ZeroTierOne/core/ECC384.cpp

// This is EASY-ECC by Kenneth MacKay with some very minor modifications for ZeroTier
// https://github.com/esxgx/easy-ecc
// This code is under the BSD 2-clause license, not ZeroTier's license

#include "Constants.hpp"
#include "ECC384.hpp"
#include "Utils.hpp"

namespace ZeroTier {

namespace {

#define uint unsigned int
#define secp384r1 48
#define ECC_CURVE secp384r1
#define ECC_BYTES ECC_CURVE
#define NUM_ECC_DIGITS (ECC_BYTES/8)
#define MAX_TRIES 1024

#ifdef ZT_HAVE_UINT128
#define SUPPORTS_INT128 1
#else
#define SUPPORTS_INT128 0
typedef struct
{
	uint64_t m_low;
	uint64_t m_high;
} uint128_t;
#endif

typedef struct EccPoint
{
	uint64_t x[NUM_ECC_DIGITS];
	uint64_t y[NUM_ECC_DIGITS];
} EccPoint;

#define CONCAT1(a, b) a##b
#define CONCAT(a, b) CONCAT1(a, b)
#define Curve_P_48 {0x00000000FFFFFFFF, 0xFFFFFFFF00000000, 0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
#define Curve_B_48 {0x2A85C8EDD3EC2AEF, 0xC656398D8A2ED19D, 0x0314088F5013875A, 0x181D9C6EFE814112, 0x988E056BE3F82D19, 0xB3312FA7E23EE7E4}
#define Curve_G_48 {{0x3A545E3872760AB7, 0x5502F25DBF55296C, 0x59F741E082542A38, 0x6E1D3B628BA79B98, 0x8EB1C71EF320AD74, 0xAA87CA22BE8B0537}, {0x7A431D7C90EA0E5F, 0x0A60B1CE1D7E819D, 0xE9DA3113B5F0B8C0, 0xF8F41DBD289A147C, 0x5D9E98BF9292DC29, 0x3617DE4A96262C6F}}
#define Curve_N_48 {0xECEC196ACCC52973, 0x581A0DB248B0A77A, 0xC7634D81F4372DDF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
const uint64_t curve_p[NUM_ECC_DIGITS] = CONCAT(Curve_P_, ECC_CURVE);
const uint64_t curve_b[NUM_ECC_DIGITS] = CONCAT(Curve_B_, ECC_CURVE);
const EccPoint curve_G = CONCAT(Curve_G_, ECC_CURVE);
const uint64_t curve_n[NUM_ECC_DIGITS] = CONCAT(Curve_N_, ECC_CURVE);

// Use ZeroTier's secure PRNG
ZT_INLINE int getRandomNumber(uint64_t *p_vli)
{
	Utils::getSecureRandom(p_vli,ECC_BYTES);
	return 1;
}

ZT_INLINE void vli_clear(uint64_t *p_vli)
{
	uint i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		p_vli[i] = 0;
	}
}

/* Returns 1 if p_vli == 0, 0 otherwise. */
ZT_INLINE int vli_isZero(const uint64_t *p_vli)
{
	uint i;
	for(i = 0; i < NUM_ECC_DIGITS; ++i)
	{
		if(p_vli[i])
		{
			return 0;
		}
	}
	return 1;
}

/* Returns nonzero if bit p_bit of p_vli is set. */
ZT_INLINE uint64_t vli_testBit(const uint64_t *p_vli,uint p_bit)
{
	return (p_vli[p_bit/64] & ((uint64_t)1 << (p_bit % 64)));
}

/* Counts the number of 64-bit "digits" in p_vli. */
ZT_INLINE uint vli_numDigits(const uint64_t *p_vli)
{
	int i;
	/* Search from the end until we find a non-zero digit.
	   We do it in reverse because we expect that most digits will be nonzero. */
	for(i = NUM_ECC_DIGITS - 1; i >= 0 && p_vli[i] == 0; --i)
	{
	}

	return (i + 1);
}

/* Counts the number of bits required for p_vli. */
ZT_INLINE uint vli_numBits(const uint64_t *p_vli)
{
	uint i;
	uint64_t l_digit;

	uint l_numDigits = vli_numDigits(p_vli);
	if(l_numDigits == 0)
	{
		return 0;
	}

	l_digit = p_vli[l_numDigits - 1];
	for(i=0; l_digit; ++i)
	{
		l_digit >>= 1;
	}

	return ((l_numDigits - 1) * 64 + i);
}

/* Sets p_dest = p_src. */
ZT_INLINE void vli_set(uint64_t *p_dest,const uint64_t *p_src)
{
	uint i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		p_dest[i] = p_src[i];
	}
}

/* Returns sign of p_left - p_right. */
ZT_INLINE int vli_cmp(const uint64_t *p_left,const uint64_t *p_right)
{
	int i;
	for(i = NUM_ECC_DIGITS-1; i >= 0; --i)
	{
		if(p_left[i] > p_right[i])
		{
			return 1;
		}
		else if(p_left[i] < p_right[i])
		{
			return -1;
		}
	}
	return 0;
}

/* Computes p_result = p_in << c, returning carry. Can modify in place (if p_result == p_in). 0 < p_shift < 64. */
ZT_INLINE uint64_t vli_lshift(uint64_t *p_result,const uint64_t *p_in,uint p_shift)
{
	uint64_t l_carry = 0;
	uint i;
	for(i = 0; i < NUM_ECC_DIGITS; ++i)
	{
		uint64_t l_temp = p_in[i];
		p_result[i] = (l_temp << p_shift) | l_carry;
		l_carry = l_temp >> (64 - p_shift);
	}

	return l_carry;
}

/* Computes p_vli = p_vli >> 1. */
ZT_INLINE void vli_rshift1(uint64_t *p_vli)
{
	uint64_t *l_end = p_vli;
	uint64_t l_carry = 0;

	p_vli += NUM_ECC_DIGITS;
	while(p_vli-- > l_end)
	{
		uint64_t l_temp = *p_vli;
		*p_vli = (l_temp >> 1) | l_carry;
		l_carry = l_temp << 63;
	}
}

/* Computes p_result = p_left + p_right, returning carry. Can modify in place. */
ZT_INLINE uint64_t vli_add(uint64_t *p_result,const uint64_t *p_left,const uint64_t *p_right)
{
	uint64_t l_carry = 0;
	uint i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		uint64_t l_sum = p_left[i] + p_right[i] + l_carry;
		if(l_sum != p_left[i])
		{
			l_carry = (l_sum < p_left[i]);
		}
		p_result[i] = l_sum;
	}
	return l_carry;
}

/* Computes p_result = p_left - p_right, returning borrow. Can modify in place. */
ZT_INLINE uint64_t vli_sub(uint64_t *p_result,const uint64_t *p_left,const uint64_t *p_right)
{
	uint64_t l_borrow = 0;
	uint i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		uint64_t l_diff = p_left[i] - p_right[i] - l_borrow;
		if(l_diff != p_left[i])
		{
			l_borrow = (l_diff > p_left[i]);
		}
		p_result[i] = l_diff;
	}
	return l_borrow;
}

#if SUPPORTS_INT128 == 1

/* Computes p_result = p_left * p_right. */
void vli_mult(uint64_t *p_result, const uint64_t *p_left, const uint64_t *p_right)
{
	uint128_t r01 = 0;
	uint64_t r2 = 0;

	uint i, k;

	/* Compute each digit of p_result in sequence, maintaining the carries. */
	for(k=0; k < NUM_ECC_DIGITS*2 - 1; ++k)
	{
		uint l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS);
		for(i=l_min; i<=k && i<NUM_ECC_DIGITS; ++i)
		{
			uint128_t l_product = (uint128_t)p_left[i] * p_right[k-i];
			r01 += l_product;
			r2 += (r01 < l_product);
		}
		p_result[k] = (uint64_t)r01;
		r01 = (r01 >> 64U) | (((uint128_t)r2) << 64U);
		r2 = 0;
	}

	p_result[NUM_ECC_DIGITS*2 - 1] = (uint64_t)r01;
}

/* Computes p_result = p_left^2. */
void vli_square(uint64_t *p_result, const uint64_t *p_left)
{
	uint128_t r01 = 0;
	uint64_t r2 = 0;

	uint i, k;
	for(k=0; k < NUM_ECC_DIGITS*2 - 1; ++k)
	{
		uint l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS);
		for(i=l_min; i<=k && i<=k-i; ++i)
		{
			uint128_t l_product = (uint128_t)p_left[i] * p_left[k-i];
			if(i < k-i)
			{
				r2 += l_product >> 127;
				l_product *= 2;
			}
			r01 += l_product;
			r2 += (r01 < l_product);
		}
		p_result[k] = (uint64_t)r01;
		r01 = (r01 >> 64) | (((uint128_t)r2) << 64);
		r2 = 0;
	}

	p_result[NUM_ECC_DIGITS*2 - 1] = (uint64_t)r01;
}

#else /* #if SUPPORTS_INT128 */

uint128_t mul_64_64(uint64_t p_left, uint64_t p_right)
{
	uint128_t l_result;

	uint64_t a0 = p_left & 0xffffffffull;
	uint64_t a1 = p_left >> 32;
	uint64_t b0 = p_right & 0xffffffffull;
	uint64_t b1 = p_right >> 32;

	uint64_t m0 = a0 * b0;
	uint64_t m1 = a0 * b1;
	uint64_t m2 = a1 * b0;
	uint64_t m3 = a1 * b1;

	m2 += (m0 >> 32);
	m2 += m1;
	if(m2 < m1)
	{ // overflow
		m3 += 0x100000000ull;
	}

	l_result.m_low = (m0 & 0xffffffffull) | (m2 << 32);
	l_result.m_high = m3 + (m2 >> 32);

	return l_result;
}

ZT_INLINE uint128_t add_128_128(uint128_t a, uint128_t b)
{
	uint128_t l_result;
	l_result.m_low = a.m_low + b.m_low;
	l_result.m_high = a.m_high + b.m_high + (l_result.m_low < a.m_low);
	return l_result;
}

void vli_mult(uint64_t *p_result, uint64_t *p_left, const uint64_t *p_right)
{
	uint128_t r01 = {0, 0};
	uint64_t r2 = 0;

	uint i, k;

	/* Compute each digit of p_result in sequence, maintaining the carries. */
	for(k=0; k < NUM_ECC_DIGITS*2 - 1; ++k)
	{
		uint l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS);
		for(i=l_min; i<=k && i<NUM_ECC_DIGITS; ++i)
		{
			uint128_t l_product = mul_64_64(p_left[i], p_right[k-i]);
			r01 = add_128_128(r01, l_product);
			r2 += (r01.m_high < l_product.m_high);
		}
		p_result[k] = r01.m_low;
		r01.m_low = r01.m_high;
		r01.m_high = r2;
		r2 = 0;
	}

	p_result[NUM_ECC_DIGITS*2 - 1] = r01.m_low;
}

void vli_square(uint64_t *p_result, uint64_t *p_left)
{
	uint128_t r01 = {0, 0};
	uint64_t r2 = 0;

	uint i, k;
	for(k=0; k < NUM_ECC_DIGITS*2 - 1; ++k)
	{
		uint l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS);
		for(i=l_min; i<=k && i<=k-i; ++i)
		{
			uint128_t l_product = mul_64_64(p_left[i], p_left[k-i]);
			if(i < k-i)
			{
				r2 += l_product.m_high >> 63;
				l_product.m_high = (l_product.m_high << 1) | (l_product.m_low >> 63);
				l_product.m_low <<= 1;
			}
			r01 = add_128_128(r01, l_product);
			r2 += (r01.m_high < l_product.m_high);
		}
		p_result[k] = r01.m_low;
		r01.m_low = r01.m_high;
		r01.m_high = r2;
		r2 = 0;
	}

	p_result[NUM_ECC_DIGITS*2 - 1] = r01.m_low;
}

#endif /* SUPPORTS_INT128 */

/* Computes p_result = (p_left + p_right) % p_mod.
   Assumes that p_left < p_mod and p_right < p_mod, p_result != p_mod. */
void vli_modAdd(uint64_t *p_result, uint64_t *p_left, const uint64_t *p_right, const uint64_t *p_mod)
{
	uint64_t l_carry = vli_add(p_result, p_left, p_right);
	if(l_carry || vli_cmp(p_result, p_mod) >= 0)
	{ /* p_result > p_mod (p_result = p_mod + remainder), so subtract p_mod to get remainder. */
		vli_sub(p_result, p_result, p_mod);
	}
}

/* Computes p_result = (p_left - p_right) % p_mod.
   Assumes that p_left < p_mod and p_right < p_mod, p_result != p_mod. */
void vli_modSub(uint64_t *p_result, uint64_t *p_left, const uint64_t *p_right, const uint64_t *p_mod)
{
	uint64_t l_borrow = vli_sub(p_result, p_left, p_right);
	if(l_borrow)
	{ /* In this case, p_result == -diff == (max int) - diff.
		 Since -x % d == d - x, we can get the correct result from p_result + p_mod (with overflow). */
		vli_add(p_result, p_result, p_mod);
	}
}

void omega_mult(uint64_t *p_result, uint64_t *p_right)
{
	uint64_t l_tmp[NUM_ECC_DIGITS];
	uint64_t l_carry, l_diff;

	/* Multiply by (2^128 + 2^96 - 2^32 + 1). */
	vli_set(p_result, p_right); /* 1 */
	l_carry = vli_lshift(l_tmp, p_right, 32);
	p_result[1 + NUM_ECC_DIGITS] = l_carry + vli_add(p_result + 1, p_result + 1, l_tmp); /* 2^96 + 1 */
	p_result[2 + NUM_ECC_DIGITS] = vli_add(p_result + 2, p_result + 2, p_right); /* 2^128 + 2^96 + 1 */
	l_carry += vli_sub(p_result, p_result, l_tmp); /* 2^128 + 2^96 - 2^32 + 1 */
	l_diff = p_result[NUM_ECC_DIGITS] - l_carry;
	if(l_diff > p_result[NUM_ECC_DIGITS])
	{ /* Propagate borrow if necessary. */
		uint i;
		for(i = 1 + NUM_ECC_DIGITS; ; ++i)
		{
			--p_result[i];
			if(p_result[i] != (uint64_t)-1)
			{
				break;
			}
		}
	}
	p_result[NUM_ECC_DIGITS] = l_diff;
}

/* Computes p_result = p_product % curve_p
	see PDF "Comparing Elliptic Curve Cryptography and RSA on 8-bit CPUs"
	section "Curve-Specific Optimizations" */
void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product)
{
	uint64_t l_tmp[2*NUM_ECC_DIGITS];

	while(!vli_isZero(p_product + NUM_ECC_DIGITS)) /* While c1 != 0 */
	{
		uint64_t l_carry = 0;
		uint i;

		vli_clear(l_tmp);
		vli_clear(l_tmp + NUM_ECC_DIGITS);
		omega_mult(l_tmp, p_product + NUM_ECC_DIGITS); /* tmp = w * c1 */
		vli_clear(p_product + NUM_ECC_DIGITS); /* p = c0 */

		/* (c1, c0) = c0 + w * c1 */
		for(i=0; i<NUM_ECC_DIGITS+3; ++i)
		{
			uint64_t l_sum = p_product[i] + l_tmp[i] + l_carry;
			if(l_sum != p_product[i])
			{
				l_carry = (l_sum < p_product[i]);
			}
			p_product[i] = l_sum;
		}
	}

	while(vli_cmp(p_product, curve_p) > 0)
	{
		vli_sub(p_product, p_product, curve_p);
	}
	vli_set(p_result, p_product);
}

/* Computes p_result = (p_left * p_right) % curve_p. */
ZT_INLINE void vli_modMult_fast(uint64_t *p_result,uint64_t *p_left,const uint64_t *p_right)
{
	uint64_t l_product[2 * NUM_ECC_DIGITS];
	vli_mult(l_product, p_left, p_right);
	vli_mmod_fast(p_result, l_product);
}

/* Computes p_result = p_left^2 % curve_p. */
ZT_INLINE void vli_modSquare_fast(uint64_t *p_result,uint64_t *p_left)
{
	uint64_t l_product[2 * NUM_ECC_DIGITS];
	vli_square(l_product, p_left);
	vli_mmod_fast(p_result, l_product);
}

#define EVEN(vli) (!(vli[0] & 1))
/* Computes p_result = (1 / p_input) % p_mod. All VLIs are the same size.
   See "From Euclid's GCD to Montgomery Multiplication to the Great Divide"
   https://labs.oracle.com/techrep/2001/smli_tr-2001-95.pdf */
void vli_modInv(uint64_t *p_result, uint64_t *p_input, const uint64_t *p_mod)
{
	uint64_t a[NUM_ECC_DIGITS], b[NUM_ECC_DIGITS], u[NUM_ECC_DIGITS], v[NUM_ECC_DIGITS];
	uint64_t l_carry;
	int l_cmpResult;

	if(vli_isZero(p_input))
	{
		vli_clear(p_result);
		return;
	}

	vli_set(a, p_input);
	vli_set(b, p_mod);
	vli_clear(u);
	u[0] = 1;
	vli_clear(v);

	while((l_cmpResult = vli_cmp(a, b)) != 0)
	{
		l_carry = 0;
		if(EVEN(a))
		{
			vli_rshift1(a);
			if(!EVEN(u))
			{
				l_carry = vli_add(u, u, p_mod);
			}
			vli_rshift1(u);
			if(l_carry)
			{
				u[NUM_ECC_DIGITS-1] |= 0x8000000000000000ull;
			}
		}
		else if(EVEN(b))
		{
			vli_rshift1(b);
			if(!EVEN(v))
			{
				l_carry = vli_add(v, v, p_mod);
			}
			vli_rshift1(v);
			if(l_carry)
			{
				v[NUM_ECC_DIGITS-1] |= 0x8000000000000000ull;
			}
		}
		else if(l_cmpResult > 0)
		{
			vli_sub(a, a, b);
			vli_rshift1(a);
			if(vli_cmp(u, v) < 0)
			{
				vli_add(u, u, p_mod);
			}
			vli_sub(u, u, v);
			if(!EVEN(u))
			{
				l_carry = vli_add(u, u, p_mod);
			}
			vli_rshift1(u);
			if(l_carry)
			{
				u[NUM_ECC_DIGITS-1] |= 0x8000000000000000ull;
			}
		}
		else
		{
			vli_sub(b, b, a);
			vli_rshift1(b);
			if(vli_cmp(v, u) < 0)
			{
				vli_add(v, v, p_mod);
			}
			vli_sub(v, v, u);
			if(!EVEN(v))
			{
				l_carry = vli_add(v, v, p_mod);
			}
			vli_rshift1(v);
			if(l_carry)
			{
				v[NUM_ECC_DIGITS-1] |= 0x8000000000000000ull;
			}
		}
	}

	vli_set(p_result, u);
}

/* ------ Point operations ------ */

/* Returns 1 if p_point is the point at infinity, 0 otherwise. */
ZT_INLINE int EccPoint_isZero(EccPoint *p_point)
{
	return (vli_isZero(p_point->x) && vli_isZero(p_point->y));
}

/* Point multiplication algorithm using Montgomery's ladder with co-Z coordinates.
From http://eprint.iacr.org/2011/338.pdf
*/

/* Double in place */
void EccPoint_double_jacobian(uint64_t *X1, uint64_t *Y1, uint64_t *Z1)
{
	/* t1 = X, t2 = Y, t3 = Z */
	uint64_t t4[NUM_ECC_DIGITS];
	uint64_t t5[NUM_ECC_DIGITS];

	if(vli_isZero(Z1))
	{
		return;
	}

	vli_modSquare_fast(t4, Y1);   /* t4 = y1^2 */
	vli_modMult_fast(t5, X1, t4); /* t5 = x1*y1^2 = A */
	vli_modSquare_fast(t4, t4);   /* t4 = y1^4 */
	vli_modMult_fast(Y1, Y1, Z1); /* t2 = y1*z1 = z3 */
	vli_modSquare_fast(Z1, Z1);   /* t3 = z1^2 */

	vli_modAdd(X1, X1, Z1, curve_p); /* t1 = x1 + z1^2 */
	vli_modAdd(Z1, Z1, Z1, curve_p); /* t3 = 2*z1^2 */
	vli_modSub(Z1, X1, Z1, curve_p); /* t3 = x1 - z1^2 */
	vli_modMult_fast(X1, X1, Z1);	/* t1 = x1^2 - z1^4 */

	vli_modAdd(Z1, X1, X1, curve_p); /* t3 = 2*(x1^2 - z1^4) */
	vli_modAdd(X1, X1, Z1, curve_p); /* t1 = 3*(x1^2 - z1^4) */
	if(vli_testBit(X1, 0))
	{
		uint64_t l_carry = vli_add(X1, X1, curve_p);
		vli_rshift1(X1);
		X1[NUM_ECC_DIGITS-1] |= l_carry << 63U;
	}
	else
	{
		vli_rshift1(X1);
	}
	/* t1 = 3/2*(x1^2 - z1^4) = B */

	vli_modSquare_fast(Z1, X1);	  /* t3 = B^2 */
	vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - A */
	vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - 2A = x3 */
	vli_modSub(t5, t5, Z1, curve_p); /* t5 = A - x3 */
	vli_modMult_fast(X1, X1, t5);	/* t1 = B * (A - x3) */
	vli_modSub(t4, X1, t4, curve_p); /* t4 = B * (A - x3) - y1^4 = y3 */

	vli_set(X1, Z1);
	vli_set(Z1, Y1);
	vli_set(Y1, t4);
}

/* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */
void apply_z(uint64_t *X1, uint64_t *Y1, uint64_t *Z)
{
	uint64_t t1[NUM_ECC_DIGITS];

	vli_modSquare_fast(t1, Z);	/* z^2 */
	vli_modMult_fast(X1, X1, t1); /* x1 * z^2 */
	vli_modMult_fast(t1, t1, Z);  /* z^3 */
	vli_modMult_fast(Y1, Y1, t1); /* y1 * z^3 */
}

/* P = (x1, y1) => 2P, (x2, y2) => P' */
void XYcZ_initial_double(uint64_t *X1, uint64_t *Y1, uint64_t *X2, uint64_t *Y2, uint64_t *p_initialZ)
{
	uint64_t z[NUM_ECC_DIGITS];

	vli_set(X2, X1);
	vli_set(Y2, Y1);

	vli_clear(z);
	z[0] = 1;
	if(p_initialZ)
	{
		vli_set(z, p_initialZ);
	}

	apply_z(X1, Y1, z);

	EccPoint_double_jacobian(X1, Y1, z);

	apply_z(X2, Y2, z);
}

/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
   Output P' = (x1', y1', Z3), P + Q = (x3, y3, Z3)
   or P => P', Q => P + Q
*/
void XYcZ_add(uint64_t *X1, uint64_t *Y1, uint64_t *X2, uint64_t *Y2)
{
	/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
	uint64_t t5[NUM_ECC_DIGITS];

	vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */
	vli_modSquare_fast(t5, t5);	  /* t5 = (x2 - x1)^2 = A */
	vli_modMult_fast(X1, X1, t5);	/* t1 = x1*A = B */
	vli_modMult_fast(X2, X2, t5);	/* t3 = x2*A = C */
	vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */
	vli_modSquare_fast(t5, Y2);	  /* t5 = (y2 - y1)^2 = D */

	vli_modSub(t5, t5, X1, curve_p); /* t5 = D - B */
	vli_modSub(t5, t5, X2, curve_p); /* t5 = D - B - C = x3 */
	vli_modSub(X2, X2, X1, curve_p); /* t3 = C - B */
	vli_modMult_fast(Y1, Y1, X2);	/* t2 = y1*(C - B) */
	vli_modSub(X2, X1, t5, curve_p); /* t3 = B - x3 */
	vli_modMult_fast(Y2, Y2, X2);	/* t4 = (y2 - y1)*(B - x3) */
	vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y3 */

	vli_set(X2, t5);
}

/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
   Output P + Q = (x3, y3, Z3), P - Q = (x3', y3', Z3)
   or P => P - Q, Q => P + Q
*/
void XYcZ_addC(uint64_t *X1, uint64_t *Y1, uint64_t *X2, uint64_t *Y2)
{
	/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
	uint64_t t5[NUM_ECC_DIGITS];
	uint64_t t6[NUM_ECC_DIGITS];
	uint64_t t7[NUM_ECC_DIGITS];

	vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */
	vli_modSquare_fast(t5, t5);	  /* t5 = (x2 - x1)^2 = A */
	vli_modMult_fast(X1, X1, t5);	/* t1 = x1*A = B */
	vli_modMult_fast(X2, X2, t5);	/* t3 = x2*A = C */
	vli_modAdd(t5, Y2, Y1, curve_p); /* t4 = y2 + y1 */
	vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */

	vli_modSub(t6, X2, X1, curve_p); /* t6 = C - B */
	vli_modMult_fast(Y1, Y1, t6);	/* t2 = y1 * (C - B) */
	vli_modAdd(t6, X1, X2, curve_p); /* t6 = B + C */
	vli_modSquare_fast(X2, Y2);	  /* t3 = (y2 - y1)^2 */
	vli_modSub(X2, X2, t6, curve_p); /* t3 = x3 */

	vli_modSub(t7, X1, X2, curve_p); /* t7 = B - x3 */
	vli_modMult_fast(Y2, Y2, t7);	/* t4 = (y2 - y1)*(B - x3) */
	vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y3 */

	vli_modSquare_fast(t7, t5);	  /* t7 = (y2 + y1)^2 = F */
	vli_modSub(t7, t7, t6, curve_p); /* t7 = x3' */
	vli_modSub(t6, t7, X1, curve_p); /* t6 = x3' - B */
	vli_modMult_fast(t6, t6, t5);	/* t6 = (y2 + y1)*(x3' - B) */
	vli_modSub(Y1, t6, Y1, curve_p); /* t2 = y3' */

	vli_set(X1, t7);
}

void EccPoint_mult(EccPoint *p_result, const EccPoint *p_point, uint64_t *p_scalar, uint64_t *p_initialZ)
{
	/* R0 and R1 */
	uint64_t Rx[2][NUM_ECC_DIGITS];
	uint64_t Ry[2][NUM_ECC_DIGITS];
	uint64_t z[NUM_ECC_DIGITS];

	int i, nb;

	vli_set(Rx[1], p_point->x);
	vli_set(Ry[1], p_point->y);

	XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], p_initialZ);

	for(i = (int)vli_numBits(p_scalar) - 2; i > 0; --i)
	{
		nb = !vli_testBit(p_scalar, i);
		XYcZ_addC(Rx[1-nb], Ry[1-nb], Rx[nb], Ry[nb]);
		XYcZ_add(Rx[nb], Ry[nb], Rx[1-nb], Ry[1-nb]);
	}

	nb = !vli_testBit(p_scalar, 0);
	XYcZ_addC(Rx[1-nb], Ry[1-nb], Rx[nb], Ry[nb]);

	/* Find final 1/Z value. */
	vli_modSub(z, Rx[1], Rx[0], curve_p); /* X1 - X0 */
	vli_modMult_fast(z, z, Ry[1-nb]);	 /* Yb * (X1 - X0) */
	vli_modMult_fast(z, z, p_point->x);   /* xP * Yb * (X1 - X0) */
	vli_modInv(z, z, curve_p);			/* 1 / (xP * Yb * (X1 - X0)) */
	vli_modMult_fast(z, z, p_point->y);   /* yP / (xP * Yb * (X1 - X0)) */
	vli_modMult_fast(z, z, Rx[1-nb]);	 /* Xb * yP / (xP * Yb * (X1 - X0)) */
	/* End 1/Z calculation */

	XYcZ_add(Rx[nb], Ry[nb], Rx[1-nb], Ry[1-nb]);

	apply_z(Rx[0], Ry[0], z);

	vli_set(p_result->x, Rx[0]);
	vli_set(p_result->y, Ry[0]);
}

ZT_INLINE void ecc_bytes2native(uint64_t p_native[NUM_ECC_DIGITS],const uint8_t p_bytes[ECC_BYTES])
{
	unsigned i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		const uint8_t *p_digit = p_bytes + 8 * (NUM_ECC_DIGITS - 1 - i);
		p_native[i] = ((uint64_t)p_digit[0] << 56) | ((uint64_t)p_digit[1] << 48) | ((uint64_t)p_digit[2] << 40) | ((uint64_t)p_digit[3] << 32) |
			((uint64_t)p_digit[4] << 24) | ((uint64_t)p_digit[5] << 16) | ((uint64_t)p_digit[6] << 8) | (uint64_t)p_digit[7];
	}
}

ZT_INLINE void ecc_native2bytes(uint8_t p_bytes[ECC_BYTES],const uint64_t p_native[NUM_ECC_DIGITS])
{
	unsigned i;
	for(i=0; i<NUM_ECC_DIGITS; ++i)
	{
		uint8_t *p_digit = p_bytes + 8 * (NUM_ECC_DIGITS - 1 - i);
		p_digit[0] = p_native[i] >> 56;
		p_digit[1] = p_native[i] >> 48;
		p_digit[2] = p_native[i] >> 40;
		p_digit[3] = p_native[i] >> 32;
		p_digit[4] = p_native[i] >> 24;
		p_digit[5] = p_native[i] >> 16;
		p_digit[6] = p_native[i] >> 8;
		p_digit[7] = p_native[i];
	}
}

/* Compute a = sqrt(a) (mod curve_p). */
void mod_sqrt(uint64_t a[NUM_ECC_DIGITS])
{
	unsigned i;
	uint64_t p1[NUM_ECC_DIGITS] = {1};
	uint64_t l_result[NUM_ECC_DIGITS] = {1};

	/* Since curve_p == 3 (mod 4) for all supported curves, we can
	   compute sqrt(a) = a^((curve_p + 1) / 4) (mod curve_p). */
	vli_add(p1, curve_p, p1); /* p1 = curve_p + 1 */
	for(i = vli_numBits(p1) - 1; i > 1; --i)
	{
		vli_modSquare_fast(l_result, l_result);
		if(vli_testBit(p1, i))
		{
			vli_modMult_fast(l_result, l_result, a);
		}
	}
	vli_set(a, l_result);
}

void ecc_point_decompress(EccPoint *p_point, const uint8_t p_compressed[ECC_BYTES+1])
{
	uint64_t _3[NUM_ECC_DIGITS] = {3}; /* -a = 3 */
	ecc_bytes2native(p_point->x, p_compressed+1);

	vli_modSquare_fast(p_point->y, p_point->x); /* y = x^2 */
	vli_modSub(p_point->y, p_point->y, _3, curve_p); /* y = x^2 - 3 */
	vli_modMult_fast(p_point->y, p_point->y, p_point->x); /* y = x^3 - 3x */
	vli_modAdd(p_point->y, p_point->y, curve_b, curve_p); /* y = x^3 - 3x + b */

	mod_sqrt(p_point->y);

	if((p_point->y[0] & 0x01) != (p_compressed[0] & 0x01))
	{
		vli_sub(p_point->y, curve_p, p_point->y);
	}
}

ZT_INLINE int ecc_make_key(uint8_t p_publicKey[ECC_BYTES + 1],uint8_t p_privateKey[ECC_BYTES])
{
	uint64_t l_private[NUM_ECC_DIGITS];
	EccPoint l_public;
	unsigned l_tries = 0;

	do
	{
		if(!getRandomNumber(l_private) || (l_tries++ >= MAX_TRIES))
		{
			return 0;
		}
		if(vli_isZero(l_private))
		{
			continue;
		}

		/* Make sure the private key is in the range [1, n-1].
		   For the supported curves, n is always large enough that we only need to subtract once at most. */
		if(vli_cmp(curve_n, l_private) != 1)
		{
			vli_sub(l_private, l_private, curve_n);
		}

		EccPoint_mult(&l_public, &curve_G, l_private, NULL);
	} while(EccPoint_isZero(&l_public));

	ecc_native2bytes(p_privateKey, l_private);
	ecc_native2bytes(p_publicKey + 1, l_public.x);
	p_publicKey[0] = 2 + (l_public.y[0] & 0x01);
	return 1;
}

ZT_INLINE int ecdh_shared_secret(const uint8_t p_publicKey[ECC_BYTES + 1],const uint8_t p_privateKey[ECC_BYTES],uint8_t p_secret[ECC_BYTES])
{
	EccPoint l_public;
	uint64_t l_private[NUM_ECC_DIGITS];
	uint64_t l_random[NUM_ECC_DIGITS];

	if(!getRandomNumber(l_random))
	{
		return 0;
	}

	ecc_point_decompress(&l_public, p_publicKey);
	ecc_bytes2native(l_private, p_privateKey);

	EccPoint l_product;
	EccPoint_mult(&l_product, &l_public, l_private, l_random);

	ecc_native2bytes(p_secret, l_product.x);

	return !EccPoint_isZero(&l_product);
}

/* -------- ECDSA code -------- */

/* Computes p_result = (p_left * p_right) % p_mod. */
void vli_modMult(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, const uint64_t *p_mod)
{
	uint64_t l_product[2 * NUM_ECC_DIGITS];
	uint64_t l_modMultiple[2 * NUM_ECC_DIGITS];
	uint l_digitShift, l_bitShift;
	uint l_productBits;
	uint l_modBits = vli_numBits(p_mod);

	vli_mult(l_product, p_left, p_right);
	l_productBits = vli_numBits(l_product + NUM_ECC_DIGITS);
	if(l_productBits)
	{
		l_productBits += NUM_ECC_DIGITS * 64;
	}
	else
	{
		l_productBits = vli_numBits(l_product);
	}

	if(l_productBits < l_modBits)
	{ /* l_product < p_mod. */
		vli_set(p_result, l_product);
		return;
	}

	/* Shift p_mod by (l_leftBits - l_modBits). This multiplies p_mod by the largest
	   power of two possible while still resulting in a number less than p_left. */
	vli_clear(l_modMultiple);
	vli_clear(l_modMultiple + NUM_ECC_DIGITS);
	l_digitShift = (l_productBits - l_modBits) / 64;
	l_bitShift = (l_productBits - l_modBits) % 64;
	if(l_bitShift)
	{
		l_modMultiple[l_digitShift + NUM_ECC_DIGITS] = vli_lshift(l_modMultiple + l_digitShift, p_mod, l_bitShift);
	}
	else
	{
		vli_set(l_modMultiple + l_digitShift, p_mod);
	}

	/* Subtract all multiples of p_mod to get the remainder. */
	vli_clear(p_result);
	p_result[0] = 1; /* Use p_result as a temp var to store 1 (for subtraction) */
	while(l_productBits > NUM_ECC_DIGITS * 64 || vli_cmp(l_modMultiple, p_mod) >= 0)
	{
		int l_cmp = vli_cmp(l_modMultiple + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS);
		if(l_cmp < 0 || (l_cmp == 0 && vli_cmp(l_modMultiple, l_product) <= 0))
		{
			if(vli_sub(l_product, l_product, l_modMultiple))
			{ /* borrow */
				vli_sub(l_product + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS, p_result);
			}
			vli_sub(l_product + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS, l_modMultiple + NUM_ECC_DIGITS);
		}
		uint64_t l_carry = (l_modMultiple[NUM_ECC_DIGITS] & 0x01) << 63;
		vli_rshift1(l_modMultiple + NUM_ECC_DIGITS);
		vli_rshift1(l_modMultiple);
		l_modMultiple[NUM_ECC_DIGITS-1] |= l_carry;

		--l_productBits;
	}
	vli_set(p_result, l_product);
}

ZT_INLINE uint umax(uint a,uint b)
{
	return (a > b ? a : b);
}

ZT_INLINE int ecdsa_sign(const uint8_t p_privateKey[ECC_BYTES],const uint8_t p_hash[ECC_BYTES],uint8_t p_signature[ECC_BYTES * 2])
{
	uint64_t k[NUM_ECC_DIGITS];
	uint64_t l_tmp[NUM_ECC_DIGITS];
	uint64_t l_s[NUM_ECC_DIGITS];
	EccPoint p;
	unsigned l_tries = 0;

	do
	{
		if(!getRandomNumber(k) || (l_tries++ >= MAX_TRIES))
		{
			return 0;
		}
		if(vli_isZero(k))
		{
			continue;
		}

		if(vli_cmp(curve_n, k) != 1)
		{
			vli_sub(k, k, curve_n);
		}

		/* tmp = k * G */
		EccPoint_mult(&p, &curve_G, k, NULL);

		/* r = x1 (mod n) */
		if(vli_cmp(curve_n, p.x) != 1)
		{
			vli_sub(p.x, p.x, curve_n);
		}
	} while(vli_isZero(p.x));

	ecc_native2bytes(p_signature, p.x);

	ecc_bytes2native(l_tmp, p_privateKey);
	vli_modMult(l_s, p.x, l_tmp, curve_n); /* s = r*d */
	ecc_bytes2native(l_tmp, p_hash);
	vli_modAdd(l_s, l_tmp, l_s, curve_n); /* s = e + r*d */
	vli_modInv(k, k, curve_n); /* k = 1 / k */
	vli_modMult(l_s, l_s, k, curve_n); /* s = (e + r*d) / k */
	ecc_native2bytes(p_signature + ECC_BYTES, l_s);

	return 1;
}

ZT_INLINE int ecdsa_verify(const uint8_t p_publicKey[ECC_BYTES + 1],const uint8_t p_hash[ECC_BYTES],const uint8_t p_signature[ECC_BYTES * 2])
{
	uint64_t u1[NUM_ECC_DIGITS], u2[NUM_ECC_DIGITS];
	uint64_t z[NUM_ECC_DIGITS];
	EccPoint l_public, l_sum;
	uint64_t rx[NUM_ECC_DIGITS];
	uint64_t ry[NUM_ECC_DIGITS];
	uint64_t tx[NUM_ECC_DIGITS];
	uint64_t ty[NUM_ECC_DIGITS];
	uint64_t tz[NUM_ECC_DIGITS];

	uint64_t l_r[NUM_ECC_DIGITS], l_s[NUM_ECC_DIGITS];

	ecc_point_decompress(&l_public, p_publicKey);
	ecc_bytes2native(l_r, p_signature);
	ecc_bytes2native(l_s, p_signature + ECC_BYTES);

	if(vli_isZero(l_r) || vli_isZero(l_s))
	{ /* r, s must not be 0. */
		return 0;
	}

	if(vli_cmp(curve_n, l_r) != 1 || vli_cmp(curve_n, l_s) != 1)
	{ /* r, s must be < n. */
		return 0;
	}

	/* Calculate u1 and u2. */
	vli_modInv(z, l_s, curve_n); /* Z = s^-1 */
	ecc_bytes2native(u1, p_hash);
	vli_modMult(u1, u1, z, curve_n); /* u1 = e/s */
	vli_modMult(u2, l_r, z, curve_n); /* u2 = r/s */

	/* Calculate l_sum = G + Q. */
	vli_set(l_sum.x, l_public.x);
	vli_set(l_sum.y, l_public.y);
	vli_set(tx, curve_G.x);
	vli_set(ty, curve_G.y);
	vli_modSub(z, l_sum.x, tx, curve_p); /* Z = x2 - x1 */
	XYcZ_add(tx, ty, l_sum.x, l_sum.y);
	vli_modInv(z, z, curve_p); /* Z = 1/Z */
	apply_z(l_sum.x, l_sum.y, z);

	/* Use Shamir's trick to calculate u1*G + u2*Q */
	const EccPoint *l_points[4] = {NULL, &curve_G, &l_public, &l_sum};
	uint l_numBits = umax(vli_numBits(u1), vli_numBits(u2));

	const EccPoint *l_point = l_points[(!!vli_testBit(u1, l_numBits-1)) | ((!!vli_testBit(u2, l_numBits-1)) << 1)];
	vli_set(rx, l_point->x);
	vli_set(ry, l_point->y);
	vli_clear(z);
	z[0] = 1;

	int i;
	for(i = l_numBits - 2; i >= 0; --i)
	{
		EccPoint_double_jacobian(rx, ry, z);

		int l_index = (!!vli_testBit(u1, i)) | ((!!vli_testBit(u2, i)) << 1);
		const EccPoint *l_point = l_points[l_index];
		if(l_point)
		{
			vli_set(tx, l_point->x);
			vli_set(ty, l_point->y);
			apply_z(tx, ty, z);
			vli_modSub(tz, rx, tx, curve_p); /* Z = x2 - x1 */
			XYcZ_add(tx, ty, rx, ry);
			vli_modMult_fast(z, z, tz);
		}
	}

	vli_modInv(z, z, curve_p); /* Z = 1/Z */
	apply_z(rx, ry, z);

	/* v = x1 (mod n) */
	if(vli_cmp(curve_n, rx) != 1)
	{
		vli_sub(rx, rx, curve_n);
	}

	/* Accept only if v == r. */
	return (vli_cmp(rx, l_r) == 0);
}

} // anonymous namespace

void ECC384GenerateKey(uint8_t pub[ZT_ECC384_PUBLIC_KEY_SIZE],uint8_t priv[ZT_ECC384_PRIVATE_KEY_SIZE])
{
	if (!ecc_make_key(pub,priv)) {
		fprintf(stderr,"FATAL: ecdsa_make_key() failed!" ZT_EOL_S);
		abort();
	}
}

void ECC384ECDSASign(const uint8_t priv[ZT_ECC384_PRIVATE_KEY_SIZE],const uint8_t hash[ZT_ECC384_SIGNATURE_HASH_SIZE],uint8_t sig[ZT_ECC384_SIGNATURE_SIZE])
{
	if (!ecdsa_sign(priv,hash,sig)) {
		fprintf(stderr,"FATAL: ecdsa_sign() failed!" ZT_EOL_S);
		abort();
	}
}

bool ECC384ECDSAVerify(const uint8_t pub[ZT_ECC384_PUBLIC_KEY_SIZE],const uint8_t hash[ZT_ECC384_SIGNATURE_HASH_SIZE],const uint8_t sig[ZT_ECC384_SIGNATURE_SIZE])
{
	return (ecdsa_verify(pub,hash,sig) != 0);
}

bool ECC384ECDH(const uint8_t theirPub[ZT_ECC384_PUBLIC_KEY_SIZE],const uint8_t ourPriv[ZT_ECC384_PRIVATE_KEY_SIZE],uint8_t secret[ZT_ECC384_SHARED_SECRET_SIZE])
{
	return (ecdh_shared_secret(theirPub,ourPriv,secret) != 0);
}

} // namespace ZeroTier