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sshecc.c
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/*
* Elliptic-curve crypto module for PuTTY
* Implements the three required curves, no optional curves
*
* NOTE: Only curves on prime field are handled by the maths functions
* in Weierstrass form using Jacobian co-ordinates.
*
* Montgomery form curves are supported for DH. (Curve25519)
*
* Edwards form curves are supported for DSA. (Ed25519)
*/
/*
* References:
*
* Elliptic curves in SSH are specified in RFC 5656:
* http://tools.ietf.org/html/rfc5656
*
* That specification delegates details of public key formatting and a
* lot of underlying mechanism to SEC 1:
* http://www.secg.org/sec1-v2.pdf
*
* Montgomery maths from:
* Handbook of elliptic and hyperelliptic curve cryptography, Chapter 13
* http://cs.ucsb.edu/~koc/ccs130h/2013/EllipticHyperelliptic-CohenFrey.pdf
*
* Curve25519 spec from libssh (with reference to other things in the
* libssh code):
* https://git.libssh.org/users/aris/libssh.git/tree/doc/[email protected]
*
* Edwards DSA:
* http://ed25519.cr.yp.to/ed25519-20110926.pdf
*/
#include <stdlib.h>
#include <assert.h>
#include "ssh.h"
/* ----------------------------------------------------------------------
* Elliptic curve definitions
*/
static void initialise_wcurve(struct ec_curve *curve, int bits,
const unsigned char *p,
const unsigned char *a, const unsigned char *b,
const unsigned char *n, const unsigned char *Gx,
const unsigned char *Gy)
{
int length = bits / 8;
if (bits % 8) ++length;
curve->type = EC_WEIERSTRASS;
curve->fieldBits = bits;
curve->p = bignum_from_bytes(p, length);
/* Curve co-efficients */
curve->w.a = bignum_from_bytes(a, length);
curve->w.b = bignum_from_bytes(b, length);
/* Group order and generator */
curve->w.n = bignum_from_bytes(n, length);
curve->w.G.x = bignum_from_bytes(Gx, length);
curve->w.G.y = bignum_from_bytes(Gy, length);
curve->w.G.curve = curve;
curve->w.G.infinity = 0;
}
static void initialise_mcurve(struct ec_curve *curve, int bits,
const unsigned char *p,
const unsigned char *a, const unsigned char *b,
const unsigned char *Gx)
{
int length = bits / 8;
if (bits % 8) ++length;
curve->type = EC_MONTGOMERY;
curve->fieldBits = bits;
curve->p = bignum_from_bytes(p, length);
/* Curve co-efficients */
curve->m.a = bignum_from_bytes(a, length);
curve->m.b = bignum_from_bytes(b, length);
/* Generator */
curve->m.G.x = bignum_from_bytes(Gx, length);
curve->m.G.y = NULL;
curve->m.G.z = NULL;
curve->m.G.curve = curve;
curve->m.G.infinity = 0;
}
static void initialise_ecurve(struct ec_curve *curve, int bits,
const unsigned char *p,
const unsigned char *l, const unsigned char *d,
const unsigned char *Bx, const unsigned char *By)
{
int length = bits / 8;
if (bits % 8) ++length;
curve->type = EC_EDWARDS;
curve->fieldBits = bits;
curve->p = bignum_from_bytes(p, length);
/* Curve co-efficients */
curve->e.l = bignum_from_bytes(l, length);
curve->e.d = bignum_from_bytes(d, length);
/* Group order and generator */
curve->e.B.x = bignum_from_bytes(Bx, length);
curve->e.B.y = bignum_from_bytes(By, length);
curve->e.B.curve = curve;
curve->e.B.infinity = 0;
}
static struct ec_curve *ec_p256(void)
{
static struct ec_curve curve = { 0 };
static unsigned char initialised = 0;
if (!initialised)
{
static const unsigned char p[] = {
0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff
};
static const unsigned char a[] = {
0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc
};
static const unsigned char b[] = {
0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b
};
static const unsigned char n[] = {
0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xbc, 0xe6, 0xfa, 0xad, 0xa7, 0x17, 0x9e, 0x84,
0xf3, 0xb9, 0xca, 0xc2, 0xfc, 0x63, 0x25, 0x51
};
static const unsigned char Gx[] = {
0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47,
0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96
};
static const unsigned char Gy[] = {
0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b,
0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5
};
initialise_wcurve(&curve, 256, p, a, b, n, Gx, Gy);
curve.textname = curve.name = "nistp256";
/* Now initialised, no need to do it again */
initialised = 1;
}
return &curve;
}
static struct ec_curve *ec_p384(void)
{
static struct ec_curve curve = { 0 };
static unsigned char initialised = 0;
if (!initialised)
{
static const unsigned char p[] = {
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe,
0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff
};
static const unsigned char a[] = {
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe,
0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xfc
};
static const unsigned char b[] = {
0xb3, 0x31, 0x2f, 0xa7, 0xe2, 0x3e, 0xe7, 0xe4,
0x98, 0x8e, 0x05, 0x6b, 0xe3, 0xf8, 0x2d, 0x19,
0x18, 0x1d, 0x9c, 0x6e, 0xfe, 0x81, 0x41, 0x12,
0x03, 0x14, 0x08, 0x8f, 0x50, 0x13, 0x87, 0x5a,
0xc6, 0x56, 0x39, 0x8d, 0x8a, 0x2e, 0xd1, 0x9d,
0x2a, 0x85, 0xc8, 0xed, 0xd3, 0xec, 0x2a, 0xef
};
static const unsigned char n[] = {
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xc7, 0x63, 0x4d, 0x81, 0xf4, 0x37, 0x2d, 0xdf,
0x58, 0x1a, 0x0d, 0xb2, 0x48, 0xb0, 0xa7, 0x7a,
0xec, 0xec, 0x19, 0x6a, 0xcc, 0xc5, 0x29, 0x73
};
static const unsigned char Gx[] = {
0xaa, 0x87, 0xca, 0x22, 0xbe, 0x8b, 0x05, 0x37,
0x8e, 0xb1, 0xc7, 0x1e, 0xf3, 0x20, 0xad, 0x74,
0x6e, 0x1d, 0x3b, 0x62, 0x8b, 0xa7, 0x9b, 0x98,
0x59, 0xf7, 0x41, 0xe0, 0x82, 0x54, 0x2a, 0x38,
0x55, 0x02, 0xf2, 0x5d, 0xbf, 0x55, 0x29, 0x6c,
0x3a, 0x54, 0x5e, 0x38, 0x72, 0x76, 0x0a, 0xb7
};
static const unsigned char Gy[] = {
0x36, 0x17, 0xde, 0x4a, 0x96, 0x26, 0x2c, 0x6f,
0x5d, 0x9e, 0x98, 0xbf, 0x92, 0x92, 0xdc, 0x29,
0xf8, 0xf4, 0x1d, 0xbd, 0x28, 0x9a, 0x14, 0x7c,
0xe9, 0xda, 0x31, 0x13, 0xb5, 0xf0, 0xb8, 0xc0,
0x0a, 0x60, 0xb1, 0xce, 0x1d, 0x7e, 0x81, 0x9d,
0x7a, 0x43, 0x1d, 0x7c, 0x90, 0xea, 0x0e, 0x5f
};
initialise_wcurve(&curve, 384, p, a, b, n, Gx, Gy);
curve.textname = curve.name = "nistp384";
/* Now initialised, no need to do it again */
initialised = 1;
}
return &curve;
}
static struct ec_curve *ec_p521(void)
{
static struct ec_curve curve = { 0 };
static unsigned char initialised = 0;
if (!initialised)
{
static const unsigned char p[] = {
0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff
};
static const unsigned char a[] = {
0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xfc
};
static const unsigned char b[] = {
0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c,
0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
0x3f, 0x00
};
static const unsigned char n[] = {
0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xfa, 0x51, 0x86, 0x87, 0x83, 0xbf, 0x2f,
0x96, 0x6b, 0x7f, 0xcc, 0x01, 0x48, 0xf7, 0x09,
0xa5, 0xd0, 0x3b, 0xb5, 0xc9, 0xb8, 0x89, 0x9c,
0x47, 0xae, 0xbb, 0x6f, 0xb7, 0x1e, 0x91, 0x38,
0x64, 0x09
};
static const unsigned char Gx[] = {
0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04,
0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
0xbd, 0x66
};
static const unsigned char Gy[] = {
0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b,
0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
0x66, 0x50
};
initialise_wcurve(&curve, 521, p, a, b, n, Gx, Gy);
curve.textname = curve.name = "nistp521";
/* Now initialised, no need to do it again */
initialised = 1;
}
return &curve;
}
static struct ec_curve *ec_curve25519(void)
{
static struct ec_curve curve = { 0 };
static unsigned char initialised = 0;
if (!initialised)
{
static const unsigned char p[] = {
0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xed
};
static const unsigned char a[] = {
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, 0x07, 0x6d, 0x06
};
static const unsigned char b[] = {
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
};
static const unsigned char gx[32] = {
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, 0x09
};
initialise_mcurve(&curve, 256, p, a, b, gx);
/* This curve doesn't need a name, because it's never used in
* any format that embeds the curve name */
curve.name = NULL;
curve.textname = "Curve25519";
/* Now initialised, no need to do it again */
initialised = 1;
}
return &curve;
}
static struct ec_curve *ec_ed25519(void)
{
static struct ec_curve curve = { 0 };
static unsigned char initialised = 0;
if (!initialised)
{
static const unsigned char q[] = {
0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xed
};
static const unsigned char l[32] = {
0x10, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x14, 0xde, 0xf9, 0xde, 0xa2, 0xf7, 0x9c, 0xd6,
0x58, 0x12, 0x63, 0x1a, 0x5c, 0xf5, 0xd3, 0xed
};
static const unsigned char d[32] = {
0x52, 0x03, 0x6c, 0xee, 0x2b, 0x6f, 0xfe, 0x73,
0x8c, 0xc7, 0x40, 0x79, 0x77, 0x79, 0xe8, 0x98,
0x00, 0x70, 0x0a, 0x4d, 0x41, 0x41, 0xd8, 0xab,
0x75, 0xeb, 0x4d, 0xca, 0x13, 0x59, 0x78, 0xa3
};
static const unsigned char Bx[32] = {
0x21, 0x69, 0x36, 0xd3, 0xcd, 0x6e, 0x53, 0xfe,
0xc0, 0xa4, 0xe2, 0x31, 0xfd, 0xd6, 0xdc, 0x5c,
0x69, 0x2c, 0xc7, 0x60, 0x95, 0x25, 0xa7, 0xb2,
0xc9, 0x56, 0x2d, 0x60, 0x8f, 0x25, 0xd5, 0x1a
};
static const unsigned char By[32] = {
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, 0x58
};
/* This curve doesn't need a name, because it's never used in
* any format that embeds the curve name */
curve.name = NULL;
initialise_ecurve(&curve, 256, q, l, d, Bx, By);
curve.textname = "Ed25519";
/* Now initialised, no need to do it again */
initialised = 1;
}
return &curve;
}
/* Return 1 if a is -3 % p, otherwise return 0
* This is used because there are some maths optimisations */
static int ec_aminus3(const struct ec_curve *curve)
{
int ret;
Bignum _p;
if (curve->type != EC_WEIERSTRASS) {
return 0;
}
_p = bignum_add_long(curve->w.a, 3);
ret = !bignum_cmp(curve->p, _p);
freebn(_p);
return ret;
}
/* ----------------------------------------------------------------------
* Elliptic curve field maths
*/
static Bignum ecf_add(const Bignum a, const Bignum b,
const struct ec_curve *curve)
{
Bignum a1, b1, ab, ret;
a1 = bigmod(a, curve->p);
b1 = bigmod(b, curve->p);
ab = bigadd(a1, b1);
freebn(a1);
freebn(b1);
ret = bigmod(ab, curve->p);
freebn(ab);
return ret;
}
static Bignum ecf_square(const Bignum a, const struct ec_curve *curve)
{
return modmul(a, a, curve->p);
}
static Bignum ecf_treble(const Bignum a, const struct ec_curve *curve)
{
Bignum ret, tmp;
/* Double */
tmp = bignum_lshift(a, 1);
/* Add itself (i.e. treble) */
ret = bigadd(tmp, a);
freebn(tmp);
/* Normalise */
while (bignum_cmp(ret, curve->p) >= 0)
{
tmp = bigsub(ret, curve->p);
assert(tmp);
freebn(ret);
ret = tmp;
}
return ret;
}
static Bignum ecf_double(const Bignum a, const struct ec_curve *curve)
{
Bignum ret = bignum_lshift(a, 1);
if (bignum_cmp(ret, curve->p) >= 0)
{
Bignum tmp = bigsub(ret, curve->p);
assert(tmp);
freebn(ret);
return tmp;
}
else
{
return ret;
}
}
/* ----------------------------------------------------------------------
* Memory functions
*/
void ec_point_free(struct ec_point *point)
{
if (point == NULL) return;
point->curve = 0;
if (point->x) freebn(point->x);
if (point->y) freebn(point->y);
if (point->z) freebn(point->z);
point->infinity = 0;
sfree(point);
}
static struct ec_point *ec_point_new(const struct ec_curve *curve,
const Bignum x, const Bignum y, const Bignum z,
unsigned char infinity)
{
struct ec_point *point = snewn(1, struct ec_point);
point->curve = curve;
point->x = x;
point->y = y;
point->z = z;
point->infinity = infinity ? 1 : 0;
return point;
}
static struct ec_point *ec_point_copy(const struct ec_point *a)
{
if (a == NULL) return NULL;
return ec_point_new(a->curve,
a->x ? copybn(a->x) : NULL,
a->y ? copybn(a->y) : NULL,
a->z ? copybn(a->z) : NULL,
a->infinity);
}
static int ec_point_verify(const struct ec_point *a)
{
if (a->infinity) {
return 1;
} else if (a->curve->type == EC_EDWARDS) {
/* Check y^2 - x^2 - 1 - d * x^2 * y^2 == 0 */
Bignum y2, x2, tmp, tmp2, tmp3;
int ret;
y2 = ecf_square(a->y, a->curve);
x2 = ecf_square(a->x, a->curve);
tmp = modmul(a->curve->e.d, x2, a->curve->p);
tmp2 = modmul(tmp, y2, a->curve->p);
freebn(tmp);
tmp = modsub(y2, x2, a->curve->p);
freebn(y2);
freebn(x2);
tmp3 = modsub(tmp, tmp2, a->curve->p);
freebn(tmp);
freebn(tmp2);
ret = !bignum_cmp(tmp3, One);
freebn(tmp3);
return ret;
} else if (a->curve->type == EC_WEIERSTRASS) {
/* Verify y^2 = x^3 + ax + b */
int ret = 0;
Bignum lhs = NULL, x3 = NULL, ax = NULL, x3ax = NULL, x3axm = NULL, x3axb = NULL, rhs = NULL;
Bignum Three = bignum_from_long(3);
lhs = modmul(a->y, a->y, a->curve->p);
/* This uses montgomery multiplication to optimise */
x3 = modpow(a->x, Three, a->curve->p);
freebn(Three);
ax = modmul(a->curve->w.a, a->x, a->curve->p);
x3ax = bigadd(x3, ax);
freebn(x3); x3 = NULL;
freebn(ax); ax = NULL;
x3axm = bigmod(x3ax, a->curve->p);
freebn(x3ax); x3ax = NULL;
x3axb = bigadd(x3axm, a->curve->w.b);
freebn(x3axm); x3axm = NULL;
rhs = bigmod(x3axb, a->curve->p);
freebn(x3axb);
ret = bignum_cmp(lhs, rhs) ? 0 : 1;
freebn(lhs);
freebn(rhs);
return ret;
} else {
return 0;
}
}
/* ----------------------------------------------------------------------
* Elliptic curve point maths
*/
/* Returns 1 on success and 0 on memory error */
static int ecp_normalise(struct ec_point *a)
{
if (!a) {
/* No point */
return 0;
}
if (a->infinity) {
/* Point is at infinity - i.e. normalised */
return 1;
}
if (a->curve->type == EC_WEIERSTRASS) {
/* In Jacobian Coordinates the triple (X, Y, Z) represents
the affine point (X / Z^2, Y / Z^3) */
Bignum Z2, Z2inv, Z3, Z3inv, tx, ty;
if (!a->x || !a->y) {
/* No point defined */
return 0;
} else if (!a->z) {
/* Already normalised */
return 1;
}
Z2 = ecf_square(a->z, a->curve);
Z2inv = modinv(Z2, a->curve->p);
if (!Z2inv) {
freebn(Z2);
return 0;
}
tx = modmul(a->x, Z2inv, a->curve->p);
freebn(Z2inv);
Z3 = modmul(Z2, a->z, a->curve->p);
freebn(Z2);
Z3inv = modinv(Z3, a->curve->p);
freebn(Z3);
if (!Z3inv) {
freebn(tx);
return 0;
}
ty = modmul(a->y, Z3inv, a->curve->p);
freebn(Z3inv);
freebn(a->x);
a->x = tx;
freebn(a->y);
a->y = ty;
freebn(a->z);
a->z = NULL;
return 1;
} else if (a->curve->type == EC_MONTGOMERY) {
/* In Montgomery (X : Z) represents the x co-ord (X / Z, ?) */
Bignum tmp, tmp2;
if (!a->x) {
/* No point defined */
return 0;
} else if (!a->z) {
/* Already normalised */
return 1;
}
tmp = modinv(a->z, a->curve->p);
if (!tmp) {
return 0;
}
tmp2 = modmul(a->x, tmp, a->curve->p);
freebn(tmp);
freebn(a->z);
a->z = NULL;
freebn(a->x);
a->x = tmp2;
return 1;
} else if (a->curve->type == EC_EDWARDS) {
/* Always normalised */
return 1;
} else {
return 0;
}
}
static struct ec_point *ecp_doublew(const struct ec_point *a, const int aminus3)
{
Bignum S, M, outx, outy, outz;
if (bignum_cmp(a->y, Zero) == 0)
{
/* Identity */
return ec_point_new(a->curve, NULL, NULL, NULL, 1);
}
/* S = 4*X*Y^2 */
{
Bignum Y2, XY2, _2XY2;
Y2 = ecf_square(a->y, a->curve);
XY2 = modmul(a->x, Y2, a->curve->p);
freebn(Y2);
_2XY2 = ecf_double(XY2, a->curve);
freebn(XY2);
S = ecf_double(_2XY2, a->curve);
freebn(_2XY2);
}
/* Faster calculation if a = -3 */
if (aminus3) {
/* if a = -3, then M can also be calculated as M = 3*(X + Z^2)*(X - Z^2) */
Bignum Z2, XpZ2, XmZ2, second;
if (a->z == NULL) {
Z2 = copybn(One);
} else {
Z2 = ecf_square(a->z, a->curve);
}
XpZ2 = ecf_add(a->x, Z2, a->curve);
XmZ2 = modsub(a->x, Z2, a->curve->p);
freebn(Z2);
second = modmul(XpZ2, XmZ2, a->curve->p);
freebn(XpZ2);
freebn(XmZ2);
M = ecf_treble(second, a->curve);
freebn(second);
} else {
/* M = 3*X^2 + a*Z^4 */
Bignum _3X2, X2, aZ4;
if (a->z == NULL) {
aZ4 = copybn(a->curve->w.a);
} else {
Bignum Z2, Z4;
Z2 = ecf_square(a->z, a->curve);
Z4 = ecf_square(Z2, a->curve);
freebn(Z2);
aZ4 = modmul(a->curve->w.a, Z4, a->curve->p);
freebn(Z4);
}
X2 = modmul(a->x, a->x, a->curve->p);
_3X2 = ecf_treble(X2, a->curve);
freebn(X2);
M = ecf_add(_3X2, aZ4, a->curve);
freebn(_3X2);
freebn(aZ4);
}
/* X' = M^2 - 2*S */
{
Bignum M2, _2S;
M2 = ecf_square(M, a->curve);
_2S = ecf_double(S, a->curve);
outx = modsub(M2, _2S, a->curve->p);
freebn(M2);
freebn(_2S);
}
/* Y' = M*(S - X') - 8*Y^4 */
{
Bignum SX, MSX, Eight, Y2, Y4, _8Y4;
SX = modsub(S, outx, a->curve->p);
freebn(S);
MSX = modmul(M, SX, a->curve->p);
freebn(SX);
freebn(M);
Y2 = ecf_square(a->y, a->curve);
Y4 = ecf_square(Y2, a->curve);
freebn(Y2);
Eight = bignum_from_long(8);
_8Y4 = modmul(Eight, Y4, a->curve->p);
freebn(Eight);
freebn(Y4);
outy = modsub(MSX, _8Y4, a->curve->p);
freebn(MSX);
freebn(_8Y4);
}
/* Z' = 2*Y*Z */
{
Bignum YZ;
if (a->z == NULL) {
YZ = copybn(a->y);
} else {
YZ = modmul(a->y, a->z, a->curve->p);
}
outz = ecf_double(YZ, a->curve);
freebn(YZ);
}
return ec_point_new(a->curve, outx, outy, outz, 0);
}
static struct ec_point *ecp_doublem(const struct ec_point *a)
{
Bignum z, outx, outz, xpz, xmz;
z = a->z;
if (!z) {
z = One;
}
/* 4xz = (x + z)^2 - (x - z)^2 */
{
Bignum tmp;
tmp = ecf_add(a->x, z, a->curve);
xpz = ecf_square(tmp, a->curve);
freebn(tmp);
tmp = modsub(a->x, z, a->curve->p);
xmz = ecf_square(tmp, a->curve);
freebn(tmp);
}
/* outx = (x + z)^2 * (x - z)^2 */
outx = modmul(xpz, xmz, a->curve->p);
/* outz = 4xz * ((x - z)^2 + ((A + 2) / 4)*4xz) */
{
Bignum _4xz, tmp, tmp2, tmp3;
tmp = bignum_from_long(2);
tmp2 = ecf_add(a->curve->m.a, tmp, a->curve);
freebn(tmp);
_4xz = modsub(xpz, xmz, a->curve->p);
freebn(xpz);
tmp = modmul(tmp2, _4xz, a->curve->p);
freebn(tmp2);
tmp2 = bignum_from_long(4);
tmp3 = modinv(tmp2, a->curve->p);
freebn(tmp2);
if (!tmp3) {
freebn(tmp);
freebn(_4xz);
freebn(outx);
freebn(xmz);
return NULL;
}
tmp2 = modmul(tmp, tmp3, a->curve->p);
freebn(tmp);
freebn(tmp3);
tmp = ecf_add(xmz, tmp2, a->curve);
freebn(xmz);
freebn(tmp2);
outz = modmul(_4xz, tmp, a->curve->p);
freebn(_4xz);
freebn(tmp);
}
return ec_point_new(a->curve, outx, NULL, outz, 0);
}
/* Forward declaration for Edwards curve doubling */
static struct ec_point *ecp_add(const struct ec_point *a,
const struct ec_point *b,
const int aminus3);
static struct ec_point *ecp_double(const struct ec_point *a, const int aminus3)
{
if (a->infinity)
{
/* Identity */
return ec_point_new(a->curve, NULL, NULL, NULL, 1);
}
if (a->curve->type == EC_EDWARDS)
{
return ecp_add(a, a, aminus3);
}
else if (a->curve->type == EC_WEIERSTRASS)
{
return ecp_doublew(a, aminus3);
}
else
{
return ecp_doublem(a);
}
}
static struct ec_point *ecp_addw(const struct ec_point *a,
const struct ec_point *b,
const int aminus3)
{
Bignum U1, U2, S1, S2, outx, outy, outz;
/* U1 = X1*Z2^2 */
/* S1 = Y1*Z2^3 */
if (b->z) {
Bignum Z2, Z3;
Z2 = ecf_square(b->z, a->curve);
U1 = modmul(a->x, Z2, a->curve->p);
Z3 = modmul(Z2, b->z, a->curve->p);
freebn(Z2);
S1 = modmul(a->y, Z3, a->curve->p);
freebn(Z3);
} else {
U1 = copybn(a->x);
S1 = copybn(a->y);
}
/* U2 = X2*Z1^2 */
/* S2 = Y2*Z1^3 */
if (a->z) {
Bignum Z2, Z3;
Z2 = ecf_square(a->z, b->curve);
U2 = modmul(b->x, Z2, b->curve->p);
Z3 = modmul(Z2, a->z, b->curve->p);
freebn(Z2);
S2 = modmul(b->y, Z3, b->curve->p);
freebn(Z3);
} else {
U2 = copybn(b->x);
S2 = copybn(b->y);
}
/* Check if multiplying by self */
if (bignum_cmp(U1, U2) == 0)
{
freebn(U1);
freebn(U2);
if (bignum_cmp(S1, S2) == 0)
{
freebn(S1);
freebn(S2);
return ecp_double(a, aminus3);
}
else
{
freebn(S1);
freebn(S2);
/* Infinity */
return ec_point_new(a->curve, NULL, NULL, NULL, 1);
}
}
{
Bignum H, R, UH2, H3;
/* H = U2 - U1 */
H = modsub(U2, U1, a->curve->p);
freebn(U2);
/* R = S2 - S1 */
R = modsub(S2, S1, a->curve->p);
freebn(S2);
/* X3 = R^2 - H^3 - 2*U1*H^2 */
{
Bignum R2, H2, _2UH2, first;
H2 = ecf_square(H, a->curve);
UH2 = modmul(U1, H2, a->curve->p);
freebn(U1);
H3 = modmul(H2, H, a->curve->p);
freebn(H2);
R2 = ecf_square(R, a->curve);
_2UH2 = ecf_double(UH2, a->curve);
first = modsub(R2, H3, a->curve->p);
freebn(R2);
outx = modsub(first, _2UH2, a->curve->p);
freebn(first);
freebn(_2UH2);
}
/* Y3 = R*(U1*H^2 - X3) - S1*H^3 */
{
Bignum RUH2mX, UH2mX, SH3;
UH2mX = modsub(UH2, outx, a->curve->p);
freebn(UH2);
RUH2mX = modmul(R, UH2mX, a->curve->p);
freebn(UH2mX);
freebn(R);
SH3 = modmul(S1, H3, a->curve->p);
freebn(S1);
freebn(H3);
outy = modsub(RUH2mX, SH3, a->curve->p);
freebn(RUH2mX);
freebn(SH3);