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ecdsa.cpp
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/** \file ecdsa.c
*
* \brief Contains functions relevant to ECDSA signing.
*
* Functions relevant to ECDSA signing include those which perform group
* operations on points of an elliptic curve (eg. pointAdd() and
* pointDouble()) and the actual signing function, ecdsaSign().
*
* The elliptic curve used is secp256k1, from the document
* "SEC 2: Recommended Elliptic Curve Domain Parameters" by Certicom
* research, obtained 11-August-2011 from:
* http://www.secg.org/collateral/sec2_final.pdf
* References to RFC 6979 refer to the version dated August 2013, obtained
* http://tools.ietf.org/html/rfc6979 on 4 April 2015.
*
* The operations here are written in a way as to encourage them to run in
* (mostly) constant time. This provides some resistance against timing
* attacks. However, the compiler may use optimisations which destroy this
* property; inspection of the generated assembly code is the only way to
* check. A disadvantage of this code is that point multiplication is slower
* than it could be.
* There are some data-dependent branches in here, but they're expected to
* only make a difference (in timing) in exceptional cases.
*
* This file is licensed as described by the file LICENCE.
*/
//#ifdef TEST_ECDSA
//#include <stdlib.h>
//#include <stdio.h>
//#include <math.h>
//#include "test_helpers.h"
//#endif // #ifdef TEST_ECDSA
#include "includeall.h"
/** A point on the elliptic curve, in Jacobian coordinates. The
* Jacobian coordinates (x, y, z) are related to affine coordinates
* (x_affine, y_affine) by:
* (x_affine, y_affine) = (x / (z ^ 2), y / (z ^ 3)).
*
* Why use Jacobian coordinates? Because then point addition and
* point doubling don't have to use inversion (division), which is very slow.
*/
typedef struct PointJacobianStruct
{
/** x component of a point in Jacobian coordinates. */
uint8_t x[32];
/** y component of a point in Jacobian coordinates. */
uint8_t y[32];
/** z component of a point in Jacobian coordinates. */
uint8_t z[32];
/** If is_point_at_infinity is non-zero, then this point represents the
* point at infinity and all other structure members are considered
* invalid. */
uint8_t is_point_at_infinity;
} PointJacobian;
/** The prime number used to define the prime finite field for secp256k1. */
static const uint8_t secp256k1_p[32] = {
0x2f, 0xfc, 0xff, 0xff, 0xfe, 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};
/** 2s complement of #secp256k1_p. */
static const uint8_t secp256k1_complement_p[5] = {
0xd1, 0x03, 0x00, 0x00, 0x01};
/** The order of the base point used in secp256k1. */
const uint8_t secp256k1_n[32] = {
0x41, 0x41, 0x36, 0xd0, 0x8c, 0x5e, 0xd2, 0xbf,
0x3b, 0xa0, 0x48, 0xaf, 0xe6, 0xdc, 0xae, 0xba,
0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff};
/** 2s complement of #secp256k1_n. */
static const uint8_t secp256k1_complement_n[17] = {
0xbf, 0xbe, 0xc9, 0x2f, 0x73, 0xa1, 0x2d, 0x40,
0xc4, 0x5f, 0xb7, 0x50, 0x19, 0x23, 0x51, 0x45,
0x01};
/** The x component of the base point G used in secp256k1. */
static const uint8_t secp256k1_Gx[32] = {
0x98, 0x17, 0xf8, 0x16, 0x5b, 0x81, 0xf2, 0x59,
0xd9, 0x28, 0xce, 0x2d, 0xdb, 0xfc, 0x9b, 0x02,
0x07, 0x0b, 0x87, 0xce, 0x95, 0x62, 0xa0, 0x55,
0xac, 0xbb, 0xdc, 0xf9, 0x7e, 0x66, 0xbe, 0x79};
/** The y component of the base point G used in secp256k1. */
static const uint8_t secp256k1_Gy[32] = {
0xb8, 0xd4, 0x10, 0xfb, 0x8f, 0xd0, 0x47, 0x9c,
0x19, 0x54, 0x85, 0xa6, 0x48, 0xb4, 0x17, 0xfd,
0xa8, 0x08, 0x11, 0x0e, 0xfc, 0xfb, 0xa4, 0x5d,
0x65, 0xc4, 0xa3, 0x26, 0x77, 0xda, 0x3a, 0x48};
/** Convert a point from affine coordinates to Jacobian coordinates. This
* is very fast.
* \param out The destination point (in Jacobian coordinates).
* \param in The source point (in affine coordinates).
*/
static void affineToJacobian(PointJacobian *out, PointAffine *in)
{
out->is_point_at_infinity = in->is_point_at_infinity;
// If out->is_point_at_infinity != 0, the rest of this function consists
// of dummy operations.
bigAssign(out->x, in->x);
bigAssign(out->y, in->y);
bigSetZero(out->z);
out->z[0] = 1;
}
/** Convert a point from Jacobian coordinates to affine coordinates. This
* is very slow because it involves inversion (division).
* \param out The destination point (in affine coordinates).
* \param in The source point (in Jacobian coordinates).
*/
static NOINLINE void jacobianToAffine(PointAffine *out, PointJacobian *in)
{
uint8_t s[32];
uint8_t t[32];
out->is_point_at_infinity = in->is_point_at_infinity;
// If out->is_point_at_infinity != 0, the rest of this function consists
// of dummy operations.
bigMultiply(s, in->z, in->z);
bigMultiply(t, s, in->z);
// Now s = z ^ 2 and t = z ^ 3.
bigInvert(s, s);
bigInvert(t, t);
bigMultiply(out->x, in->x, s);
bigMultiply(out->y, in->y, t);
}
/** Double (p = 2 x p) the point p (which is in Jacobian coordinates), placing
* the result back into p.
* The formulae for this function were obtained from the article:
* "Software Implementation of the NIST Elliptic Curves Over Prime Fields",
* obtained from:
* http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.25.8619&rep=rep1&type=pdf
* on 16-August-2011. See equations (2) ("doubling in Jacobian coordinates")
* from section 4 of that article.
* \param p The point (in Jacobian coordinates) to double.
*/
static NOINLINE void pointDouble(PointJacobian *p)
{
uint8_t t[32];
uint8_t u[32];
// If p->is_point_at_infinity != 0, then the rest of this function will
// consist of dummy operations. Nothing else needs to be done since
// 2O = O.
// If y is zero then the tangent line is vertical and never hits the
// curve, therefore the result should be O. If y is zero, the rest of this
// function will consist of dummy operations.
p->is_point_at_infinity |= bigIsZero(p->y);
bigMultiply(p->z, p->z, p->y);
bigAdd(p->z, p->z, p->z);
bigMultiply(p->y, p->y, p->y);
bigMultiply(t, p->y, p->x);
bigAdd(t, t, t);
bigAdd(t, t, t);
// t is now 4.0 * p->x * p->y ^ 2.
bigMultiply(p->x, p->x, p->x);
bigAssign(u, p->x);
bigAdd(u, u, u);
bigAdd(u, u, p->x);
// u is now 3.0 * p->x ^ 2.
// For curves with a != 0, a * p->z ^ 4 needs to be added to u.
// But since a == 0 in secp256k1, we save 2 squarings and 1
// multiplication.
bigMultiply(p->x, u, u);
bigSubtract(p->x, p->x, t);
bigSubtract(p->x, p->x, t);
bigSubtract(t, t, p->x);
bigMultiply(t, t, u);
bigMultiply(p->y, p->y, p->y);
bigAdd(p->y, p->y, p->y);
bigAdd(p->y, p->y, p->y);
bigAdd(p->y, p->y, p->y);
bigSubtract(p->y, t, p->y);
}
/** Add (p1 = p1 + p2) the point p2 to the point p1, storing the result back
* into p1.
* Mixed coordinates are used because it reduces the number of squarings and
* multiplications from 16 to 11.
* See equations (3) ("addition in mixed Jacobian-affine coordinates") from
* section 4 of that article described in the comments to pointDouble().
* junk must point at some memory area to redirect dummy writes to. The dummy
* writes are used to encourage this function's completion time to be
* independent of its parameters.
* \param p1 The point (in Jacobian coordinates) to add p2 to.
* \param junk Pointer to a dummy variable which may receive dummy writes.
* \param p2 The point (in affine coordinates) to add to p1.
*/
static NOINLINE void pointAdd(PointJacobian *p1, PointJacobian *junk, PointAffine *p2)
{
uint8_t s[32];
uint8_t t[32];
uint8_t u[32];
uint8_t v[32];
uint8_t is_O;
uint8_t is_O2;
uint8_t cmp_xs;
uint8_t cmp_yt;
PointJacobian *lookup[2];
lookup[0] = p1;
lookup[1] = junk;
// O + p2 == p2.
// If p1 is O, then copy p2 into p1 and redirect all writes to the dummy
// write area.
// The following line does: "is_O = p1->is_point_at_infinity ? 1 : 0;".
is_O = (uint8_t)((((uint16_t)(-(int)p1->is_point_at_infinity)) >> 8) & 1);
affineToJacobian(lookup[1 - is_O], p2);
p1 = lookup[is_O];
lookup[0] = p1; // p1 might have changed
// p1 + O == p1.
// If p2 is O, then redirect all writes to the dummy write area. This
// preserves the value of p1.
// The following line does: "is_O2 = p2->is_point_at_infinity ? 1 : 0;".
is_O2 = (uint8_t)((((uint16_t)(-(int)p2->is_point_at_infinity)) >> 8) & 1);
p1 = lookup[is_O2];
lookup[0] = p1; // p1 might have changed
bigMultiply(s, p1->z, p1->z);
bigMultiply(t, s, p1->z);
bigMultiply(t, t, p2->y);
bigMultiply(s, s, p2->x);
// The following two lines do: "cmp_xs = bigCompare(p1->x, s) == BIGCMP_EQUAL ? 0 : 0xff;".
cmp_xs = (uint8_t)(bigCompare(p1->x, s) ^ BIGCMP_EQUAL);
cmp_xs = (uint8_t)(((uint16_t)(-(int)cmp_xs)) >> 8);
// The following two lines do: "cmp_yt = bigCompare(p1->y, t) == BIGCMP_EQUAL ? 0 : 0xff;".
cmp_yt = (uint8_t)(bigCompare(p1->y, t) ^ BIGCMP_EQUAL);
cmp_yt = (uint8_t)(((uint16_t)(-(int)cmp_yt)) >> 8);
// The following branch can never be taken when calling pointMultiply(),
// so its existence doesn't compromise timing regularity.
if ((cmp_xs | cmp_yt | is_O | is_O2) == 0)
{
// Points are actually the same; use point doubling.
pointDouble(p1);
return;
}
// p2 == -p1 when p1->x == s and p1->y != t.
// If p1->is_point_at_infinity is set, then all subsequent operations in
// this function become dummy operations.
p1->is_point_at_infinity = (uint8_t)(p1->is_point_at_infinity | (~cmp_xs & cmp_yt & 1));
bigSubtract(s, s, p1->x);
// s now contains p2->x * p1->z ^ 2 - p1->x.
bigSubtract(t, t, p1->y);
// t now contains p2->y * p1->z ^ 3 - p1->y.
bigMultiply(p1->z, p1->z, s);
bigMultiply(v, s, s);
bigMultiply(u, v, p1->x);
bigMultiply(p1->x, t, t);
bigMultiply(s, s, v);
bigSubtract(p1->x, p1->x, s);
bigSubtract(p1->x, p1->x, u);
bigSubtract(p1->x, p1->x, u);
bigSubtract(u, u, p1->x);
bigMultiply(u, u, t);
bigMultiply(s, s, p1->y);
bigSubtract(p1->y, u, s);
}
/** Set field parameters to be those defined by the prime number p which
* is used in secp256k1. */
static void setFieldToP(void)
{
bigSetField(secp256k1_p, secp256k1_complement_p, sizeof(secp256k1_complement_p));
}
/** Set field parameters to be those defined by the prime number n which
* is used in secp256k1. */
void setFieldToN(void)
{
bigSetField(secp256k1_n, secp256k1_complement_n, sizeof(secp256k1_complement_n));
}
/** Perform scalar multiplication (p = k x p) of the point p by the scalar k.
* The result will be stored back into p. The multiplication is
* accomplished by repeated point doubling and adding of the
* original point. All multi-precision integer operations are done under
* the prime finite field specified by #secp256k1_p.
* \param p The point (in affine coordinates) to multiply.
* \param k The 32 byte multi-precision scalar to multiply p by.
*/
void pointMultiply(PointAffine *p, BigNum256 k)
{
PointJacobian accumulator;
PointJacobian junk;
PointAffine always_point_at_infinity; // for dummy operations
uint8_t i;
uint8_t j;
uint8_t one_byte;
uint8_t one_bit;
PointAffine *lookup_affine[2];
memset(&accumulator, 0, sizeof(PointJacobian));
memset(&junk, 0, sizeof(PointJacobian));
memset(&always_point_at_infinity, 0, sizeof(PointAffine));
setFieldToP();
// The Montgomery ladder method can't be used here because it requires
// point addition to be done in pure Jacobian coordinates. Point addition
// in pure Jacobian coordinates would make point multiplication about
// 26% slower. Instead, dummy operations are used to make point
// multiplication a constant time operation. However, the use of dummy
// operations does make this code more susceptible to fault analysis -
// by introducing faults where dummy operations may occur, an attacker
// can determine whether bits in the private key are set or not.
// So the use of this code is not appropriate in situations where fault
// analysis can occur.
accumulator.is_point_at_infinity = 1;
always_point_at_infinity.is_point_at_infinity = 1;
lookup_affine[1] = p;
lookup_affine[0] = &always_point_at_infinity;
for (i = 31; i < 32; i--)
{
one_byte = k[i];
for (j = 0; j < 8; j++)
{
pointDouble(&accumulator);
one_bit = (uint8_t)((one_byte & 0x80) >> 7);
pointAdd(&accumulator, &junk, lookup_affine[one_bit]);
one_byte = (uint8_t)(one_byte << 1);
}
}
jacobianToAffine(p, &accumulator);
}
/** Set a point to the base point of secp256k1.
* \param p The point to set.
*/
void setToG(PointAffine *p)
{
uint8_t buffer[32];
uint8_t i;
p->is_point_at_infinity = 0;
for (i = 0; i < 32; i++)
{
buffer[i] = LOOKUP_BYTE(secp256k1_Gx[i]);
}
bigAssign(p->x, (BigNum256)buffer);
for (i = 0; i < 32; i++)
{
buffer[i] = LOOKUP_BYTE(secp256k1_Gy[i]);
}
bigAssign(p->y, (BigNum256)buffer);
}
/** Create a deterministic ECDSA signature of a given message (digest) and
* private key.
* This is an implementation of the algorithm described in the document
* "SEC 1: Elliptic Curve Cryptography" by Certicom research, obtained
* 15-August-2011 from: http://www.secg.org/collateral/sec1_final.pdf
* section 4.1.3 ("Signing Operation"). The ephemeral private key "k" will
* be deterministically generated according to RFC 6979.
* \param r The "r" component of the signature will be written to here as
* a 32 byte multi-precision number.
* \param s The "s" component of the signature will be written to here, as
* a 32 byte multi-precision number.
* \param hash The message digest of the message to sign, represented as a
* 32 byte multi-precision number.
* \param private_key The private key to use in the signing operation,
* represented as a 32 byte multi-precision number.
*/
void ecdsaSign(BigNum256 r, BigNum256 s, const BigNum256 hash, const BigNum256 private_key)
{
PointAffine big_r;
uint8_t k[32];
uint8_t seed_material[32 + SHA256_HASH_LENGTH];
HMACDRBGState state;
// From RFC 6979, section 3.3a:
// seed_material = int2octets(private_key) || bits2octets(hash)
// int2octets and bits2octets both interpret the number as big-endian.
// However, both the private_key and hash parameters are BigNum256, which
// is little-endian.
bigAssign(seed_material, private_key);
swapEndian256(seed_material); // little-endian -> big-endian
bigAssign(&(seed_material[32]), hash);
swapEndian256(&(seed_material[32])); // little-endian -> big-endian
drbgInstantiate(&state, seed_material, sizeof(seed_material));
while (true)
{
drbgGenerate(k, &state, 32, NULL, 0);
// From RFC 6979, section 3.3b, the output of the DRBG is run through
// the bits2int function, which interprets the output as a big-endian
// integer. However, functions in bignum256.c expect a little-endian
// integer.
swapEndian256(k); // big-endian -> little-endian
// This is one of many data-dependent branches in this function. They do
// not compromise timing attack resistance because these branches are
// expected to occur extremely infrequently.
if (bigIsZero(k))
{
continue;
}
if (bigCompare(k, (BigNum256)secp256k1_n) != BIGCMP_LESS)
{
continue;
}
// Compute ephemeral elliptic curve key pair (k, big_r).
setToG(&big_r);
pointMultiply(&big_r, k);
// big_r now contains k * G.
setFieldToN();
bigModulo(r, big_r.x);
// r now contains (k * G).x (mod n).
if (bigIsZero(r))
{
continue;
}
bigMultiply(s, r, private_key);
bigModulo(big_r.y, hash); // use big_r.y as temporary
bigAdd(s, s, big_r.y);
bigInvert(big_r.y, k);
bigMultiply(s, s, big_r.y);
// s now contains (hash + (r * private_key)) / k (mod n).
if (bigIsZero(s))
{
continue;
}
// Canonicalise s by negating it if s > secp256k1_n / 2.
// See https://github.com/bitcoin/bitcoin/pull/3016 for more info.
bigShiftRightNoModulo(k, (const BigNum256)secp256k1_n); // use k as temporary
if (bigCompare(s, k) == BIGCMP_GREATER)
{
bigSubtractNoModulo(s, (BigNum256)secp256k1_n, s);
}
break;
}
}
/** Serialise an elliptic curve point in a manner which is Bitcoin-compatible.
* This means using the serialisation rules in:
* "SEC 1: Elliptic Curve Cryptography" by Certicom research, obtained
* 15-August-2011 from: http://www.secg.org/collateral/sec1_final.pdf
* sections 2.3.2 ("OctetString-to-BitString Conversion") and
* 2.3.3 ("EllipticCurvePoint-to-OctetString Conversion").
* The document basically says that integers should be represented big-endian
* and that a prefix byte should be prepended to indicate that the public key
* is compressed or not.
* \param out Where the serialised point will be written to. This must be a
* byte array with space for at least #ECDSA_MAX_SERIALISE_SIZE
* bytes.
* \param point The elliptic point curve to serialise.
* \param do_compress Whether to apply point compression - this will reduce
* the size of public keys and hence transactions.
* As of 2014, all Bitcoin clients out there are able to
* decompress points, so it should be safe to always
* compress points.
* \return The number of bytes written to out.
*/
uint8_t ecdsaSerialise(uint8_t *out, const PointAffine *point, const bool do_compress)
{
PointAffine temp;
memcpy(&temp, point, sizeof(temp)); // need temp for endian reversing
if (temp.is_point_at_infinity)
{
// Special case for point at infinity.
out[0] = 0x00;
return 1;
}
else if (!do_compress)
{
// Uncompressed point.
out[0] = 0x04;
swapEndian256(temp.x);
swapEndian256(temp.y);
memcpy(&(out[1]), temp.x, 32);
memcpy(&(out[33]), temp.y, 32);
return 65;
}
else
{
// Compressed point.
if ((temp.y[0] & 1) != 0)
{
out[0] = 0x03; // is odd
}
else
{
out[0] = 0x02; // is not odd
}
swapEndian256(temp.x);
memcpy(&(out[1]), temp.x, 32);
return 33;
}
}
//#ifdef TEST_ECDSA
//
///** The curve parameter b of secp256k1. The other parameter, a, is zero. */
//static const uint8_t secp256k1_b[32] = {
//0x07, 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};
//
///** This is #secp256k1_p plus 1, then divided by 4. It is a constant used for
// * decompressing elliptic curve points. */
//static const uint8_t secp256k1_p_plus1over4[32] = {
//0x0c, 0xff, 0xff, 0xbf, 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, 0x3f};
//
///** Test vector generated using https://brainwallet.github.io/, which is a
// * convenient way to generate serialised public keys. */
//struct BrainwalletTestCase
//{
// /** Private key (brainwallet.org calls this the private exponent. This is
// * big-endian. */
// uint8_t private_exponent[32];
// /** Whether the public key should be compressed. */
// bool is_compressed;
// /** Serialised public key. */
// uint8_t serialised[ECDSA_MAX_SERIALISE_SIZE];
// /** Size of serialised public key, in bytes. */
// uint8_t serialised_size;
//};
//
//static const struct BrainwalletTestCase brainwallet_test_cases[] = {
//
//{ // uncompressed example 1
//{0x9f, 0x79, 0xfd, 0x6c, 0xdc, 0x88, 0x0e, 0x39, // private exponent
//0x14, 0xac, 0x75, 0xb5, 0x0d, 0x71, 0x22, 0x2d,
//0x29, 0xd4, 0xe5, 0xe8, 0x68, 0x0b, 0xc1, 0x4e,
//0x18, 0xe5, 0xef, 0x28, 0xc1, 0x98, 0x14, 0x84},
//false, // is compressed?
//{0x04, 0x99, 0xf9, 0xe0, 0x0e, 0x30, 0x53, 0x1d, // public key
//0x93, 0x12, 0x72, 0x8c, 0x37, 0x7a, 0x56, 0x5c,
//0x8f, 0xef, 0x86, 0x2a, 0x6e, 0xbc, 0x10, 0x77,
//0x33, 0x27, 0x70, 0xae, 0x79, 0xf5, 0xd6, 0x82,
//0xfb, 0xae, 0x86, 0x1a, 0x5e, 0x55, 0x55, 0xc0,
//0x1f, 0x65, 0x37, 0x3d, 0xd6, 0xea, 0xb4, 0x7b,
//0xee, 0xe0, 0x2d, 0x7a, 0xb7, 0x62, 0x6a, 0x00,
//0xd3, 0x82, 0xd4, 0x34, 0xfa, 0xba, 0x84, 0xfe,
//0x60},
//65 // public key length
//},
//
//{ // uncompressed example 2 (only 1 different from above)
//{0x9f, 0x79, 0xfd, 0x6c, 0xdc, 0x88, 0x0e, 0x39, // private exponent
//0x14, 0xac, 0x75, 0xb5, 0x0d, 0x71, 0x22, 0x2d,
//0x29, 0xd4, 0xe5, 0xe8, 0x68, 0x0b, 0xc1, 0x4e,
//0x18, 0xe5, 0xef, 0x28, 0xc1, 0x98, 0x14, 0x85},
//false, // is compressed?
//{0x04, 0x3b, 0x70, 0x76, 0x2c, 0xfa, 0xda, 0xbd, // public key
//0x45, 0x03, 0xdc, 0x5b, 0x71, 0x8c, 0xae, 0x17,
//0xff, 0xa0, 0x9e, 0x0c, 0xc8, 0xd8, 0x0b, 0xa7,
//0xb9, 0xf8, 0x6a, 0x4a, 0x7c, 0xe5, 0xc8, 0x72,
//0x55, 0x65, 0x2c, 0xd6, 0x5d, 0x60, 0xc7, 0x30,
//0x8f, 0x27, 0x61, 0xd1, 0xca, 0xdd, 0x2a, 0x0c,
//0x69, 0x23, 0xfa, 0x24, 0x11, 0x9d, 0x03, 0xb5,
//0x5e, 0xf6, 0xb2, 0xd5, 0xbc, 0x9a, 0xeb, 0x1f,
//0xdf},
//65 // public key length
//},
//
//{ // compressed example with 0x02 prefix
//{0x9f, 0x79, 0xfd, 0x6c, 0xdc, 0x88, 0x0e, 0x39, // private exponent
//0x14, 0xac, 0x75, 0xb5, 0x0d, 0x71, 0x22, 0x2d,
//0x29, 0xd4, 0xe5, 0xe8, 0x68, 0x0b, 0xc1, 0x4e,
//0x18, 0xe5, 0xef, 0x28, 0xc1, 0x98, 0x14, 0x84},
//true, // is compressed?
//{0x02, 0x99, 0xf9, 0xe0, 0x0e, 0x30, 0x53, 0x1d, // public key
//0x93, 0x12, 0x72, 0x8c, 0x37, 0x7a, 0x56, 0x5c,
//0x8f, 0xef, 0x86, 0x2a, 0x6e, 0xbc, 0x10, 0x77,
//0x33, 0x27, 0x70, 0xae, 0x79, 0xf5, 0xd6, 0x82,
//0xfb},
//33 // public key length
//},
//
//{ // compressed example with 0x03 prefix
//{0x9f, 0x79, 0xfd, 0x6c, 0xdc, 0x88, 0x0e, 0x39, // private exponent
//0x14, 0xac, 0x75, 0xb5, 0x0d, 0x71, 0x22, 0x2d,
//0x29, 0xd4, 0xe5, 0xe8, 0x68, 0x0b, 0xc1, 0x4e,
//0x18, 0xe5, 0xef, 0x28, 0xc1, 0x98, 0x14, 0x85},
//true, // is compressed?
//{0x03, 0x3b, 0x70, 0x76, 0x2c, 0xfa, 0xda, 0xbd, // public key
//0x45, 0x03, 0xdc, 0x5b, 0x71, 0x8c, 0xae, 0x17,
//0xff, 0xa0, 0x9e, 0x0c, 0xc8, 0xd8, 0x0b, 0xa7,
//0xb9, 0xf8, 0x6a, 0x4a, 0x7c, 0xe5, 0xc8, 0x72,
//0x55},
//33 // public key length
//},
//
//{ // another compressed example with 0x03 prefix
//{0x9a, 0xbf, 0x38, 0x28, 0xda, 0xad, 0xb8, 0x73, // private exponent
//0xea, 0xc9, 0xff, 0x3a, 0xeb, 0x79, 0xc9, 0x3e,
//0x03, 0xca, 0x9c, 0x28, 0x6c, 0x63, 0x44, 0xf9,
//0x37, 0x62, 0x27, 0x99, 0x04, 0x0c, 0x5d, 0x74},
//true, // is compressed?
//{0x03, 0x68, 0xd6, 0xaf, 0xa4, 0xe1, 0x62, 0xa1, // public key
//0xa2, 0x46, 0x94, 0x30, 0xf9, 0x2d, 0xee, 0x74,
//0x10, 0xf9, 0x4d, 0xd9, 0x9c, 0xa8, 0xca, 0x29,
//0x8a, 0x2b, 0xcc, 0x6c, 0x5a, 0xb7, 0x92, 0xfc,
//0xa3},
//33 // public key length
//},
//
//{ // another compressed example with 0x02 prefix
//{0x3c, 0x01, 0xb9, 0xbf, 0x95, 0x9a, 0x97, 0x35, // private exponent
//0x35, 0x06, 0x81, 0xdc, 0xba, 0x7b, 0xe7, 0xe6,
//0x62, 0xc2, 0x43, 0x9c, 0x1b, 0xa7, 0xb5, 0x9a,
//0xc0, 0x71, 0x32, 0x44, 0xf3, 0x03, 0x95, 0x24},
//true, // is compressed?
//{0x02, 0xdc, 0x07, 0x31, 0xa3, 0x17, 0x61, 0x9c,
//0xfd, 0x7c, 0x40, 0xf1, 0x7f, 0xa2, 0x0e, 0x7b,
//0xd4, 0x4f, 0x2b, 0x5c, 0x68, 0x52, 0x5c, 0x1a,
//0x09, 0xdb, 0x54, 0x41, 0xa4, 0xfb, 0xb6, 0xd6,
//0xa6},
//33 // public key length
//}
//};
//
///** Test vectors for RFC 6979 (deterministic signatures) test cases. */
//struct RFC6979TestCase
//{
// /** Private key. This is big-endian. */
// uint8_t private_key[32];
// /** Message to sign. */
// const char *message;
// /** Expected signature, as r concatenated with s, big-endian. */
// uint8_t expected_signature[32 + 32];
// /** Whether to hash message twice or not. */
// bool double_hash;
//};
//
//static const struct RFC6979TestCase rfc6979_test_cases[] = {
//// These next 5 test vectors are from user fpgaminer on the bitcointalk
//// forums. These were obtained from
//// https://bitcointalk.org/index.php?topic=285142.msg3299061#msg3299061
//// on 9 April 2015.
//{ // fpgaminer test vector 1
//{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // private key
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
//"Satoshi Nakamoto", // message
//{0x93, 0x4b, 0x1e, 0xa1, 0x0a, 0x4b, 0x3c, 0x17, // expected signature: r || s
//0x57, 0xe2, 0xb0, 0xc0, 0x17, 0xd0, 0xb6, 0x14,
//0x3c, 0xe3, 0xc9, 0xa7, 0xe6, 0xa4, 0xa4, 0x98,
//0x60, 0xd7, 0xa6, 0xab, 0x21, 0x0e, 0xe3, 0xd8,
//0x24, 0x42, 0xce, 0x9d, 0x2b, 0x91, 0x60, 0x64,
//0x10, 0x80, 0x14, 0x78, 0x3e, 0x92, 0x3e, 0xc3,
//0x6b, 0x49, 0x74, 0x3e, 0x2f, 0xfa, 0x1c, 0x44,
//0x96, 0xf0, 0x1a, 0x51, 0x2a, 0xaf, 0xd9, 0xe5},
//false // use SHA-256 once
//},
//{ // fpgaminer test vector 2
//{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // private key
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
//"All those moments will be lost in time, like tears in rain. Time to die...", // message
//{0x86, 0x00, 0xdb, 0xd4, 0x1e, 0x34, 0x8f, 0xe5, // expected signature: r || s
//0xc9, 0x46, 0x5a, 0xb9, 0x2d, 0x23, 0xe3, 0xdb,
//0x8b, 0x98, 0xb8, 0x73, 0xbe, 0xec, 0xd9, 0x30,
//0x73, 0x64, 0x88, 0x69, 0x64, 0x38, 0xcb, 0x6b,
//0x54, 0x7f, 0xe6, 0x44, 0x27, 0x49, 0x6d, 0xb3,
//0x3b, 0xf6, 0x60, 0x19, 0xda, 0xcb, 0xf0, 0x03,
//0x9c, 0x04, 0x19, 0x9a, 0xbb, 0x01, 0x22, 0x91,
//0x86, 0x01, 0xdb, 0x38, 0xa7, 0x2c, 0xfc, 0x21},
//false // use SHA-256 once
//},
//{ // fpgaminer test vector 3
//{0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, // private key
//0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE,
//0xBA, 0xAE, 0xDC, 0xE6, 0xAF, 0x48, 0xA0, 0x3B,
//0xBF, 0xD2, 0x5E, 0x8C, 0xD0, 0x36, 0x41, 0x40},
//"Satoshi Nakamoto", // message
//{0xfd, 0x56, 0x7d, 0x12, 0x1d, 0xb6, 0x6e, 0x38, // expected signature: r || s
//0x29, 0x91, 0x53, 0x4a, 0xda, 0x77, 0xa6, 0xbd,
//0x31, 0x06, 0xf0, 0xa1, 0x09, 0x8c, 0x23, 0x1e,
//0x47, 0x99, 0x34, 0x47, 0xcd, 0x6a, 0xf2, 0xd0,
//0x6b, 0x39, 0xcd, 0x0e, 0xb1, 0xbc, 0x86, 0x03,
//0xe1, 0x59, 0xef, 0x5c, 0x20, 0xa5, 0xc8, 0xad,
//0x68, 0x5a, 0x45, 0xb0, 0x6c, 0xe9, 0xbe, 0xbe,
//0xd3, 0xf1, 0x53, 0xd1, 0x0d, 0x93, 0xbe, 0xd5},
//false // use SHA-256 once
//},
//{ // fpgaminer test vector 4
//{0xf8, 0xb8, 0xaf, 0x8c, 0xe3, 0xc7, 0xcc, 0xa5, // private key
//0xe3, 0x00, 0xd3, 0x39, 0x39, 0x54, 0x0c, 0x10,
//0xd4, 0x5c, 0xe0, 0x01, 0xb8, 0xf2, 0x52, 0xbf,
//0xbc, 0x57, 0xba, 0x03, 0x42, 0x90, 0x41, 0x81},
//"Alan Turing", // message
//{0x70, 0x63, 0xae, 0x83, 0xe7, 0xf6, 0x2b, 0xbb, // expected signature: r || s
//0x17, 0x17, 0x98, 0x13, 0x1b, 0x4a, 0x05, 0x64,
//0xb9, 0x56, 0x93, 0x00, 0x92, 0xb3, 0x3b, 0x07,
//0xb3, 0x95, 0x61, 0x5d, 0x9e, 0xc7, 0xe1, 0x5c,
//0x58, 0xdf, 0xcc, 0x1e, 0x00, 0xa3, 0x5e, 0x15,
//0x72, 0xf3, 0x66, 0xff, 0xe3, 0x4b, 0xa0, 0xfc,
//0x47, 0xdb, 0x1e, 0x71, 0x89, 0x75, 0x9b, 0x9f,
//0xb2, 0x33, 0xc5, 0xb0, 0x5a, 0xb3, 0x88, 0xea},
//false // use SHA-256 once
//},
//{ // fpgaminer test vector 5
//{0xe9, 0x16, 0x71, 0xc4, 0x62, 0x31, 0xf8, 0x33, // private key
//0xa6, 0x40, 0x6c, 0xcb, 0xea, 0x0e, 0x3e, 0x39,
//0x2c, 0x76, 0xc1, 0x67, 0xba, 0xc1, 0xcb, 0x01,
//0x3f, 0x6f, 0x10, 0x13, 0x98, 0x04, 0x55, 0xc2},
//"There is a computer disease that anybody who works with computers knows about. It's a very serious disease and it interferes completely with the work. The trouble with computers is that you 'play' with them!", // message
//{0xb5, 0x52, 0xed, 0xd2, 0x75, 0x80, 0x14, 0x1f, // expected signature: r || s
//0x3b, 0x2a, 0x54, 0x63, 0x04, 0x8c, 0xb7, 0xcd,
//0x3e, 0x04, 0x7b, 0x97, 0xc9, 0xf9, 0x80, 0x76,
//0xc3, 0x2d, 0xbd, 0xf8, 0x5a, 0x68, 0x71, 0x8b,
//0x27, 0x9f, 0xa7, 0x2d, 0xd1, 0x9b, 0xfa, 0xe0,
//0x55, 0x77, 0xe0, 0x6c, 0x7c, 0x0c, 0x19, 0x00,
//0xc3, 0x71, 0xfc, 0xd5, 0x89, 0x3f, 0x7e, 0x1d,
//0x56, 0xa3, 0x7d, 0x30, 0x17, 0x46, 0x71, 0xf6},
//false // use SHA-256 once
//},
//
//// These next 2 test vectors are from user plaprade on the bitcointalk
//// forums. These were obtained from
//// https://bitcointalk.org/index.php?topic=285142.msg3300992#msg3300992
//// on 9 April 2015. One requires s to be negated and the other doesn't.
//{ // plaprade test vector 1
//{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, // private key
//0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe,
//0xba, 0xae, 0xdc, 0xe6, 0xaf, 0x48, 0xa0, 0x3b,
//0xbf, 0xd2, 0x5e, 0x8c, 0xd0, 0x36, 0x41, 0x40},
//"Equations are more important to me, because politics is for the present, but an equation is something for eternity.", // message
//{0x54, 0xc4, 0xa3, 0x3c, 0x64, 0x23, 0xd6, 0x89, // expected signature: r || s
//0x37, 0x8f, 0x16, 0x0a, 0x7f, 0xf8, 0xb6, 0x13,
//0x30, 0x44, 0x4a, 0xbb, 0x58, 0xfb, 0x47, 0x0f,
//0x96, 0xea, 0x16, 0xd9, 0x9d, 0x4a, 0x2f, 0xed,
//0x07, 0x08, 0x23, 0x04, 0x41, 0x0e, 0xfa, 0x6b,
//0x29, 0x43, 0x11, 0x1b, 0x6a, 0x4e, 0x0a, 0xaa,
//0x7b, 0x7d, 0xb5, 0x5a, 0x07, 0xe9, 0x86, 0x1d,
//0x1f, 0xb3, 0xcb, 0x1f, 0x42, 0x10, 0x44, 0xa5},
//false // use SHA-256 once
//},
//{ // plaprade test vector 2
//{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // private key
//0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
//0x00, 0x00, 0x00, 0x00, 0x00, 0x05, 0x69, 0x16,
//0xd0, 0xf9, 0xb3, 0x1d, 0xc9, 0xb6, 0x37, 0xf3},
//"The question of whether computers can think is like the question of whether submarines can swim.", // message
//{0xcd, 0xe1, 0x30, 0x2d, 0x83, 0xf8, 0xdd, 0x83, // expected signature: r || s
//0x5d, 0x89, 0xae, 0xf8, 0x03, 0xc7, 0x4a, 0x11,
//0x9f, 0x56, 0x1f, 0xba, 0xef, 0x3e, 0xb9, 0x12,
//0x9e, 0x45, 0xf3, 0x0d, 0xe8, 0x6a, 0xbb, 0xf9,
//0x06, 0xce, 0x64, 0x3f, 0x50, 0x49, 0xee, 0x1f,
//0x27, 0x89, 0x04, 0x67, 0xb7, 0x7a, 0x6a, 0x8e,
//0x11, 0xec, 0x46, 0x61, 0xcc, 0x38, 0xcd, 0x8b,
//0xad, 0xf9, 0x01, 0x15, 0xfb, 0xd0, 0x3c, 0xef},
//false // use SHA-256 once
//},
//
//// These next 2 test vectors are from Bitcoin Core's test suite. They were
//// obtained from:
//// https://github.com/bitcoin/bitcoin/blob/master/src/test/key_tests.cpp
//// on 10 April 2015.
//// Commit: 92fd887fd42a61e95f716d3193104827f60f856c
//{ // Bitcoin Core: strSecret1
//{0x12, 0xb0, 0x04, 0xff, 0xf7, 0xf4, 0xb6, 0x9e, // private key = 5HxWvvfubhXpYYpS3tJkw6fq9jE9j18THftkZjHHfmFiWtmAbrj
//0xf8, 0x65, 0x0e, 0x76, 0x7f, 0x18, 0xf1, 0x1e,
//0xde, 0x15, 0x81, 0x48, 0xb4, 0x25, 0x66, 0x07,
//0x23, 0xb9, 0xf9, 0xa6, 0x6e, 0x61, 0xf7, 0x47},
//"Very deterministic message", // message
//{0x5d, 0xbb, 0xdd, 0xda, 0x71, 0x77, 0x2d, 0x95, // expected signature: r || s
//0xce, 0x91, 0xcd, 0x2d, 0x14, 0xb5, 0x92, 0xcf,
//0xbc, 0x1d, 0xd0, 0xaa, 0xbd, 0x6a, 0x39, 0x4b,
//0x6c, 0x2d, 0x37, 0x7b, 0xbe, 0x59, 0xd3, 0x1d,
//0x14, 0xdd, 0xda, 0x21, 0x49, 0x4a, 0x4e, 0x22,
//0x1f, 0x08, 0x24, 0xf0, 0xb8, 0xb9, 0x24, 0xc4,
//0x3f, 0xa4, 0x3c, 0x0a, 0xd5, 0x7d, 0xcc, 0xda,
//0xa1, 0x1f, 0x81, 0xa6, 0xbd, 0x45, 0x82, 0xf6},
//true // use SHA-256 twice
//},
//{ // Bitcoin Core: strSecret2
//{0xb5, 0x24, 0xc2, 0x8b, 0x61, 0xc9, 0xb2, 0xc4, // private key = 5KC4ejrDjv152FGwP386VD1i2NYc5KkfSMyv1nGy1VGDxGHqVY3
//0x9b, 0x2c, 0x7d, 0xd4, 0xc2, 0xd7, 0x58, 0x87,
//0xab, 0xb7, 0x87, 0x68, 0xc0, 0x54, 0xbd, 0x7c,
//0x01, 0xaf, 0x40, 0x29, 0xf6, 0xc0, 0xd1, 0x17},
//"Very deterministic message", // message
//{0x52, 0xd8, 0xa3, 0x20, 0x79, 0xc1, 0x1e, 0x79, // expected signature: r || s
//0xdb, 0x95, 0xaf, 0x63, 0xbb, 0x96, 0x00, 0xc5,
//0xb0, 0x4f, 0x21, 0xa9, 0xca, 0x33, 0xdc, 0x12,
//0x9c, 0x2b, 0xfa, 0x8a, 0xc9, 0xdc, 0x1c, 0xd5,
//0x61, 0xd8, 0xae, 0x5e, 0x0f, 0x6c, 0x1a, 0x16,
//0xbd, 0xe3, 0x71, 0x9c, 0x64, 0xc2, 0xfd, 0x70,
//0xe4, 0x04, 0xb6, 0x42, 0x8a, 0xb9, 0xa6, 0x95,
//0x66, 0x96, 0x2e, 0x87, 0x71, 0xb5, 0x94, 0x4d},
//true // use SHA-256 twice
//}
//
//};
//
///** Order ("n") divided by 2. Obtained from BIP 0062, from the
// * section "Low S values in signatures". */
//static const uint8_t halforder[32] = {
//0xA0, 0x20, 0x1B, 0x68, 0x46, 0x2F, 0xE9, 0xDF,
//0x1D, 0x50, 0xA4, 0x57, 0x73, 0x6E, 0x57, 0x5D,
//0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
//0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x7F};
//
///** Check if a point is on the elliptic curve. This signals success/failure
// * by calling reportSuccess() or reportFailure().
// * \param p The point to check.
// */
//static void checkPointIsOnCurve(PointAffine *p)
//{
// uint8_t y_squared[32];
// uint8_t x_cubed[32];
//
// if (p->is_point_at_infinity)
// {
// // O is always on the curve.
// reportSuccess();
// return;
// }
// bigMultiply(y_squared, p->y, p->y);
// bigMultiply(x_cubed, p->x, p->x);
// bigMultiply(x_cubed, x_cubed, p->x);
// bigAdd(x_cubed, x_cubed, (BigNum256)secp256k1_b);
// if (bigCompare(y_squared, x_cubed) != BIGCMP_EQUAL)
// {
// printf("Point is not on curve\n");
// printf("x = ");
// printLittleEndian32(p->x);
// printf("\n");
// printf("y = ");
// printLittleEndian32(p->y);
// printf("\n");
// reportFailure();
// }
// else
// {
// reportSuccess();
// }
//}
//
///** Decompress an elliptic curve point - that is, given only the x value of
// * a point, this will calculate the y value. This means that only the x value
// * needs to be stored, which decreases memory use at the expense of time.
// * \param point The point to decompress. Only the x field needs to be filled
// * in - the y field will be ignored and overwritten.
// * \param is_odd For any x value, there are two valid y values - one odd and
// * one even. This parameter instructs the function to pick
// * one of them. Use 0 to pick the even one, 1 to pick the odd
// * one.
// * \return false on success, true if point could not be decompressed.
// */
//static bool ecdsaPointDecompress(PointAffine *point, uint8_t is_odd)
//{
// uint8_t temp[32];
// uint8_t sqrt_y_squared[32];
// uint8_t x_cubed_plus_b[32];
// uint8_t is_sqrt_y_squared_odd;
// uint8_t supposed_to_be_odd;
// BigNum256 lookup[2];
// unsigned int i;
// unsigned int byte_num;
// unsigned int bit_num;
//
// setFieldToP();
// bigMultiply(x_cubed_plus_b, point->x, point->x);
// bigMultiply(x_cubed_plus_b, x_cubed_plus_b, point->x);
// bigAdd(x_cubed_plus_b, x_cubed_plus_b, (BigNum256)secp256k1_b); // x_cubed_plus_b = x^3 + b = y^2
// // Since y^2 = x^3 + b in secp256k1, y = sqrt(x^3 + b). The square
// // root can be performed using the Tonelli-Shanks algorithm. Here, a special
// // case is used, which only works for p = 3 (mod 4) - this is satisfied for
// // the secp256k1 curve. For more information see:
// // http://point-at-infinity.org/ecc/Algorithm_of_Shanks_&_Tonelli.html
// // Exponentiation is done by a standard binary square-and-multiply
// // algorithm.
// bigSetZero(sqrt_y_squared);
// sqrt_y_squared[0] = 1;
// for (i = 255; i < 256; i--)
// {
// bigMultiply(sqrt_y_squared, sqrt_y_squared, sqrt_y_squared);
// byte_num = i >> 3;
// bit_num = i & 7;
// // Yes, this is a data-dependent branch, but it is based on
// // secp256k1_p_plus1over4, which is a (public) constant.
// if (((secp256k1_p_plus1over4[byte_num] >> bit_num) & 1) != 0)
// {
// bigMultiply(sqrt_y_squared, sqrt_y_squared, x_cubed_plus_b);
// }
// }
// // sqrt(y^2) has two solutions ("positive" and "negative"). One of the
// // solutions is odd and the other even. The is_odd parameter controls
// // which one is picked.
// bigSubtractNoModulo(temp, (BigNum256)secp256k1_p, sqrt_y_squared); // temp = -sqrt_y_squared
// is_sqrt_y_squared_odd = (uint8_t)(sqrt_y_squared[0] & 1);
// supposed_to_be_odd = (uint8_t)(is_odd & 1);
// lookup[0] = sqrt_y_squared; // sqrt_y_squared has correct least significant bit
// lookup[1] = temp; // sqrt_y_squared has incorrect least significant bit; use -sqrt_y_squared
// memcpy(point->y, lookup[is_sqrt_y_squared_odd ^ supposed_to_be_odd], sizeof(point->y));
//
// // Check that y^2 does actually equal x^3 + b (i.e. the point is on the
// // curve).
// bigMultiply(temp, point->y, point->y);
// if (bigCompare(temp, x_cubed_plus_b) == BIGCMP_EQUAL)
// {
// return false; // success
// }
// else
// {
// return true; // could not decompress (resulting point is not on curve)
// }
//}
//
///** Read hex string containing a little-endian 256 bit integer from a file.
// * \param r Where the number will be stored into after it is read. This must
// * be a byte array with space for 32 bytes.
// * \param f The file to read from.
// */
//static void bigFRead(BigNum256 r, FILE *f)
//{
// int i;
// int value;
//
// for (i = 0; i < 32; i++)
// {
// fscanf(f, "%02x", &value);
// r[i] = (uint8_t)value;
// }
//}
//
///** Verify an ECDSA signature.
// * \param r One half of the signature (see ecdsaSign()).
// * \param s The other half of the signature (see ecdsaSign()).
// * \param hash The message digest of the message that was signed, represented
// * as a 32 byte little-endian multi-precision integer.
// * \param public_key_x x component of public key, represented as a 32 byte
// * little-endian multi-precision integer.
// * \param public_key_y y component of public key, represented as a 32 byte
// * little-endian multi-precision integer.
// * \return 0 if signature is good, 1 otherwise.
// * \warning Use this for testing only. It's called "crappy" for a reason.
// */
//static int crappyVerifySignature(BigNum256 r, BigNum256 s, BigNum256 hash, BigNum256 public_key_x, BigNum256 public_key_y)
//{
// PointAffine p;
// PointAffine p2;
// PointJacobian pj;
// PointJacobian junk;
// PointAffine result;
// uint8_t temp1[32];
// uint8_t temp2[32];
// uint8_t k1[32];
// uint8_t k2[32];
//
// setFieldToN();
// bigModulo(temp1, hash);
// bigInvert(temp2, s);
// bigMultiply(k1, temp2, temp1);
// bigMultiply(k2, temp2, r);
// setFieldToP();
// bigModulo(k1, k1);
// bigModulo(k2, k2);
// setToG(&p);
// pointMultiply(&p, k1);
// p2.is_point_at_infinity = 0;
// bigAssign(p2.x, public_key_x);
// bigAssign(p2.y, public_key_y);
// pointMultiply(&p2, k2);
// affineToJacobian(&pj, &p);
// pointAdd(&pj, &junk, &p2);
// jacobianToAffine(&result, &pj);
// setFieldToN();
// bigModulo(result.x, result.x);
// if (bigCompare(result.x, r) == BIGCMP_EQUAL)
// {
// return 0;
// }
// else
// {
// return 1;
// }
//}
//
//int main(void)
//{
// PointAffine p;
// PointJacobian p2;
// PointJacobian junk;
// PointAffine compare;
// uint8_t temp[32];
// uint8_t r[32];
// uint8_t s[32];
// uint8_t r_again[32];