Nist P-384 guts and glory

make-linux.mk

@@ -14,7 +14,6 @@ DEFS?=
 LDLIBS?=
 DESTDIR?=
 
-
 include objects.mk
 ONE_OBJS+=osdep/LinuxEthernetTap.o
 ONE_OBJS+=osdep/LinuxNetLink.o

node/C25519.cpp

@@ -2687,7 +2687,7 @@ void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
 	}
 }
 
-void get_hram(unsigned char *hram, const unsigned char *sm, const unsigned char *pk, unsigned char *playground, unsigned long long smlen)
+void get_hram(unsigned char *hram, const unsigned char *sm, const unsigned char *pk, unsigned char *playground, unsigned long smlen)
 {
 	unsigned long long i;
 
@@ -2778,13 +2778,22 @@ void C25519::sign(const C25519::Private &myPrivate,const C25519::Public &myPubli
 #endif
 }
 
-bool C25519::verify(const C25519::Public &their,const void *msg,unsigned int len,const void *signature)
+bool C25519::verify(const C25519::Public &their,const void *msg,unsigned int len,const void *signature,const unsigned int siglen)
 {
-	const unsigned char *const sig = (const unsigned char *)signature;
+	if (siglen < 64) return false;
+
+	const unsigned char *sig = (const unsigned char *)signature;
 	unsigned char digest[64]; // we sign the first 32 bytes of SHA-512(msg)
+	unsigned char sigtmp[96];
 	SHA512::hash(digest,msg,len);
-	if (!Utils::secureEq(sig + 64,digest,32))
+
+	if ((siglen == 96)&&(!Utils::secureEq(sig+64,digest,32))) {
 		return false;
+	} else if (siglen == 64) {
+		memcpy(sigtmp,sig,64);
+		memcpy(sigtmp+64,digest,32);
+		sig = sigtmp;
+	}
 
 	unsigned char t2[32];
 	ge25519 get1, get2;

node/C25519.hpp

@@ -125,6 +125,11 @@ public:
 	/**
 	 * Sign a message with a sender's key pair
 	 *
+	 * Note that this generates a 96-byte signature that contains an extra 32 bytes
+	 * of hash data. This data is included for historical reasons and is optional. The
+	 * verify function here will take the first 64 bytes only (normal ed25519 signature)
+	 * or a 96-byte length signature with the extra input hash data.
+	 * 
 	 * @param myPrivate My private key
 	 * @param myPublic My public key
 	 * @param msg Message to sign
@@ -150,10 +155,11 @@ public:
 	 * @param their Public key to verify against
 	 * @param msg Message to verify signature integrity against
 	 * @param len Length of message in bytes
-	 * @param signature 96-byte signature
+	 * @param signature Signature bytes
+	 * @param siglen Length of signature in bytes
 	 * @return True if signature is valid and the message is authentic and unmodified
 	 */
-	static bool verify(const Public &their,const void *msg,unsigned int len,const void *signature);
+	static bool verify(const Public &their,const void *msg,unsigned int len,const void *signature,const unsigned int siglen);
 
 	/**
 	 * Verify a message's signature
@@ -164,10 +170,7 @@ public:
 	 * @param signature 96-byte signature
 	 * @return True if signature is valid and the message is authentic and unmodified
 	 */
-	static inline bool verify(const Public &their,const void *msg,unsigned int len,const Signature &signature)
-	{
-		return verify(their,msg,len,signature.data);
-	}
+	static inline bool verify(const Public &their,const void *msg,unsigned int len,const Signature &signature) { return verify(their,msg,len,signature.data,96); }
 
 private:
 	// derive first 32 bytes of kp.pub from first 32 bytes of kp.priv

node/ECC384.cpp

@@ -0,0 +1,1430 @@
+/*
+ * ZeroTier One - Network Virtualization Everywhere
+ * Copyright (C) 2011-2019  ZeroTier, Inc.  https://www.zerotier.com/
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
+ *
+ * --
+ *
+ * You can be released from the requirements of the license by purchasing
+ * a commercial license. Buying such a license is mandatory as soon as you
+ * develop commercial closed-source software that incorporates or links
+ * directly against ZeroTier software without disclosing the source code
+ * of your own application.
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdint.h>
+#include <string.h>
+
+#include "Constants.hpp"
+#include "ECC384.hpp"
+#include "Utils.hpp"
+
+namespace ZeroTier {
+
+namespace {
+//////////////////////////////////////////////////////////////////////////////
+// This is EASY-ECC by Kenneth MacKay
+// https://github.com/esxgx/easy-ecc
+// This code is under the BSD 2-clause license, not ZeroTier's license
+//////////////////////////////////////////////////////////////////////////////
+
+//////////////////////////////////////////////////////////////////////////////
+// ecc.h from easy-ecc
+//////////////////////////////////////////////////////////////////////////////
+
+#define secp128r1 16
+#define secp192r1 24
+#define secp256r1 32
+#define secp384r1 48
+
+//#ifndef ECC_CURVE
+//	#define ECC_CURVE secp256r1
+//#endif
+#define ECC_CURVE secp384r1
+
+//#if (ECC_CURVE != secp128r1 && ECC_CURVE != secp192r1 && ECC_CURVE != secp256r1 && ECC_CURVE != secp384r1)
+//	#error "Must define ECC_CURVE to one of the available curves"
+//#endif
+
+#define ECC_BYTES ECC_CURVE
+
+//////////////////////////////////////////////////////////////////////////////
+// ecc.c from easy-ecc
+//////////////////////////////////////////////////////////////////////////////
+
+//#include "ecc.h"
+//#include <string.h>
+
+#define NUM_ECC_DIGITS (ECC_BYTES/8)
+#define MAX_TRIES 16
+
+typedef unsigned int uint;
+
+#if defined(__SIZEOF_INT128__) || ((__clang_major__ * 100 + __clang_minor__) >= 302)
+	#define SUPPORTS_INT128 1
+#else
+	#define SUPPORTS_INT128 0
+#endif
+
+#if SUPPORTS_INT128
+typedef unsigned __int128 uint128_t;
+#else
+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_16 {0xFFFFFFFFFFFFFFFF, 0xFFFFFFFDFFFFFFFF}
+#define Curve_P_24 {0xFFFFFFFFFFFFFFFFull, 0xFFFFFFFFFFFFFFFEull, 0xFFFFFFFFFFFFFFFFull}
+#define Curve_P_32 {0xFFFFFFFFFFFFFFFFull, 0x00000000FFFFFFFFull, 0x0000000000000000ull, 0xFFFFFFFF00000001ull}
+#define Curve_P_48 {0x00000000FFFFFFFF, 0xFFFFFFFF00000000, 0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
+
+#define Curve_B_16 {0xD824993C2CEE5ED3, 0xE87579C11079F43D}
+#define Curve_B_24 {0xFEB8DEECC146B9B1ull, 0x0FA7E9AB72243049ull, 0x64210519E59C80E7ull}
+#define Curve_B_32 {0x3BCE3C3E27D2604Bull, 0x651D06B0CC53B0F6ull, 0xB3EBBD55769886BCull, 0x5AC635D8AA3A93E7ull}
+#define Curve_B_48 {0x2A85C8EDD3EC2AEF, 0xC656398D8A2ED19D, 0x0314088F5013875A, 0x181D9C6EFE814112, 0x988E056BE3F82D19, 0xB3312FA7E23EE7E4}
+
+#define Curve_G_16 { \
+	{0x0C28607CA52C5B86, 0x161FF7528B899B2D}, \
+	{0xC02DA292DDED7A83, 0xCF5AC8395BAFEB13}}
+
+#define Curve_G_24 { \
+	{0xF4FF0AFD82FF1012ull, 0x7CBF20EB43A18800ull, 0x188DA80EB03090F6ull}, \
+	{0x73F977A11E794811ull, 0x631011ED6B24CDD5ull, 0x07192B95FFC8DA78ull}}
+	
+#define Curve_G_32 { \
+	{0xF4A13945D898C296ull, 0x77037D812DEB33A0ull, 0xF8BCE6E563A440F2ull, 0x6B17D1F2E12C4247ull}, \
+	{0xCBB6406837BF51F5ull, 0x2BCE33576B315ECEull, 0x8EE7EB4A7C0F9E16ull, 0x4FE342E2FE1A7F9Bull}}
+
+#define Curve_G_48 { \
+	{0x3A545E3872760AB7, 0x5502F25DBF55296C, 0x59F741E082542A38, 0x6E1D3B628BA79B98, 0x8EB1C71EF320AD74, 0xAA87CA22BE8B0537}, \
+	{0x7A431D7C90EA0E5F, 0x0A60B1CE1D7E819D, 0xE9DA3113B5F0B8C0, 0xF8F41DBD289A147C, 0x5D9E98BF9292DC29, 0x3617DE4A96262C6F}}
+
+#define Curve_N_16 {0x75A30D1B9038A115, 0xFFFFFFFE00000000}
+#define Curve_N_24 {0x146BC9B1B4D22831ull, 0xFFFFFFFF99DEF836ull, 0xFFFFFFFFFFFFFFFFull}
+#define Curve_N_32 {0xF3B9CAC2FC632551ull, 0xBCE6FAADA7179E84ull, 0xFFFFFFFFFFFFFFFFull, 0xFFFFFFFF00000000ull}
+#define Curve_N_48 {0xECEC196ACCC52973, 0x581A0DB248B0A77A, 0xC7634D81F4372DDF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
+
+static uint64_t curve_p[NUM_ECC_DIGITS] = CONCAT(Curve_P_, ECC_CURVE);
+static uint64_t curve_b[NUM_ECC_DIGITS] = CONCAT(Curve_B_, ECC_CURVE);
+static EccPoint curve_G = CONCAT(Curve_G_, ECC_CURVE);
+static uint64_t curve_n[NUM_ECC_DIGITS] = CONCAT(Curve_N_, ECC_CURVE);
+
+#if 0
+#if (defined(_WIN32) || defined(_WIN64))
+/* Windows */
+
+#define WIN32_LEAN_AND_MEAN
+#include <windows.h>
+#include <wincrypt.h>
+
+static int getRandomNumber(uint64_t *p_vli)
+{
+	HCRYPTPROV l_prov;
+	if(!CryptAcquireContext(&l_prov, NULL, NULL, PROV_RSA_FULL, CRYPT_VERIFYCONTEXT))
+	{
+		return 0;
+	}
+
+	CryptGenRandom(l_prov, ECC_BYTES, (BYTE *)p_vli);
+	CryptReleaseContext(l_prov, 0);
+	
+	return 1;
+}
+
+#else /* _WIN32 */
+
+/* Assume that we are using a POSIX-like system with /dev/urandom or /dev/random. */
+#include <sys/types.h>
+#include <fcntl.h>
+#include <unistd.h>
+
+#ifndef O_CLOEXEC
+	#define O_CLOEXEC 0
+#endif
+
+static int getRandomNumber(uint64_t *p_vli)
+{
+	int l_fd = open("/dev/urandom", O_RDONLY | O_CLOEXEC);
+	if(l_fd == -1)
+	{
+		l_fd = open("/dev/random", O_RDONLY | O_CLOEXEC);
+		if(l_fd == -1)
+		{
+			return 0;
+		}
+	}
+	
+	char *l_ptr = (char *)p_vli;
+	size_t l_left = ECC_BYTES;
+	while(l_left > 0)
+	{
+		int l_read = read(l_fd, l_ptr, l_left);
+		if(l_read <= 0)
+		{ // read failed
+			close(l_fd);
+			return 0;
+		}
+		l_left -= l_read;
+		l_ptr += l_read;
+	}
+	
+	close(l_fd);
+	return 1;
+}
+
+#endif /* _WIN32 */
+#endif
+
+// Use ZeroTier's secure PRNG
+static inline int getRandomNumber(uint64_t *p_vli)
+{
+	Utils::getSecureRandom(p_vli,ECC_BYTES);
+	return 1;
+}
+
+static 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. */
+static inline int vli_isZero(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. */
+static inline uint64_t vli_testBit(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. */
+static inline uint vli_numDigits(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. */
+static inline uint vli_numBits(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. */
+static inline void vli_set(uint64_t *p_dest, 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. */
+static inline int vli_cmp(uint64_t *p_left, 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. */
+static inline uint64_t vli_lshift(uint64_t *p_result, 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. */
+static 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. */
+static inline uint64_t vli_add(uint64_t *p_result, uint64_t *p_left, 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. */
+static inline uint64_t vli_sub(uint64_t *p_result, uint64_t *p_left, 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
+
+/* Computes p_result = p_left * p_right. */
+static inline void vli_mult(uint64_t *p_result, uint64_t *p_left, 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 >> 64) | (((uint128_t)r2) << 64);
+		r2 = 0;
+	}
+	
+	p_result[NUM_ECC_DIGITS*2 - 1] = (uint64_t)r01;
+}
+
+/* Computes p_result = p_left^2. */
+static inline void vli_square(uint64_t *p_result, 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 */
+
+static inline 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;
+}
+
+static 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;
+}
+
+static inline void vli_mult(uint64_t *p_result, uint64_t *p_left, 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;
+}
+
+static inline 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. */
+static inline void vli_modAdd(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, 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. */
+static inline void vli_modSub(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, 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);
+	}
+}
+
+#if ECC_CURVE == secp128r1
+
+/* Computes p_result = p_product % curve_p.
+   See algorithm 5 and 6 from http://www.isys.uni-klu.ac.at/PDF/2001-0126-MT.pdf */
+static void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product)
+{
+	uint64_t l_tmp[NUM_ECC_DIGITS];
+	int l_carry;
+	
+	vli_set(p_result, p_product);
+	
+	l_tmp[0] = p_product[2];
+	l_tmp[1] = (p_product[3] & 0x1FFFFFFFFull) | (p_product[2] << 33);
+	l_carry = vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = (p_product[2] >> 31) | (p_product[3] << 33);
+	l_tmp[1] = (p_product[3] >> 31) | ((p_product[2] & 0xFFFFFFFF80000000ull) << 2);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = (p_product[2] >> 62) | (p_product[3] << 2);
+	l_tmp[1] = (p_product[3] >> 62) | ((p_product[2] & 0xC000000000000000ull) >> 29) | (p_product[3] << 35);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = (p_product[3] >> 29);
+	l_tmp[1] = ((p_product[3] & 0xFFFFFFFFE0000000ull) << 4);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = (p_product[3] >> 60);
+	l_tmp[1] = (p_product[3] & 0xFFFFFFFE00000000ull);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = 0;
+	l_tmp[1] = ((p_product[3] & 0xF000000000000000ull) >> 27);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	while(l_carry || vli_cmp(curve_p, p_result) != 1)
+	{
+		l_carry -= vli_sub(p_result, p_result, curve_p);
+	}
+}
+
+#elif ECC_CURVE == secp192r1
+
+/* Computes p_result = p_product % curve_p.
+   See algorithm 5 and 6 from http://www.isys.uni-klu.ac.at/PDF/2001-0126-MT.pdf */
+static void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product)
+{
+	uint64_t l_tmp[NUM_ECC_DIGITS];
+	int l_carry;
+	
+	vli_set(p_result, p_product);
+	
+	vli_set(l_tmp, &p_product[3]);
+	l_carry = vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = 0;
+	l_tmp[1] = p_product[3];
+	l_tmp[2] = p_product[4];
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	l_tmp[0] = l_tmp[1] = p_product[5];
+	l_tmp[2] = 0;
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	while(l_carry || vli_cmp(curve_p, p_result) != 1)
+	{
+		l_carry -= vli_sub(p_result, p_result, curve_p);
+	}
+}
+
+#elif ECC_CURVE == secp256r1
+
+/* Computes p_result = p_product % curve_p
+   from http://www.nsa.gov/ia/_files/nist-routines.pdf */
+static void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product)
+{
+	uint64_t l_tmp[NUM_ECC_DIGITS];
+	int l_carry;
+	
+	/* t */
+	vli_set(p_result, p_product);
+	
+	/* s1 */
+	l_tmp[0] = 0;
+	l_tmp[1] = p_product[5] & 0xffffffff00000000ull;
+	l_tmp[2] = p_product[6];
+	l_tmp[3] = p_product[7];
+	l_carry = vli_lshift(l_tmp, l_tmp, 1);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	/* s2 */
+	l_tmp[1] = p_product[6] << 32;
+	l_tmp[2] = (p_product[6] >> 32) | (p_product[7] << 32);
+	l_tmp[3] = p_product[7] >> 32;
+	l_carry += vli_lshift(l_tmp, l_tmp, 1);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	/* s3 */
+	l_tmp[0] = p_product[4];
+	l_tmp[1] = p_product[5] & 0xffffffff;
+	l_tmp[2] = 0;
+	l_tmp[3] = p_product[7];
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	/* s4 */
+	l_tmp[0] = (p_product[4] >> 32) | (p_product[5] << 32);
+	l_tmp[1] = (p_product[5] >> 32) | (p_product[6] & 0xffffffff00000000ull);
+	l_tmp[2] = p_product[7];
+	l_tmp[3] = (p_product[6] >> 32) | (p_product[4] << 32);
+	l_carry += vli_add(p_result, p_result, l_tmp);
+	
+	/* d1 */
+	l_tmp[0] = (p_product[5] >> 32) | (p_product[6] << 32);
+	l_tmp[1] = (p_product[6] >> 32);
+	l_tmp[2] = 0;
+	l_tmp[3] = (p_product[4] & 0xffffffff) | (p_product[5] << 32);
+	l_carry -= vli_sub(p_result, p_result, l_tmp);
+	
+	/* d2 */
+	l_tmp[0] = p_product[6];
+	l_tmp[1] = p_product[7];
+	l_tmp[2] = 0;
+	l_tmp[3] = (p_product[4] >> 32) | (p_product[5] & 0xffffffff00000000ull);
+	l_carry -= vli_sub(p_result, p_result, l_tmp);
+	
+	/* d3 */
+	l_tmp[0] = (p_product[6] >> 32) | (p_product[7] << 32);
+	l_tmp[1] = (p_product[7] >> 32) | (p_product[4] << 32);
+	l_tmp[2] = (p_product[4] >> 32) | (p_product[5] << 32);
+	l_tmp[3] = (p_product[6] << 32);
+	l_carry -= vli_sub(p_result, p_result, l_tmp);
+	
+	/* d4 */
+	l_tmp[0] = p_product[7];
+	l_tmp[1] = p_product[4] & 0xffffffff00000000ull;
+	l_tmp[2] = p_product[5];
+	l_tmp[3] = p_product[6] & 0xffffffff00000000ull;
+	l_carry -= vli_sub(p_result, p_result, l_tmp);
+	
+	if(l_carry < 0)
+	{
+		do
+		{
+			l_carry += vli_add(p_result, p_result, curve_p);
+		} while(l_carry < 0);
+	}
+	else
+	{
+		while(l_carry || vli_cmp(curve_p, p_result) != 1)
+		{
+			l_carry -= vli_sub(p_result, p_result, curve_p);
+		}
+	}
+}
+
+#elif ECC_CURVE == secp384r1
+
+static inline 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" */
+static inline 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);
+}
+
+#endif
+
+/* Computes p_result = (p_left * p_right) % curve_p. */
+static inline void vli_modMult_fast(uint64_t *p_result, uint64_t *p_left, 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. */
+static 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 */
+static inline void vli_modInv(uint64_t *p_result, uint64_t *p_input, 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. */
+static 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 */
+static inline 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 << 63;
+	}
+	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) */
+static inline 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' */
+static inline 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
+*/
+static inline 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
+*/
+static inline 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);
+}
+
+static inline void EccPoint_mult(EccPoint *p_result, 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 = 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]);
+}
+
+static 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];
+	}
+}
+
+static 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). */
+static inline 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);
+}
+
+static inline 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);
+	}
+}
+
+static 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;
+}
+
+static 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. */
+static inline void vli_modMult(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, 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);
+}
+
+static inline uint umax(uint a, uint b)
+{
+	return (a > b ? a : b);
+}
+
+static 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;
+}
+
+static 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 */
+	EccPoint *l_points[4] = {NULL, &curve_G, &l_public, &l_sum};
+	uint l_numBits = umax(vli_numBits(u1), vli_numBits(u2));
+	
+	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);
+		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

node/ECC384.hpp

@@ -0,0 +1,74 @@
+/*
+ * ZeroTier One - Network Virtualization Everywhere
+ * Copyright (C) 2011-2019  ZeroTier, Inc.  https://www.zerotier.com/
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program. If not, see <http://www.gnu.org/licenses/>.
+ *
+ * --
+ *
+ * You can be released from the requirements of the license by purchasing
+ * a commercial license. Buying such a license is mandatory as soon as you
+ * develop commercial closed-source software that incorporates or links
+ * directly against ZeroTier software without disclosing the source code
+ * of your own application.
+ */
+
+// This is glue code to ease the use of the NIST P-384 elliptic curve.
+
+// Note that some of the code inside ECC384.cpp is third party code and
+// is under the BSD 2-clause license rather than ZeroTier's license.
+
+#ifndef ZT_ECC384_HPP
+#define ZT_ECC384_HPP
+
+#include "Constants.hpp"
+
+/**
+ * Size of a (point compressed) P-384 public key
+ */
+#define ZT_ECC384_PUBLIC_KEY_SIZE 49
+
+/**
+ * Size of a P-384 private key
+ */
+#define ZT_ECC384_PRIVATE_KEY_SIZE 48
+
+/**
+ * Size of the hash that should be signed using P-384
+ */
+#define ZT_ECC384_SIGNATURE_HASH_SIZE 48
+
+/**
+ * Size of a P-384 signature
+ */
+#define ZT_ECC384_SIGNATURE_SIZE 96
+
+/**
+ * Size of shared secret generated by ECDH key agreement
+ */
+#define ZT_ECC384_SHARED_SECRET_SIZE 48
+
+namespace ZeroTier {
+
+void ECC384GenerateKey(uint8_t pub[ZT_ECC384_PUBLIC_KEY_SIZE],uint8_t priv[ZT_ECC384_PRIVATE_KEY_SIZE]);
+
+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]);
+
+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]);
+
+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]);
+
+} // namespace ZeroTier
+
+#endif

node/Identity.hpp

@@ -159,12 +159,7 @@ public:
 	 * @param siglen Length of signature in bytes
 	 * @return True if signature validates and data integrity checks
 	 */
-	inline bool verify(const void *data,unsigned int len,const void *signature,unsigned int siglen) const
-	{
-		if (siglen != ZT_C25519_SIGNATURE_LEN)
-			return false;
-		return C25519::verify(_publicKey,data,len,signature);
-	}
+	inline bool verify(const void *data,unsigned int len,const void *signature,unsigned int siglen) const { return C25519::verify(_publicKey,data,len,signature,siglen); }
 
 	/**
 	 * Verify a message signature against this identity
@@ -174,10 +169,7 @@ public:
 	 * @param signature Signature
 	 * @return True if signature validates and data integrity checks
 	 */
-	inline bool verify(const void *data,unsigned int len,const C25519::Signature &signature) const
-	{
-		return C25519::verify(_publicKey,data,len,signature);
-	}
+	inline bool verify(const void *data,unsigned int len,const C25519::Signature &signature) const { return C25519::verify(_publicKey,data,len,signature); }
 
 	/**
 	 * Shortcut method to perform key agreement with another identity

objects.mk

@@ -3,6 +3,7 @@ CORE_OBJS=\
 	node/Capability.o \
 	node/CertificateOfMembership.o \
 	node/CertificateOfOwnership.o \
+	node/ECC384.o \
 	node/Identity.o \
 	node/IncomingPacket.o \
 	node/InetAddress.o \

selftest.cpp

@@ -50,6 +50,7 @@
 #include "node/Dictionary.hpp"
 #include "node/SHA512.hpp"
 #include "node/C25519.hpp"
+#include "node/ECC384.hpp"
 #include "node/Poly1305.hpp"
 #include "node/CertificateOfMembership.hpp"
 #include "node/Node.hpp"
@@ -305,18 +306,35 @@ static int testCrypto()
 		::free((void *)bb);
 	}
 
-	/*
-	for(unsigned int d=8;d<=10;++d) {
-		for(int k=0;k<8;++k) {
-			std::cout << "[crypto] computeSalsa2012Sha512ProofOfWork(" << d << ",\"foobarbaz\",9) == "; std::cout.flush();
-			unsigned char result[16];
-			uint64_t start = OSUtils::now();
-			IncomingPacket::computeSalsa2012Sha512ProofOfWork(d,"foobarbaz",9,result);
-			uint64_t end = OSUtils::now();
-			std::cout << Utils::hex(result,16) << " -- valid: " << IncomingPacket::testSalsa2012Sha512ProofOfWorkResult(d,"foobarbaz",9,result) << ", " << (end - start) << "ms" << std::endl;
+	std::cout << "[crypto] Testing ECC384 (NIST P-384)..." << std::endl;
+	{
+		uint8_t p384pub[ZT_ECC384_PUBLIC_KEY_SIZE],p384priv[ZT_ECC384_PRIVATE_KEY_SIZE],p384sig[ZT_ECC384_SIGNATURE_SIZE],p384hash[ZT_ECC384_SIGNATURE_HASH_SIZE];
+		char p384hex[256];
+		ECC384GenerateKey(p384pub,p384priv);
+		std::cout << "[crypto]   Public Key: " << Utils::hex(p384pub,sizeof(p384pub),p384hex) << std::endl;
+		Utils::getSecureRandom(p384hash,sizeof(p384hash));
+		ECC384ECDSASign(p384priv,p384hash,p384sig);
+		if (!ECC384ECDSAVerify(p384pub,p384hash,p384sig)) {
+			std::cout << "[crypto]   Signature: FAILED (verify good signature)" << std::endl;
+			return -1;
 		}
+		++p384sig[0];
+		if (ECC384ECDSAVerify(p384pub,p384hash,p384sig)) {
+			std::cout << "[crypto]   Signature: FAILED (verify bad signature)" << std::endl;
+			return -1;
+		}
+		--p384sig[0];
+		std::cout << "[crypto]   Signature: " << Utils::hex(p384sig,sizeof(p384sig),p384hex) << std::endl;
+		uint8_t p384pub2[ZT_ECC384_PUBLIC_KEY_SIZE],p384priv2[ZT_ECC384_PRIVATE_KEY_SIZE],p384sec[ZT_ECC384_SHARED_SECRET_SIZE],p384sec2[ZT_ECC384_SHARED_SECRET_SIZE];
+		ECC384GenerateKey(p384pub2,p384priv2);
+		ECC384ECDH(p384pub,p384priv2,p384sec);
+		ECC384ECDH(p384pub2,p384priv,p384sec2);
+		if (memcmp(p384sec,p384sec2,ZT_ECC384_SHARED_SECRET_SIZE)) {
+			std::cout << "[crypto]   ECDH Agree: FAILED (secrets do not match)" << std::endl;
+			return -1;
+		}
+		std::cout << "[crypto]   ECDH Agree: " << Utils::hex(p384sec,sizeof(p384sec),p384hex) << std::endl;
 	}
-	*/
 
 	std::cout << "[crypto] Testing C25519 and Ed25519 against test vectors... "; std::cout.flush();
 	for(int k=0;k<ZT_NUM_C25519_TEST_VECTORS;++k) {