
	{ { { 0x1f, 0x96, 0xff, 0x34, 0x2c, 0x13, 0x21, 0xcb, 0x0a, 0x89, 0x85, 0xbe, 0xb3, 0x70, 0x9e, 0x1e, 0xde, 0x97, 0xaf, 0x96, 0x30, 0xf7, 0x48, 0x89, 0x40, 0x8d, 0x07, 0xf1, 0x25, 0xf0, 0x30, 0x58 } },
	  { { 0x1e, 0xd4, 0x93, 0x57, 0xe2, 0x17, 0xe7, 0x9d, 0xab, 0x3c, 0x55, 0x03, 0x82, 0x2f, 0x2b, 0xdb, 0x56, 0x1e, 0x30, 0x2e, 0x24, 0x47, 0x6e, 0xe6, 0xff, 0x33, 0x24, 0x2c, 0x75, 0x51, 0xd4, 0x67 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0x2b, 0x06, 0xd9, 0xa1, 0x5d, 0xe1, 0xf4, 0xd1, 0x1e, 0x3c, 0x9a, 0xc6, 0x29, 0x2b, 0x13, 0x13, 0x78, 0xc0, 0xd8, 0x16, 0x17, 0x2d, 0x9e, 0xa9, 0xc9, 0x79, 0x57, 0xab, 0x24, 0x91, 0x92, 0x19 } },
	  { { 0x69, 0xfb, 0xa1, 0x9c, 0xa6, 0x75, 0x49, 0x7d, 0x60, 0x73, 0x40, 0x42, 0xc4, 0x13, 0x0a, 0x95, 0x79, 0x1e, 0x04, 0x83, 0x94, 0x99, 0x9b, 0x1e, 0x0c, 0xe8, 0x1f, 0x54, 0xef, 0xcb, 0xc0, 0x52 } } },
	{ { { 0x14, 0x89, 0x73, 0xa1, 0x37, 0x87, 0x6a, 0x7a, 0xcf, 0x1d, 0xd9, 0x2e, 0x1a, 0x67, 0xed, 0x74, 0xc0, 0xf0, 0x9c, 0x33, 0xdd, 0xdf, 0x08, 0xbf, 0x7b, 0xd1, 0x66, 0xda, 0xe6, 0xc9, 0x49, 0x08 } },
	  { { 0xe9, 0xdd, 0x5e, 0x55, 0xb0, 0x0a, 0xde, 0x21, 0x4c, 0x5a, 0x2e, 0xd4, 0x80, 0x3a, 0x57, 0x92, 0x7a, 0xf1, 0xc4, 0x2c, 0x40, 0xaf, 0x2f, 0xc9, 0x92, 0x03, 0xe5, 0x5a, 0xbc, 0xdc, 0xf4, 0x09 } } },
	{ { { 0xf3, 0xe1, 0x2b, 0x7c, 0x05, 0x86, 0x80, 0x93, 0x4a, 0xad, 0xb4, 0x8f, 0x7e, 0x99, 0x0c, 0xfd, 0xcd, 0xef, 0xd1, 0xff, 0x2c, 0x69, 0x34, 0x13, 0x41, 0x64, 0xcf, 0x3b, 0xd0, 0x90, 0x09, 0x1e } },
	  { { 0x9d, 0x45, 0xd6, 0x80, 0xe6, 0x45, 0xaa, 0xf4, 0x15, 0xaa, 0x5c, 0x34, 0x87, 0x99, 0xa2, 0x8c, 0x26, 0x84, 0x62, 0x7d, 0xb6, 0x29, 0xc0, 0x52, 0xea, 0xf5, 0x81, 0x18, 0x0f, 0x35, 0xa9, 0x0e } } },
	{ { { 0xe7, 0x20, 0x72, 0x7c, 0x6d, 0x94, 0x5f, 0x52, 0x44, 0x54, 0xe3, 0xf1, 0xb2, 0xb0, 0x36, 0x46, 0x0f, 0xae, 0x92, 0xe8, 0x70, 0x9d, 0x6e, 0x79, 0xb1, 0xad, 0x37, 0xa9, 0x5f, 0xc0, 0xde, 0x03 } },
	  { { 0x15, 0x55, 0x37, 0xc6, 0x1c, 0x27, 0x1c, 0x6d, 0x14, 0x4f, 0xca, 0xa4, 0xc4, 0x88, 0x25, 0x46, 0x39, 0xfc, 0x5a, 0xe5, 0xfe, 0x29, 0x11, 0x69, 0xf5, 0x72, 0x84, 0x4d, 0x78, 0x9f, 0x94, 0x15 } } },
	{ { { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } },
	  { { 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00 } } },
	{ { { 0xec, 0xd3, 0xff, 0x57, 0x0b, 0xb0, 0xb2, 0xdc, 0xf8, 0x4f, 0xe2, 0x12, 0xd5, 0x36, 0xbe, 0x6b, 0x09, 0x43, 0x6d, 0xa3, 0x4d, 0x90, 0x2d, 0xb8, 0x74, 0xe8, 0x71, 0x45, 0x19, 0x8b, 0x0c, 0x6a } },
	  { { 0xb8, 0x42, 0x1c, 0x03, 0xad, 0x2c, 0x03, 0x8e, 0xac, 0xd7, 0x98, 0x29, 0x13, 0xc6, 0x02, 0x29, 0xb5, 0xd4, 0xe7, 0xcf, 0xcc, 0x8b, 0x83, 0xec, 0x35, 0xc7, 0x9c, 0x74, 0xb7, 0xad, 0x85, 0x5f } } },
	{ { { 0x78, 0x84, 0xe1, 0x56, 0x45, 0x69, 0x68, 0x5a, 0x4f, 0xb8, 0xb1, 0x29, 0xff, 0x33, 0x03, 0x31, 0xb7, 0xcb, 0x96, 0x25, 0xe6, 0xe6, 0x41, 0x98, 0x1a, 0xbb, 0x03, 0x56, 0xf2, 0xb2, 0x91, 0x34 } },
	  { { 0x2c, 0x6c, 0xf7, 0x66, 0xa4, 0x62, 0x6b, 0x39, 0xb3, 0xba, 0x65, 0xd3, 0x1c, 0xf8, 0x11, 0xaa, 0xbe, 0xdc, 0x80, 0x59, 0x87, 0xf5, 0x7b, 0xe5, 0xe3, 0xb3, 0x3e, 0x39, 0xda, 0xbe, 0x88, 0x09 } } },
	{ { { 0x8b, 0xf1, 0xa0, 0xf5, 0xdc, 0x29, 0xb4, 0xe2, 0x07, 0xc6, 0x7a, 0x00, 0xd0, 0x89, 0x17, 0x51, 0xd4, 0xbb, 0xd4, 0x22, 0xea, 0x7e, 0x7d, 0x7c, 0x24, 0xea, 0xf2, 0xe8, 0x22, 0x12, 0x95, 0x06 } },
	  { { 0xda, 0x7c, 0xa4, 0x0c, 0xf4, 0xba, 0x6e, 0xe1, 0x89, 0xb5, 0x59, 0xca, 0xf1, 0xc0, 0x29, 0x36, 0x09, 0x44, 0xe2, 0x7f, 0xd1, 0x63, 0x15, 0x99, 0xea, 0x25, 0xcf, 0x0c, 0x9d, 0xc0, 0x44, 0x6f } } },
	{ { { 0x1d, 0x86, 0x4e, 0xcf, 0xf7, 0x37, 0x10, 0x25, 0x8f, 0x12, 0xfb, 0x19, 0xfb, 0xe0, 0xed, 0x10, 0xc8, 0xe2, 0xf5, 0x75, 0xb1, 0x33, 0xc0, 0x96, 0x0d, 0xfb, 0x15, 0x6c, 0x0d, 0x07, 0x5f, 0x05 } },
	  { { 0x69, 0x3e, 0x47, 0x97, 0x2c, 0xaf, 0x52, 0x7c, 0x78, 0x83, 0xad, 0x1b, 0x39, 0x82, 0x2f, 0x02, 0x6f, 0x47, 0xdb, 0x2a, 0xb0, 0xe1, 0x91, 0x99, 0x55, 0xb8, 0x99, 0x3a, 0xa0, 0x44, 0x11, 0x51 } } }
};

static inline void p1p1_to_p2(ge25519_p2* r, const ge25519_p1p1* p)
{
	fe25519_mul(&r->x, &p->x, &p->t);
	fe25519_mul(&r->y, &p->y, &p->z);
	fe25519_mul(&r->z, &p->z, &p->t);
}

static inline void p1p1_to_p2_2(ge25519_p3* r, const ge25519_p1p1* p)
{
	fe25519_mul(&r->x, &p->x, &p->t);
	fe25519_mul(&r->y, &p->y, &p->z);
	fe25519_mul(&r->z, &p->z, &p->t);
}

static inline void p1p1_to_p3(ge25519_p3* r, const ge25519_p1p1* p)
{
	p1p1_to_p2_2(r, p);
	fe25519_mul(&r->t, &p->x, &p->y);
}

static inline void ge25519_mixadd2(ge25519_p3* r, const ge25519_aff* q)
{
	fe25519 a, b, t1, t2, c, d, e, f, g, h, qt;
	fe25519_mul(&qt, &q->x, &q->y);
	fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
	fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
	fe25519_sub(&t1, &q->y, &q->x);
	fe25519_add(&t2, &q->y, &q->x);
	fe25519_mul(&a, &a, &t1);
	fe25519_mul(&b, &b, &t2);
	fe25519_sub(&e, &b, &a);	 /* E = B-A */
	fe25519_add(&h, &b, &a);	 /* H = B+A */
	fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
	fe25519_sub(&f, &d, &c);	   /* F = D-C */
	fe25519_add(&g, &d, &c);	   /* G = D+C */
	fe25519_mul(&r->x, &e, &f);
	fe25519_mul(&r->y, &h, &g);
	fe25519_mul(&r->z, &g, &f);
	fe25519_mul(&r->t, &e, &h);
}

static inline void add_p1p1(ge25519_p1p1* r, const ge25519_p3* p, const ge25519_p3* q)
{
	fe25519 a, b, c, d, t;

	fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
	fe25519_sub(&t, &q->y, &q->x);
	fe25519_mul(&a, &a, &t);
	fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
	fe25519_add(&t, &q->x, &q->y);
	fe25519_mul(&b, &b, &t);
	fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
	fe25519_add(&d, &d, &d);
	fe25519_sub(&r->x, &b, &a); /* E = B-A */
	fe25519_sub(&r->t, &d, &c); /* F = D-C */
	fe25519_add(&r->z, &d, &c); /* G = D+C */
	fe25519_add(&r->y, &b, &a); /* H = B+A */
}

/* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
static inline void dbl_p1p1(ge25519_p1p1* r, const ge25519_p2* p)
{
	fe25519 a, b, c, d;
	fe25519_square(&a, &p->x);
	fe25519_square(&b, &p->y);
	fe25519_square(&c, &p->z);
	fe25519_add(&c, &c, &c);
	fe25519_neg(&d, &a);

	fe25519_add(&r->x, &p->x, &p->y);
	fe25519_square(&r->x, &r->x);
	fe25519_sub(&r->x, &r->x, &a);
	fe25519_sub(&r->x, &r->x, &b);
	fe25519_add(&r->z, &d, &b);
	fe25519_sub(&r->t, &r->z, &c);
	fe25519_sub(&r->y, &d, &b);
}

/* Constant-time version of: if(b) r = p */
static inline void cmov_aff(ge25519_aff* r, const ge25519_aff* p, unsigned char b)
{
	fe25519_cmov(&r->x, &p->x, b);
	fe25519_cmov(&r->y, &p->y, b);
}

static inline unsigned char equal(signed char b, signed char c)
{
	unsigned char ub = b;
	unsigned char uc = c;
	unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
	crypto_uint32 y = x;	   /* 0: yes; 1..255: no */
	y -= 1;					   /* 4294967295: yes; 0..254: no */
	y >>= 31;				   /* 1: yes; 0: no */
	return (unsigned char)y;
}

static inline unsigned char negative(signed char b)
{
	unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
	x >>= 63;				  /* 1: yes; 0: no */
	return (unsigned char)x;
}

static inline void choose_t(ge25519_aff* t, unsigned long long pos, signed char b)
{
	/* constant time */
	fe25519 v;
	*t = ge25519_base_multiples_affine[5 * pos + 0];
	cmov_aff(t, &ge25519_base_multiples_affine[5 * pos + 1], equal(b, 1) | equal(b, -1));
	cmov_aff(t, &ge25519_base_multiples_affine[5 * pos + 2], equal(b, 2) | equal(b, -2));
	cmov_aff(t, &ge25519_base_multiples_affine[5 * pos + 3], equal(b, 3) | equal(b, -3));
	cmov_aff(t, &ge25519_base_multiples_affine[5 * pos + 4], equal(b, -4));
	fe25519_neg(&v, &t->x);
	fe25519_cmov(&t->x, &v, negative(b));
}

static inline void setneutral(ge25519* r)
{
	fe25519_setzero(&r->x);
	fe25519_setone(&r->y);
	fe25519_setone(&r->z);
	fe25519_setzero(&r->t);
}

/* return 0 on success, -1 otherwise */
static inline int ge25519_unpackneg_vartime(ge25519_p3* r, const unsigned char p[32])
{
	unsigned char par;
	fe25519 t, chk, num, den, den2, den4, den6;
	fe25519_setone(&r->z);
	par = p[31] >> 7;
	fe25519_unpack(&r->y, p);
	fe25519_square(&num, &r->y);		   /* x = y^2 */
	fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
	fe25519_sub(&num, &num, &r->z);		   /* x = y^2-1 */
	fe25519_add(&den, &r->z, &den);		   /* den = dy^2+1 */

	/* Computation of sqrt(num/den) */
	/* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
	fe25519_square(&den2, &den);
	fe25519_square(&den4, &den2);
	fe25519_mul(&den6, &den4, &den2);
	fe25519_mul(&t, &den6, &num);
	fe25519_mul(&t, &t, &den);

	fe25519_pow2523(&t, &t);
	/* 2. computation of r->x = t * num * den^3 */
	fe25519_mul(&t, &t, &num);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&r->x, &t, &den);

	/* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (! fe25519_iseq_vartime(&chk, &num)) {
		fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);
	}

	/* 4. Now we have one of the two square roots, except if input was not a square */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (! fe25519_iseq_vartime(&chk, &num)) {
		return -1;
	}

	/* 5. Choose the desired square root according to parity: */
	if (fe25519_getparity(&r->x) != (1 - par)) {
		fe25519_neg(&r->x, &r->x);
	}

	fe25519_mul(&r->t, &r->x, &r->y);
	return 0;
}

static inline void ge25519_pack(unsigned char r[32], const ge25519_p3* p)
{
	fe25519 tx, ty, zi;
	fe25519_invert(&zi, &p->z);
	fe25519_mul(&tx, &p->x, &zi);
	fe25519_mul(&ty, &p->y, &zi);
	fe25519_pack(r, &ty);
	r[31] ^= fe25519_getparity(&tx) << 7;
}

/* computes [s1]p1 + [s2]p2 */
static inline void ge25519_double_scalarmult_vartime(ge25519_p3* r, const ge25519_p3* p1, const sc25519* s1, const ge25519_p3* p2, const sc25519* s2)
{
	ge25519_p1p1 tp1p1;
	ge25519_p3 pre[16];
	char* pre5 = (char*)(&(pre[5]));   // eliminate type punning warning
	unsigned char b[127];
	int i;

	/* precomputation                                                        s2 s1 */
	setneutral(pre); /* 00 00 */
	pre[1] = *p1;	 /* 00 01 */
	dbl_p1p1(&tp1p1, (ge25519_p2*)p1);
	p1p1_to_p3(&pre[2], &tp1p1); /* 00 10 */
	add_p1p1(&tp1p1, &pre[1], &pre[2]);
	p1p1_to_p3(&pre[3], &tp1p1); /* 00 11 */
	pre[4] = *p2;				 /* 01 00 */
	add_p1p1(&tp1p1, &pre[1], &pre[4]);
	p1p1_to_p3(&pre[5], &tp1p1); /* 01 01 */
	add_p1p1(&tp1p1, &pre[2], &pre[4]);
	p1p1_to_p3(&pre[6], &tp1p1); /* 01 10 */
	add_p1p1(&tp1p1, &pre[3], &pre[4]);
	p1p1_to_p3(&pre[7], &tp1p1); /* 01 11 */
	dbl_p1p1(&tp1p1, (ge25519_p2*)p2);
	p1p1_to_p3(&pre[8], &tp1p1); /* 10 00 */
	add_p1p1(&tp1p1, &pre[1], &pre[8]);
	p1p1_to_p3(&pre[9], &tp1p1); /* 10 01 */
	dbl_p1p1(&tp1p1, (ge25519_p2*)pre5);
	p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
	add_p1p1(&tp1p1, &pre[3], &pre[8]);
	p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
	add_p1p1(&tp1p1, &pre[4], &pre[8]);
	p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
	add_p1p1(&tp1p1, &pre[1], &pre[12]);
	p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
	add_p1p1(&tp1p1, &pre[2], &pre[12]);
	p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
	add_p1p1(&tp1p1, &pre[3], &pre[12]);
	p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */

	sc25519_2interleave2(b, s1, s2);

	/* scalar multiplication */
	*r = pre[b[126]];
	for (i = 125; i >= 0; i--) {
		dbl_p1p1(&tp1p1, (ge25519_p2*)r);
		p1p1_to_p2((ge25519_p2*)r, &tp1p1);
		dbl_p1p1(&tp1p1, (ge25519_p2*)r);
		if (b[i] != 0) {
			p1p1_to_p3(r, &tp1p1);
			add_p1p1(&tp1p1, r, &pre[b[i]]);
		}
		if (i != 0) {
			p1p1_to_p2((ge25519_p2*)r, &tp1p1);
		}
		else {
			p1p1_to_p3(r, &tp1p1);
		}
	}
}

static inline void ge25519_scalarmult_base(ge25519_p3* r, const sc25519* s)
{
	signed char b[85];
	int i;
	ge25519_aff t;
	sc25519_window3(b, s);

	choose_t((ge25519_aff*)r, 0, b[0]);
	fe25519_setone(&r->z);
	fe25519_mul(&r->t, &r->x, &r->y);
	for (i = 1; i < 85; i++) {
		choose_t(&t, (unsigned long long)i, b[i]);
		ge25519_mixadd2(r, &t);
	}
}

static inline void get_hram(unsigned char* hram, const unsigned char* sm, const unsigned char* pk, unsigned char* playground, unsigned long long smlen)
{
	unsigned long long i;

	for (i = 0; i < 32; ++i) {
		playground[i] = sm[i];
	}
	for (i = 32; i < 64; ++i) {
		playground[i] = pk[i - 32];
	}
	for (i = 64; i < smlen; ++i) {
		playground[i] = sm[i];
	}

	ZeroTier::SHA512(hram, playground, (unsigned int)smlen);
}

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////

}	// anonymous namespace

#ifdef ZT_USE_FAST_X64_ED25519
extern "C" void ed25519_amd64_asm_sign(const unsigned char* sk, const unsigned char* pk, const unsigned char* digest, unsigned char* sig);
#endif

namespace ZeroTier {

void ECC::agree(const ECC::Private& mine, const ECC::Public& their, void* keybuf, unsigned int keylen)
{
	unsigned char rawkey[32];
	unsigned char digest[64];

	crypto_scalarmult(rawkey, mine.data, their.data);
	SHA512(digest, rawkey, 32);
	for (unsigned int i = 0, k = 0; i < keylen;) {
		if (k == 64) {
			k = 0;
			SHA512(digest, digest, 64);
		}
		((unsigned char*)keybuf)[i++] = digest[k++];
	}
}

void ECC::sign(const ECC::Private& myPrivate, const ECC::Public& myPublic, const void* msg, unsigned int len, void* signature)
{
	unsigned char digest[64];	// we sign the first 32 bytes of SHA-512(msg)
	SHA512(digest, msg, len);

#ifdef ZT_USE_FAST_X64_ED25519
	ed25519_amd64_asm_sign(myPrivate.data + 32, myPublic.data + 32, digest, (unsigned char*)signature);
#else
	sc25519 sck, scs, scsk;
	ge25519 ger;
	unsigned char r[32];
	unsigned char s[32];
	unsigned char extsk[64];
	unsigned char hmg[crypto_hash_sha512_BYTES];
	unsigned char hram[crypto_hash_sha512_BYTES];
	unsigned char* sig = (unsigned char*)signature;

	SHA512(extsk, myPrivate.data + 32, 32);
	extsk[0] &= 248;
	extsk[31] &= 127;
	extsk[31] |= 64;

	for (unsigned int i = 0; i < 32; i++) {
		sig[32 + i] = extsk[32 + i];
	}
	for (unsigned int i = 0; i < 32; i++) {
		sig[64 + i] = digest[i];
	}

	SHA512(hmg, sig + 32, 64);

	/* Computation of R */
	sc25519_from64bytes(&sck, hmg);
	ge25519_scalarmult_base(&ger, &sck);
	ge25519_pack(r, &ger);

	/* Computation of s */
	for (unsigned int i = 0; i < 32; i++) {
		sig[i] = r[i];
	}

	get_hram(hram, sig, myPublic.data + 32, sig, 96);

	sc25519_from64bytes(&scs, hram);
	sc25519_from32bytes(&scsk, extsk);
	sc25519_mul(&scs, &scs, &scsk);

	sc25519_add(&scs, &scs, &sck);

	sc25519_to32bytes(s, &scs); /* cat s */
	for (unsigned int i = 0; i < 32; i++) {
		sig[32 + i] = s[i];
	}
#endif
}

bool ECC::verify(const ECC::Public& their, const void* msg, unsigned int len, const void* signature)
{
	const unsigned char* const sig = (const unsigned char*)signature;
	unsigned char digest[64];	// we sign the first 32 bytes of SHA-512(msg)
	SHA512(digest, msg, len);
	if (! Utils::secureEq(sig + 64, digest, 32)) {
		return false;
	}

	unsigned char t2[32];
	ge25519 get1, get2;
	sc25519 schram, scs;
	unsigned char hram[crypto_hash_sha512_BYTES];
	unsigned char m[96];

	if (ge25519_unpackneg_vartime(&get1, their.data + 32)) {
		return false;
	}

	get_hram(hram, sig, their.data + 32, m, 96);

	sc25519_from64bytes(&schram, hram);

	sc25519_from32bytes(&scs, sig + 32);

	ge25519_double_scalarmult_vartime(&get2, &get1, &schram, &ge25519_base, &scs);
	ge25519_pack(t2, &get2);

	return Utils::secureEq(sig, t2, 32);
}

void ECC::_calcPubDH(ECC::Pair& kp)
{
	// First 32 bytes of pub and priv are the keys for ECDH key
	// agreement. This generates the public portion from the private.
	crypto_scalarmult_base(kp.pub.data, kp.priv.data);
}

void ECC::_calcPubED(ECC::Pair& kp)
{
	unsigned char extsk[64];
	sc25519 scsk;
	ge25519 gepk;

	// Second 32 bytes of pub and priv are the keys for ed25519
	// signing and verification.
	SHA512(extsk, kp.priv.data + 32, 32);
	extsk[0] &= 248;
	extsk[31] &= 127;
	extsk[31] |= 64;
	sc25519_from32bytes(&scsk, extsk);
	ge25519_scalarmult_base(&gepk, &scsk);
	ge25519_pack(kp.pub.data + 32, &gepk);
	// In NaCl, the public key is crammed into the next 32 bytes
	// of the private key for signing since both keys are required
	// to sign. In this version we just get it from kp.pub, so we
	// leave that out of private.
}

}	// namespace ZeroTier