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arduinolibs/libraries/Crypto/P521.cpp
Rhys Weatherley 5c4d7ce69a Port the big number routines to 64-bit systems
2016-03-27 07:52:55 +10:00

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/*
* Copyright (C) 2016 Southern Storm Software, Pty Ltd.
*
* Permission is hereby granted, free of charge, to any person obtaining a
* copy of this software and associated documentation files (the "Software"),
* to deal in the Software without restriction, including without limitation
* the rights to use, copy, modify, merge, publish, distribute, sublicense,
* and/or sell copies of the Software, and to permit persons to whom the
* Software is furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included
* in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
* OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
* DEALINGS IN THE SOFTWARE.
*/
#include "P521.h"
#include "Crypto.h"
#include "RNG.h"
#include "SHA512.h"
#include "utility/LimbUtil.h"
#include <string.h>
/**
* \class P521 P521.h <P521.h>
* \brief Elliptic curve operations with the NIST P-521 curve.
*
* This class supports both ECDH key exchange and ECDSA signatures.
*
* \note The public functions in this class need a substantial amount of
* stack space to store intermediate results while the curve function is
* being evaluated. About 2k of free stack space is recommended for safety.
*
* References: NIST FIPS 186-4,
* <a href="http://tools.ietf.org/html/rfc6090">RFC 6090</a>,
* <a href="http://tools.ietf.org/html/rfc6979">RFC 6979</a>,
* <a href="http://tools.ietf.org/html/rfc6090">RFC 5903</a>
*
* \sa Curve25519
*/
// Number of limbs that are needed to represent a 521-bit number.
#define NUM_LIMBS_521BIT NUM_LIMBS_BITS(521)
// Number of limbs that are needed to represent a 1042-bit number.
// To simply things we also require that this be twice the size of
// NUM_LIMB_521BIT which involves a little wastage at the high end
// of one extra limb for 8-bit and 32-bit limbs. There is no
// wastage for 16-bit limbs.
#define NUM_LIMBS_1042BIT (NUM_LIMBS_BITS(521) * 2)
// The overhead of clean() calls in mul(), etc can add up to a lot of
// processing time. Only do such cleanups if strict mode has been enabled.
#if defined(P521_STRICT_CLEAN)
#define strict_clean(x) clean(x)
#else
#define strict_clean(x) do { ; } while (0)
#endif
// Expand the partial 9-bit left over limb at the top of a 521-bit number.
#if BIGNUMBER_LIMB_8BIT
#define LIMB_PARTIAL(value) ((uint8_t)(value)), \
((uint8_t)((value) >> 8))
#else
#define LIMB_PARTIAL(value) (value)
#endif
/** @cond */
// The group order "q" value from RFC 4754 and RFC 5903. This is the
// same as the "n" value from Appendix D.1.2.5 of NIST FIPS 186-4.
static limb_t const P521_q[NUM_LIMBS_521BIT] PROGMEM = {
LIMB_PAIR(0x91386409, 0xbb6fb71e), LIMB_PAIR(0x899c47ae, 0x3bb5c9b8),
LIMB_PAIR(0xf709a5d0, 0x7fcc0148), LIMB_PAIR(0xbf2f966b, 0x51868783),
LIMB_PAIR(0xfffffffa, 0xffffffff), LIMB_PAIR(0xffffffff, 0xffffffff),
LIMB_PAIR(0xffffffff, 0xffffffff), LIMB_PAIR(0xffffffff, 0xffffffff),
LIMB_PARTIAL(0x1ff)
};
// The "b" value from Appendix D.1.2.5 of NIST FIPS 186-4.
static limb_t const P521_b[NUM_LIMBS_521BIT] PROGMEM = {
LIMB_PAIR(0x6b503f00, 0xef451fd4), LIMB_PAIR(0x3d2c34f1, 0x3573df88),
LIMB_PAIR(0x3bb1bf07, 0x1652c0bd), LIMB_PAIR(0xec7e937b, 0x56193951),
LIMB_PAIR(0x8ef109e1, 0xb8b48991), LIMB_PAIR(0x99b315f3, 0xa2da725b),
LIMB_PAIR(0xb68540ee, 0x929a21a0), LIMB_PAIR(0x8e1c9a1f, 0x953eb961),
LIMB_PARTIAL(0x051)
};
// The "Gx" value from Appendix D.1.2.5 of NIST FIPS 186-4.
static limb_t const P521_Gx[NUM_LIMBS_521BIT] PROGMEM = {
LIMB_PAIR(0xc2e5bd66, 0xf97e7e31), LIMB_PAIR(0x856a429b, 0x3348b3c1),
LIMB_PAIR(0xa2ffa8de, 0xfe1dc127), LIMB_PAIR(0xefe75928, 0xa14b5e77),
LIMB_PAIR(0x6b4d3dba, 0xf828af60), LIMB_PAIR(0x053fb521, 0x9c648139),
LIMB_PAIR(0x2395b442, 0x9e3ecb66), LIMB_PAIR(0x0404e9cd, 0x858e06b7),
LIMB_PARTIAL(0x0c6)
};
// The "Gy" value from Appendix D.1.2.5 of NIST FIPS 186-4.
static limb_t const P521_Gy[NUM_LIMBS_521BIT] PROGMEM = {
LIMB_PAIR(0x9fd16650, 0x88be9476), LIMB_PAIR(0xa272c240, 0x353c7086),
LIMB_PAIR(0x3fad0761, 0xc550b901), LIMB_PAIR(0x5ef42640, 0x97ee7299),
LIMB_PAIR(0x273e662c, 0x17afbd17), LIMB_PAIR(0x579b4468, 0x98f54449),
LIMB_PAIR(0x2c7d1bd9, 0x5c8a5fb4), LIMB_PAIR(0x9a3bc004, 0x39296a78),
LIMB_PARTIAL(0x118)
};
/** @endcond */
/**
* \brief Evaluates the curve function.
*
* \param result The result of applying the curve function, which consists
* of the x and y values of the result point encoded in big-endian order.
* \param f The scalar value to multiply by \a point to create the \a result.
* This is assumed to be be a 521-bit number in big-endian order.
* \param point The curve point to multiply consisting of the x and y
* values encoded in big-endian order. If \a point is NULL, then the
* generator Gx and Gy values for the curve will be used instead.
*
* \return Returns true if \a f * \a point could be evaluated, or false if
* \a point is not a point on the curve.
*
* This function provides access to the raw curve operation for testing
* purposes. Normally an application would use a higher-level function
* like dh1(), dh2(), sign(), or verify().
*
* \sa dh1(), sign()
*/
bool P521::eval(uint8_t result[132], const uint8_t f[66], const uint8_t point[132])
{
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
bool ok;
// Unpack the curve point from the parameters and validate it.
if (point) {
BigNumberUtil::unpackBE(x, NUM_LIMBS_521BIT, point, 66);
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, point + 66, 66);
ok = validate(x, y);
} else {
memcpy_P(x, P521_Gx, sizeof(x));
memcpy_P(y, P521_Gy, sizeof(y));
ok = true;
}
// Evaluate the curve function.
evaluate(x, y, f);
// Pack the answer into the result array.
BigNumberUtil::packBE(result, 66, x, NUM_LIMBS_521BIT);
BigNumberUtil::packBE(result + 66, 66, y, NUM_LIMBS_521BIT);
// Clean up.
clean(x);
clean(y);
return ok;
}
/**
* \brief Performs phase 1 of an ECDH key exchange using P-521.
*
* \param k The key value to send to the other party as part of the exchange.
* \param f The generated secret value for this party. This must not be
* transmitted to any party or stored in permanent storage. It only needs
* to be kept in memory until dh2() is called.
*
* The \a f value is generated with \link RNGClass::rand() RNG.rand()\endlink.
* It is the caller's responsibility to ensure that the global random number
* pool has sufficient entropy to generate the 66 bytes of \a f safely
* before calling this function.
*
* The following example demonstrates how to perform a full ECDH
* key exchange using dh1() and dh2():
*
* \code
* uint8_t f[66];
* uint8_t k[132];
*
* // Generate the secret value "f" and the public value "k".
* P521::dh1(k, f);
*
* // Send "k" to the other party.
* ...
*
* // Read the "k" value that the other party sent to us.
* ...
*
* // Generate the shared secret in "f".
* if (!P521::dh2(k, f)) {
* // The received "k" value was invalid - abort the session.
* ...
* }
*
* // The "f" value can now be used to generate session keys for encryption.
* ...
* \endcode
*
* Reference: <a href="http://tools.ietf.org/html/rfc6090">RFC 6090</a>
*
* \sa dh2()
*/
void P521::dh1(uint8_t k[132], uint8_t f[66])
{
generatePrivateKey(f);
derivePublicKey(k, f);
}
/**
* \brief Performs phase 2 of an ECDH key exchange using P-521.
*
* \param k The public key value that was received from the other
* party as part of the exchange.
* \param f On entry, this is the secret value for this party that was
* generated by dh1(). On exit, this will be the shared secret.
*
* \return Returns true if the key exchange was successful, or false if
* the \a k value is invalid.
*
* Reference: <a href="http://tools.ietf.org/html/rfc6090">RFC 6090</a>
*
* \sa dh1()
*/
bool P521::dh2(const uint8_t k[132], uint8_t f[66])
{
// Unpack the (x, y) point from k.
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
BigNumberUtil::unpackBE(x, NUM_LIMBS_521BIT, k, 66);
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, k + 66, 66);
// Validate the curve point. We keep going to preserve the timing.
bool ok = validate(x, y);
// Evaluate the curve function.
evaluate(x, y, f);
// The secret key is the x component of the final value.
BigNumberUtil::packBE(f, 66, x, NUM_LIMBS_521BIT);
// Clean up.
clean(x);
clean(y);
return ok;
}
/**
* \brief Signs a message using a specific P-521 private key.
*
* \param signature The signature value.
* \param privateKey The private key to use to sign the message.
* \param message Points to the message to be signed.
* \param len The length of the \a message to be signed.
* \param hash The hash algorithm to use to hash the \a message before signing.
* If \a hash is NULL, then the \a message is assumed to already be a hash
* value from some previous process.
*
* This function generates deterministic ECDSA signatures according to
* RFC 6979. The \a hash function is used to generate the k value for
* the signature. If \a hash is NULL, then SHA512 is used.
* The \a hash object must be capable of HMAC mode.
*
* The length of the hashed message must be less than or equal to 64
* bytes in size. Longer messages will be truncated to 64 bytes.
*
* References: <a href="http://tools.ietf.org/html/rfc6090">RFC 6090</a>,
* <a href="http://tools.ietf.org/html/rfc6979">RFC 6979</a>
*
* \sa verify(), generatePrivateKey()
*/
void P521::sign(uint8_t signature[132], const uint8_t privateKey[66],
const void *message, size_t len, Hash *hash)
{
uint8_t hm[66];
uint8_t k[66];
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
limb_t t[NUM_LIMBS_521BIT];
uint64_t count = 0;
// Format the incoming message, hashing it if necessary.
if (hash) {
// Hash the message.
hash->reset();
hash->update(message, len);
len = hash->hashSize();
if (len > 64)
len = 64;
memset(hm, 0, 66 - len);
hash->finalize(hm + 66 - len, len);
} else {
// The message is the hash.
if (len > 64)
len = 64;
memset(hm, 0, 66 - len);
memcpy(hm + 66 - len, message, len);
}
// Keep generating k values until both r and s are non-zero.
for (;;) {
// Generate the k value deterministically according to RFC 6979.
if (hash)
generateK(k, hm, privateKey, hash, count);
else
generateK(k, hm, privateKey, count);
// Generate r = kG.x mod q.
memcpy_P(x, P521_Gx, sizeof(x));
memcpy_P(y, P521_Gy, sizeof(y));
evaluate(x, y, k);
BigNumberUtil::reduceQuick_P(x, x, P521_q, NUM_LIMBS_521BIT);
BigNumberUtil::packBE(signature, 66, x, NUM_LIMBS_521BIT);
// If r is zero, then we need to generate a new k value.
// This is utterly improbable, but let's be safe anyway.
if (BigNumberUtil::isZero(x, NUM_LIMBS_521BIT)) {
++count;
continue;
}
// Generate s = (privateKey * r + hm) / k mod q.
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, privateKey, 66);
mulQ(y, y, x);
BigNumberUtil::unpackBE(x, NUM_LIMBS_521BIT, hm, 66);
BigNumberUtil::add(x, x, y, NUM_LIMBS_521BIT);
BigNumberUtil::reduceQuick_P(x, x, P521_q, NUM_LIMBS_521BIT);
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, k, 66);
recipQ(t, y);
mulQ(x, x, t);
BigNumberUtil::packBE(signature + 66, 66, x, NUM_LIMBS_521BIT);
// Exit the loop if s is non-zero.
if (!BigNumberUtil::isZero(x, NUM_LIMBS_521BIT))
break;
// We need to generate a new k value according to RFC 6979.
// This is utterly improbable, but let's be safe anyway.
++count;
}
// Clean up.
clean(hm);
clean(k);
clean(x);
clean(y);
clean(t);
}
/**
* \brief Verifies a signature using a specific P-521 public key.
*
* \param signature The signature value to be verified.
* \param publicKey The public key to use to verify the signature.
* \param message The message whose signature is to be verified.
* \param len The length of the \a message to be verified.
* \param hash The hash algorithm to use to hash the \a message before
* verification. If \a hash is NULL, then the \a message is assumed to
* already be a hash value from some previous process.
*
* The length of the hashed message must be less than or equal to 64
* bytes in size. Longer messages will be truncated to 64 bytes.
*
* \return Returns true if the \a signature is valid for \a message;
* or false if the \a publicKey or \a signature is not valid.
*
* \sa sign()
*/
bool P521::verify(const uint8_t signature[132],
const uint8_t publicKey[132],
const void *message, size_t len, Hash *hash)
{
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
limb_t r[NUM_LIMBS_521BIT];
limb_t s[NUM_LIMBS_521BIT];
limb_t u1[NUM_LIMBS_521BIT];
limb_t u2[NUM_LIMBS_521BIT];
uint8_t t[66];
bool ok = false;
// Because we are operating on public values, we don't need to
// be as strict about constant time. Bail out early if there
// is a problem with the parameters.
// Unpack the signature. The values must be between 1 and q - 1.
BigNumberUtil::unpackBE(r, NUM_LIMBS_521BIT, signature, 66);
BigNumberUtil::unpackBE(s, NUM_LIMBS_521BIT, signature + 66, 66);
if (BigNumberUtil::isZero(r, NUM_LIMBS_521BIT) ||
BigNumberUtil::isZero(s, NUM_LIMBS_521BIT) ||
!BigNumberUtil::sub_P(x, r, P521_q, NUM_LIMBS_521BIT) ||
!BigNumberUtil::sub_P(x, s, P521_q, NUM_LIMBS_521BIT)) {
goto failed;
}
// Unpack the public key and check that it is a valid curve point.
BigNumberUtil::unpackBE(x, NUM_LIMBS_521BIT, publicKey, 66);
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, publicKey + 66, 66);
if (!validate(x, y)) {
goto failed;
}
// Hash the message to generate hm, which we store into u1.
if (hash) {
// Hash the message.
hash->reset();
hash->update(message, len);
len = hash->hashSize();
if (len > 64)
len = 64;
hash->finalize(u2, len);
BigNumberUtil::unpackBE(u1, NUM_LIMBS_521BIT, (uint8_t *)u2, len);
} else {
// The message is the hash.
if (len > 64)
len = 64;
BigNumberUtil::unpackBE(u1, NUM_LIMBS_521BIT, (uint8_t *)message, len);
}
// Compute u1 = hm * s^-1 mod q and u2 = r * s^-1 mod q.
recipQ(u2, s);
mulQ(u1, u1, u2);
mulQ(u2, r, u2);
// Compute the curve point R = u2 * publicKey + u1 * G.
BigNumberUtil::packBE(t, 66, u2, NUM_LIMBS_521BIT);
evaluate(x, y, t);
memcpy_P(u2, P521_Gx, sizeof(x));
memcpy_P(s, P521_Gy, sizeof(y));
BigNumberUtil::packBE(t, 66, u1, NUM_LIMBS_521BIT);
evaluate(u2, s, t);
addAffine(u2, s, x, y);
// If R.x = r mod q, then the signature is valid.
BigNumberUtil::reduceQuick_P(u1, u2, P521_q, NUM_LIMBS_521BIT);
ok = secure_compare(u1, r, NUM_LIMBS_521BIT * sizeof(limb_t));
// Clean up and exit.
failed:
clean(x);
clean(y);
clean(r);
clean(s);
clean(u1);
clean(u2);
clean(t);
return ok;
}
/**
* \brief Generates a private key for P-521 signing operations.
*
* \param privateKey The resulting private key.
*
* The private key is generated with \link RNGClass::rand() RNG.rand()\endlink.
* It is the caller's responsibility to ensure that the global random number
* pool has sufficient entropy to generate the 521 bits of the key safely
* before calling this function.
*
* \sa derivePublicKey(), sign()
*/
void P521::generatePrivateKey(uint8_t privateKey[66])
{
// Generate a random 521-bit value for the private key. The value
// must be generated uniformly at random between 1 and q - 1 where q
// is the group order (RFC 6090). We use the recommended algorithm
// from Appendix B of RFC 6090: generate a random 521-bit value
// and discard it if it is not within the range 1 to q - 1.
limb_t x[NUM_LIMBS_521BIT];
do {
RNG.rand((uint8_t *)x, sizeof(x));
#if BIGNUMBER_LIMB_8BIT
x[NUM_LIMBS_521BIT - 1] &= 0x01;
#else
x[NUM_LIMBS_521BIT - 1] &= 0x1FF;
#endif
BigNumberUtil::packBE(privateKey, 66, x, NUM_LIMBS_521BIT);
} while (BigNumberUtil::isZero(x, NUM_LIMBS_521BIT) ||
!BigNumberUtil::sub_P(x, x, P521_q, NUM_LIMBS_521BIT));
clean(x);
}
/**
* \brief Derives the public key from a private key for P-521
* signing operations.
*
* \param publicKey The public key.
* \param privateKey The private key, which is assumed to have been
* created by generatePrivateKey().
*
* \sa generatePrivateKey(), verify()
*/
void P521::derivePublicKey(uint8_t publicKey[132], const uint8_t privateKey[66])
{
// Evaluate the curve function starting with the generator.
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
memcpy_P(x, P521_Gx, sizeof(x));
memcpy_P(y, P521_Gy, sizeof(y));
evaluate(x, y, privateKey);
// Pack the (x, y) point into the public key.
BigNumberUtil::packBE(publicKey, 66, x, NUM_LIMBS_521BIT);
BigNumberUtil::packBE(publicKey + 66, 66, y, NUM_LIMBS_521BIT);
// Clean up.
clean(x);
clean(y);
}
/**
* \brief Validates a private key value to ensure that it is
* between 1 and q - 1.
*
* \param privateKey The private key value to validate.
* \return Returns true if \a privateKey is valid, false if not.
*
* \sa isValidPublicKey()
*/
bool P521::isValidPrivateKey(const uint8_t privateKey[66])
{
// The value "q" as a byte array from most to least significant.
static uint8_t const P521_q_bytes[66] PROGMEM = {
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
};
uint8_t zeroTest = 0;
uint8_t posn = 66;
uint16_t borrow = 0;
while (posn > 0) {
--posn;
// Check for zero.
zeroTest |= privateKey[posn];
// Subtract P521_q_bytes from the key. If there is no borrow,
// then the key value was greater than or equal to q.
borrow = ((uint16_t)(privateKey[posn])) -
pgm_read_byte(&(P521_q_bytes[posn])) -
((borrow >> 8) & 0x01);
}
return zeroTest != 0 && borrow != 0;
}
/**
* \brief Validates a public key to ensure that it is a valid curve point.
*
* \param publicKey The public key value to validate.
* \return Returns true if \a publicKey is valid, false if not.
*
* \sa isValidPrivateKey()
*/
bool P521::isValidPublicKey(const uint8_t publicKey[132])
{
limb_t x[NUM_LIMBS_521BIT];
limb_t y[NUM_LIMBS_521BIT];
BigNumberUtil::unpackBE(x, NUM_LIMBS_521BIT, publicKey, 66);
BigNumberUtil::unpackBE(y, NUM_LIMBS_521BIT, publicKey + 66, 66);
bool ok = validate(x, y);
clean(x);
clean(y);
return ok;
}
/**
* \fn bool P521::isValidCurvePoint(const uint8_t point[132])
* \brief Validates a point to ensure that it is on the curve.
*
* \param point The point to validate.
* \return Returns true if \a point is valid and on the curve, false if not.
*
* This is a convenience function that calls isValidPublicKey() as the
* two operations are equivalent.
*/
/**
* \brief Evaluates the curve function by multiplying (x, y) by f.
*
* \param x The X co-ordinate of the curve point. Replaced with the X
* co-ordinate of the result on exit.
* \param y The Y co-ordinate of the curve point. Replaced with the Y
* co-ordinate of the result on exit.
* \param f The 521-bit scalar to multiply (x, y) by, most significant
* bit first.
*/
void P521::evaluate(limb_t *x, limb_t *y, const uint8_t f[66])
{
limb_t x1[NUM_LIMBS_521BIT];
limb_t y1[NUM_LIMBS_521BIT];
limb_t z1[NUM_LIMBS_521BIT];
limb_t x2[NUM_LIMBS_521BIT];
limb_t y2[NUM_LIMBS_521BIT];
limb_t z2[NUM_LIMBS_521BIT];
// We want the input in Jacobian co-ordinates. The point (x, y, z)
// corresponds to the affine point (x / z^2, y / z^3), so if we set z
// to 1 we end up with Jacobian co-ordinates. Remember that z is 1
// and continue on.
// Set the answer to the point-at-infinity initially (z = 0).
memset(x1, 0, sizeof(x1));
memset(y1, 0, sizeof(y1));
memset(z1, 0, sizeof(z1));
// Special handling for the highest bit. We can skip dblPoint()/addPoint()
// and simply conditionally move (x, y, z) into (x1, y1, z1).
uint8_t select = (f[0] & 0x01);
cmove(select, x1, x);
cmove(select, y1, y);
cmove1(select, z1); // z = 1
// Iterate over the remaining 520 bits of f from highest to lowest.
uint8_t mask = 0x80;
uint8_t fposn = 1;
for (uint16_t t = 520; t > 0; --t) {
// Double the answer.
dblPoint(x1, y1, z1, x1, y1, z1);
// Add (x, y, z) to (x1, y1, z1) for the next 1 bit.
// We must always do this to preserve the overall timing.
// The z value is always 1 so we can omit that argument.
addPoint(x2, y2, z2, x1, y1, z1, x, y/*, z*/);
// If the bit was 1, then move (x2, y2, z2) into (x1, y1, z1).
select = (f[fposn] & mask);
cmove(select, x1, x2);
cmove(select, y1, y2);
cmove(select, z1, z2);
// Move onto the next bit.
mask >>= 1;
if (!mask) {
++fposn;
mask = 0x80;
}
}
// Convert from Jacobian co-ordinates back into affine co-ordinates.
// x = x1 * (z1^2)^-1, y = y1 * (z1^3)^-1.
recip(x2, z1);
square(y2, x2);
mul(x, x1, y2);
mul(y2, y2, x2);
mul(y, y1, y2);
// Clean up.
clean(x1);
clean(y1);
clean(z1);
clean(x2);
clean(y2);
clean(z2);
}
/**
* \brief Adds two affine points.
*
* \param x1 The X value for the first point to add, and the result.
* \param y1 The Y value for the first point to add, and the result.
* \param x2 The X value for the second point to add.
* \param y2 The Y value for the second point to add.
*
* The Z values for the two points are assumed to be 1.
*/
void P521::addAffine(limb_t *x1, limb_t *y1, const limb_t *x2, const limb_t *y2)
{
limb_t xout[NUM_LIMBS_521BIT];
limb_t yout[NUM_LIMBS_521BIT];
limb_t zout[NUM_LIMBS_521BIT];
limb_t z1[NUM_LIMBS_521BIT];
// z1 = 1
z1[0] = 1;
memset(z1 + 1, 0, (NUM_LIMBS_521BIT - 1) * sizeof(limb_t));
// Add the two points.
addPoint(xout, yout, zout, x1, y1, z1, x2, y2/*, z2*/);
// Convert from Jacobian co-ordinates back into affine co-ordinates.
// x1 = xout * (zout^2)^-1, y1 = yout * (zout^3)^-1.
recip(z1, zout);
square(zout, z1);
mul(x1, xout, zout);
mul(zout, zout, z1);
mul(y1, yout, zout);
// Clean up.
clean(xout);
clean(yout);
clean(zout);
clean(z1);
}
/**
* \brief Validates that (x, y) is actually a point on the curve.
*
* \param x The X co-ordinate of the point to test.
* \param y The Y co-ordinate of the point to test.
* \return Returns true if (x, y) is on the curve, or false if not.
*
* \sa inRange()
*/
bool P521::validate(const limb_t *x, const limb_t *y)
{
bool result;
// If x or y is greater than or equal to 2^521 - 1, then the
// point is definitely not on the curve. Preserve timing by
// delaying the reporting of the result until later.
result = inRange(x);
result &= inRange(y);
// We need to check that y^2 = x^3 - 3 * x + b mod 2^521 - 1.
limb_t t1[NUM_LIMBS_521BIT];
limb_t t2[NUM_LIMBS_521BIT];
square(t1, x);
mul(t1, t1, x);
mulLiteral(t2, x, 3);
sub(t1, t1, t2);
memcpy_P(t2, P521_b, sizeof(t2));
add(t1, t1, t2);
square(t2, y);
result &= secure_compare(t1, t2, sizeof(t1));
clean(t1);
clean(t2);
return result;
}
/**
* \brief Determines if a value is between 0 and 2^521 - 2.
*
* \param x The value to test.
* \return Returns true if \a x is in range, false if not.
*
* \sa validate()
*/
bool P521::inRange(const limb_t *x)
{
// Do a trial subtraction of 2^521 - 1 from x, which is equivalent
// to adding 1 and subtracting 2^521. We only need the carry.
dlimb_t carry = 1;
limb_t word = 0;
for (uint8_t index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry += *x++;
word = (limb_t)carry;
carry >>= LIMB_BITS;
}
// Determine the carry out from the low 521 bits.
#if BIGNUMBER_LIMB_8BIT
carry = (carry << 7) + (word >> 1);
#else
carry = (carry << (LIMB_BITS - 9)) + (word >> 9);
#endif
// If the carry is zero, then x was in range. Otherwise it is out
// of range. Check for zero in a way that preserves constant timing.
word = (limb_t)(carry | (carry >> LIMB_BITS));
word = (limb_t)(((((dlimb_t)1) << LIMB_BITS) - word) >> LIMB_BITS);
return (bool)word;
}
/**
* \brief Reduces a number modulo 2^521 - 1.
*
* \param result The array that will contain the result when the
* function exits. Must be NUM_LIMBS_521BIT limbs in size.
* \param x The number to be reduced, which must be NUM_LIMBS_1042BIT
* limbs in size and less than square(2^521 - 1). This array can be
* the same as \a result.
*/
void P521::reduce(limb_t *result, const limb_t *x)
{
#if BIGNUMBER_LIMB_16BIT || BIGNUMBER_LIMB_32BIT || BIGNUMBER_LIMB_64BIT
// According to NIST FIPS 186-4, we add the high 521 bits to the
// low 521 bits and then do a trial subtraction of 2^521 - 1.
// We do both in a single step. Subtracting 2^521 - 1 is equivalent
// to adding 1 and subtracting 2^521.
uint8_t index;
const limb_t *xl = x;
const limb_t *xh = x + NUM_LIMBS_521BIT;
limb_t *rr = result;
dlimb_t carry;
limb_t word = x[NUM_LIMBS_521BIT - 1];
carry = (word >> 9) + 1;
word &= 0x1FF;
for (index = 0; index < (NUM_LIMBS_521BIT - 1); ++index) {
carry += *xl++;
carry += ((dlimb_t)(*xh++)) << (LIMB_BITS - 9);
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
carry += word;
carry += ((dlimb_t)(x[NUM_LIMBS_1042BIT - 1])) << (LIMB_BITS - 9);
word = (limb_t)carry;
*rr = word;
// If the carry out was 1, then mask it off and we have the answer.
// If the carry out was 0, then we need to add 2^521 - 1 back again.
// To preserve the timing we perform a conditional subtract of 1 and
// then mask off the high bits.
carry = ((word >> 9) ^ 0x01) & 0x01;
rr = result;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*rr)) - carry;
*rr++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--rr) &= 0x1FF;
#elif BIGNUMBER_LIMB_8BIT
// Same as above, but for 8-bit limbs.
uint8_t index;
const limb_t *xl = x;
const limb_t *xh = x + NUM_LIMBS_521BIT;
limb_t *rr = result;
dlimb_t carry;
limb_t word = x[NUM_LIMBS_521BIT - 1];
carry = (word >> 1) + 1;
word &= 0x01;
for (index = 0; index < (NUM_LIMBS_521BIT - 1); ++index) {
carry += *xl++;
carry += ((dlimb_t)(*xh++)) << 7;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
carry += word;
carry += ((dlimb_t)(x[NUM_LIMBS_1042BIT - 1])) << 1;
word = (limb_t)carry;
*rr = word;
carry = ((word >> 1) ^ 0x01) & 0x01;
rr = result;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*rr)) - carry;
*rr++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--rr) &= 0x01;
#else
#error "Don't know how to reduce values mod 2^521 - 1"
#endif
}
/**
* \brief Quickly reduces a number modulo 2^521 - 1.
*
* \param x The number to be reduced, which must be NUM_LIMBS_521BIT
* limbs in size and less than or equal to 2 * (2^521 - 2).
*
* The answer is also put into \a x and will consist of NUM_LIMBS_521BIT limbs.
*
* This function is intended for reducing the result of additions where
* the caller knows that \a x is within the described range. A single
* trial subtraction is all that is needed to reduce the number.
*/
void P521::reduceQuick(limb_t *x)
{
// Perform a trial subtraction of 2^521 - 1 from x. This is
// equivalent to adding 1 and subtracting 2^521 - 1.
uint8_t index;
limb_t *xx = x;
dlimb_t carry = 1;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry += *xx;
*xx++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
// If the carry out was 1, then mask it off and we have the answer.
// If the carry out was 0, then we need to add 2^521 - 1 back again.
// To preserve the timing we perform a conditional subtract of 1 and
// then mask off the high bits.
#if BIGNUMBER_LIMB_16BIT || BIGNUMBER_LIMB_32BIT || BIGNUMBER_LIMB_64BIT
carry = ((x[NUM_LIMBS_521BIT - 1] >> 9) ^ 0x01) & 0x01;
xx = x;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*xx)) - carry;
*xx++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--xx) &= 0x1FF;
#elif BIGNUMBER_LIMB_8BIT
carry = ((x[NUM_LIMBS_521BIT - 1] >> 1) ^ 0x01) & 0x01;
xx = x;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*xx)) - carry;
*xx++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--xx) &= 0x01;
#endif
}
/**
* \brief Multiplies two 521-bit values to produce a 1042-bit result.
*
* \param result The result, which must be NUM_LIMBS_1042BIT limbs in size
* and must not overlap with \a x or \a y.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size.
* \param y The second value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size.
*
* \sa mul()
*/
void P521::mulNoReduce(limb_t *result, const limb_t *x, const limb_t *y)
{
uint8_t i, j;
dlimb_t carry;
limb_t word;
const limb_t *yy;
limb_t *rr;
// Multiply the lowest word of x by y.
carry = 0;
word = x[0];
yy = y;
rr = result;
for (i = 0; i < NUM_LIMBS_521BIT; ++i) {
carry += ((dlimb_t)(*yy++)) * word;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
*rr = (limb_t)carry;
// Multiply and add the remaining words of x by y.
for (i = 1; i < NUM_LIMBS_521BIT; ++i) {
word = x[i];
carry = 0;
yy = y;
rr = result + i;
for (j = 0; j < NUM_LIMBS_521BIT; ++j) {
carry += ((dlimb_t)(*yy++)) * word;
carry += *rr;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
*rr = (limb_t)carry;
}
}
/**
* \brief Multiplies two values and then reduces the result modulo 2^521 - 1.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size
* and can be the same array as \a x or \a y.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT limbs
* in size and less than 2^521 - 1.
* \param y The second value to multiply, which must be NUM_LIMBS_521BIT limbs
* in size and less than 2^521 - 1. This can be the same array as \a x.
*/
void P521::mul(limb_t *result, const limb_t *x, const limb_t *y)
{
limb_t temp[NUM_LIMBS_1042BIT];
mulNoReduce(temp, x, y);
reduce(result, temp);
strict_clean(temp);
}
/**
* \fn void P521::square(limb_t *result, const limb_t *x)
* \brief Squares a value and then reduces it modulo 2^521 - 1.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size and
* can be the same array as \a x.
* \param x The value to square, which must be NUM_LIMBS_521BIT limbs in size
* and less than 2^521 - 1.
*/
/**
* \brief Multiply a value by a single-limb literal modulo 2^521 - 1.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size and
* can be the same array as \a x.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT limbs
* in size and less than 2^521 - 1.
* \param y The second value to multiply, which must be less than 128.
*/
void P521::mulLiteral(limb_t *result, const limb_t *x, limb_t y)
{
uint8_t index;
dlimb_t carry = 0;
const limb_t *xx = x;
limb_t *rr = result;
// Multiply x by the literal and put it into the result array.
// We assume that y is small enough that overflow from the
// highest limb will not occur during this process.
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry += ((dlimb_t)(*xx++)) * y;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
// Reduce the value modulo 2^521 - 1. The high half is only a
// single limb, so we can short-cut some of reduce() here.
#if BIGNUMBER_LIMB_16BIT || BIGNUMBER_LIMB_32BIT || BIGNUMBER_LIMB_64BIT
limb_t word = result[NUM_LIMBS_521BIT - 1];
carry = (word >> 9) + 1;
word &= 0x1FF;
rr = result;
for (index = 0; index < (NUM_LIMBS_521BIT - 1); ++index) {
carry += *rr;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
carry += word;
word = (limb_t)carry;
*rr = word;
// If the carry out was 1, then mask it off and we have the answer.
// If the carry out was 0, then we need to add 2^521 - 1 back again.
// To preserve the timing we perform a conditional subtract of 1 and
// then mask off the high bits.
carry = ((word >> 9) ^ 0x01) & 0x01;
rr = result;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*rr)) - carry;
*rr++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--rr) &= 0x1FF;
#elif BIGNUMBER_LIMB_8BIT
// Same as above, but for 8-bit limbs.
limb_t word = result[NUM_LIMBS_521BIT - 1];
carry = (word >> 1) + 1;
word &= 0x01;
rr = result;
for (index = 0; index < (NUM_LIMBS_521BIT - 1); ++index) {
carry += *rr;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
carry += word;
word = (limb_t)carry;
*rr = word;
carry = ((word >> 1) ^ 0x01) & 0x01;
rr = result;
for (index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry = ((dlimb_t)(*rr)) - carry;
*rr++ = (limb_t)carry;
carry = (carry >> LIMB_BITS) & 0x01;
}
*(--rr) &= 0x01;
#endif
}
/**
* \brief Adds two values and then reduces the result modulo 2^521 - 1.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size
* and can be the same array as \a x or \a y.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size and less than 2^521 - 1.
* \param y The second value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size and less than 2^521 - 1.
*/
void P521::add(limb_t *result, const limb_t *x, const limb_t *y)
{
dlimb_t carry = 0;
limb_t *rr = result;
for (uint8_t posn = 0; posn < NUM_LIMBS_521BIT; ++posn) {
carry += *x++;
carry += *y++;
*rr++ = (limb_t)carry;
carry >>= LIMB_BITS;
}
reduceQuick(result);
}
/**
* \brief Subtracts two values and then reduces the result modulo 2^521 - 1.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size
* and can be the same array as \a x or \a y.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size and less than 2^521 - 1.
* \param y The second value to multiply, which must be NUM_LIMBS_521BIT
* limbs in size and less than 2^521 - 1.
*/
void P521::sub(limb_t *result, const limb_t *x, const limb_t *y)
{
dlimb_t borrow;
uint8_t posn;
limb_t *rr = result;
// Subtract y from x to generate the intermediate result.
borrow = 0;
for (posn = 0; posn < NUM_LIMBS_521BIT; ++posn) {
borrow = ((dlimb_t)(*x++)) - (*y++) - ((borrow >> LIMB_BITS) & 0x01);
*rr++ = (limb_t)borrow;
}
// If we had a borrow, then the result has gone negative and we
// have to add 2^521 - 1 to the result to make it positive again.
// The top bits of "borrow" will be all 1's if there is a borrow
// or it will be all 0's if there was no borrow. Easiest is to
// conditionally subtract 1 and then mask off the high bits.
rr = result;
borrow = (borrow >> LIMB_BITS) & 1U;
borrow = ((dlimb_t)(*rr)) - borrow;
*rr++ = (limb_t)borrow;
for (posn = 1; posn < NUM_LIMBS_521BIT; ++posn) {
borrow = ((dlimb_t)(*rr)) - ((borrow >> LIMB_BITS) & 0x01);
*rr++ = (limb_t)borrow;
}
#if BIGNUMBER_LIMB_8BIT
*(--rr) &= 0x01;
#else
*(--rr) &= 0x1FF;
#endif
}
/**
* \brief Doubles a point represented in Jacobian co-ordinates.
*
* \param xout The X value for the result.
* \param yout The Y value for the result.
* \param zout The Z value for the result.
* \param xin The X value for the point to be doubled.
* \param yin The Y value for the point to be doubled.
* \param zin The Z value for the point to be doubled.
*
* The output parameters can be the same as the input parameters
* to double in-place.
*
* Reference: http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
*/
void P521::dblPoint(limb_t *xout, limb_t *yout, limb_t *zout,
const limb_t *xin, const limb_t *yin,
const limb_t *zin)
{
limb_t alpha[NUM_LIMBS_521BIT];
limb_t beta[NUM_LIMBS_521BIT];
limb_t gamma[NUM_LIMBS_521BIT];
limb_t delta[NUM_LIMBS_521BIT];
limb_t tmp[NUM_LIMBS_521BIT];
// Double the point. If it is the point at infinity (z = 0),
// then zout will still be zero at the end of this process so
// we don't need any special handling for that case.
square(delta, zin); // delta = z^2
square(gamma, yin); // gamma = y^2
mul(beta, xin, gamma); // beta = x * gamma
sub(tmp, xin, delta); // alpha = 3 * (x - delta) * (x + delta)
mulLiteral(alpha, tmp, 3);
add(tmp, xin, delta);
mul(alpha, alpha, tmp);
square(xout, alpha); // xout = alpha^2 - 8 * beta
mulLiteral(tmp, beta, 8);
sub(xout, xout, tmp);
add(zout, yin, zin); // zout = (y + z)^2 - gamma - delta
square(zout, zout);
sub(zout, zout, gamma);
sub(zout, zout, delta);
mulLiteral(yout, beta, 4);// yout = alpha * (4 * beta - xout) - 8 * gamma^2
sub(yout, yout, xout);
mul(yout, alpha, yout);
square(gamma, gamma);
mulLiteral(gamma, gamma, 8);
sub(yout, yout, gamma);
// Clean up.
strict_clean(alpha);
strict_clean(beta);
strict_clean(gamma);
strict_clean(delta);
strict_clean(tmp);
}
/**
* \brief Adds two curve points, one represented in Jacobian co-ordinates,
* and the other represented in affine co-ordinates.
*
* \param xout The X value for the result.
* \param yout The Y value for the result.
* \param zout The Z value for the result.
* \param x1 The X value for the first point to add.
* \param y1 The Y value for the first point to add.
* \param z1 The Z value for the first point to add.
* \param x2 The X value for the second point to add.
* \param y2 The Y value for the second point to add.
*
* The output parameters must not overlap with either of the inputs.
*
* The Z value of the second point is implicitly assumed to be 1.
*
* Reference: http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
*/
void P521::addPoint(limb_t *xout, limb_t *yout, limb_t *zout,
const limb_t *x1, const limb_t *y1,
const limb_t *z1, const limb_t *x2,
const limb_t *y2)
{
limb_t z1z1[NUM_LIMBS_521BIT];
limb_t u2[NUM_LIMBS_521BIT];
limb_t s2[NUM_LIMBS_521BIT];
limb_t h[NUM_LIMBS_521BIT];
limb_t i[NUM_LIMBS_521BIT];
limb_t j[NUM_LIMBS_521BIT];
limb_t r[NUM_LIMBS_521BIT];
limb_t v[NUM_LIMBS_521BIT];
// Determine if the first value is the point-at-infinity identity element.
// The second z value is always 1 so it cannot be the point-at-infinity.
limb_t p1IsIdentity = BigNumberUtil::isZero(z1, NUM_LIMBS_521BIT);
// Multiply the points, assuming that z2 = 1.
square(z1z1, z1); // z1z1 = z1^2
mul(u2, x2, z1z1); // u2 = x2 * z1z1
mul(s2, y2, z1); // s2 = y2 * z1 * z1z1
mul(s2, s2, z1z1);
sub(h, u2, x1); // h = u2 - x1
mulLiteral(i, h, 2); // i = (2 * h)^2
square(i, i);
sub(r, s2, y1); // r = 2 * (s2 - y1)
add(r, r, r);
mul(j, h, i); // j = h * i
mul(v, x1, i); // v = x1 * i
square(xout, r); // xout = r^2 - j - 2 * v
sub(xout, xout, j);
sub(xout, xout, v);
sub(xout, xout, v);
sub(yout, v, xout); // yout = r * (v - xout) - 2 * y1 * j
mul(yout, r, yout);
mul(j, y1, j);
sub(yout, yout, j);
sub(yout, yout, j);
mul(zout, z1, h); // zout = 2 * z1 * h
add(zout, zout, zout);
// Select the answer to return. If (x1, y1, z1) was the identity,
// then the answer is (x2, y2, z2). Otherwise it is (xout, yout, zout).
// Conditionally move the second argument over the output if necessary.
cmove(p1IsIdentity, xout, x2);
cmove(p1IsIdentity, yout, y2);
cmove1(p1IsIdentity, zout); // z2 = 1
// Clean up.
strict_clean(z1z1);
strict_clean(u2);
strict_clean(s2);
strict_clean(h);
strict_clean(i);
strict_clean(j);
strict_clean(r);
strict_clean(v);
}
/**
* \brief Conditionally moves \a y into \a x if a selection value is non-zero.
*
* \param select Non-zero to move \a y into \a x, zero to leave \a x unchanged.
* \param x The destination to move into.
* \param y The value to conditionally move.
*
* The move is performed in a way that it should take the same amount of
* time irrespective of the value of \a select.
*
* \sa cmove1()
*/
void P521::cmove(limb_t select, limb_t *x, const limb_t *y)
{
uint8_t posn;
limb_t dummy;
limb_t sel;
// Turn "select" into an all-zeroes or all-ones mask. We don't care
// which bit or bits is set in the original "select" value.
sel = (limb_t)(((((dlimb_t)1) << LIMB_BITS) - select) >> LIMB_BITS);
--sel;
// Move y into x based on "select".
for (posn = 0; posn < NUM_LIMBS_521BIT; ++posn) {
dummy = sel & (*x ^ *y++);
*x++ ^= dummy;
}
}
/**
* \brief Conditionally moves 1 into \a x if a selection value is non-zero.
*
* \param select Non-zero to move 1 into \a x, zero to leave \a x unchanged.
* \param x The destination to move into.
*
* The move is performed in a way that it should take the same amount of
* time irrespective of the value of \a select.
*
* \sa cmove()
*/
void P521::cmove1(limb_t select, limb_t *x)
{
uint8_t posn;
limb_t dummy;
limb_t sel;
// Turn "select" into an all-zeroes or all-ones mask. We don't care
// which bit or bits is set in the original "select" value.
sel = (limb_t)(((((dlimb_t)1) << LIMB_BITS) - select) >> LIMB_BITS);
--sel;
// Move 1 into x based on "select".
dummy = sel & (*x ^ 1);
*x++ ^= dummy;
for (posn = 1; posn < NUM_LIMBS_521BIT; ++posn) {
dummy = sel & *x;
*x++ ^= dummy;
}
}
/**
* \brief Computes the reciprocal of a number modulo 2^521 - 1.
*
* \param result The result as a array of NUM_LIMBS_521BIT limbs in size.
* This cannot be the same array as \a x.
* \param x The number to compute the reciprocal for, also NUM_LIMBS_521BIT
* limbs in size.
*/
void P521::recip(limb_t *result, const limb_t *x)
{
limb_t t1[NUM_LIMBS_521BIT];
// The reciprocal is the same as x ^ (p - 2) where p = 2^521 - 1.
// The big-endian hexadecimal expansion of (p - 2) is:
// 01FF FFFFFFF FFFFFFFF ... FFFFFFFF FFFFFFFD
//
// The naive implementation needs to do 2 multiplications per 1 bit and
// 1 multiplication per 0 bit. We can improve upon this by creating a
// pattern 1111 and then shifting and multiplying to create 11111111,
// and then 1111111111111111, and so on for the top 512-bits.
// Build a 4-bit pattern 1111 in the result.
square(result, x);
mul(result, result, x);
square(result, result);
mul(result, result, x);
square(result, result);
mul(result, result, x);
// Shift and multiply by increasing powers of two. This turns
// 1111 into 11111111, and then 1111111111111111, and so on.
for (size_t power = 4; power <= 256; power <<= 1) {
square(t1, result);
for (size_t temp = 1; temp < power; ++temp)
square(t1, t1);
mul(result, result, t1);
}
// Handle the 9 lowest bits of (p - 2), 111111101, from highest to lowest.
for (uint8_t index = 0; index < 7; ++index) {
square(result, result);
mul(result, result, x);
}
square(result, result);
square(result, result);
mul(result, result, x);
// Clean up.
clean(t1);
}
/**
* \brief Reduces a number modulo q.
*
* \param result The result array, which must be NUM_LIMBS_521BIT limbs in size.
* \param r The value to reduce, which must be NUM_LIMBS_1042BIT limbs in size.
*
* It is allowed for \a result to be the same as \a r.
*/
void P521::reduceQ(limb_t *result, const limb_t *r)
{
// Algorithm from: http://en.wikipedia.org/wiki/Barrett_reduction
//
// We assume that r is less than or equal to (q - 1)^2.
//
// We want to compute result = r mod q. Find the smallest k such
// that 2^k > q. In our case, k = 521. Then set m = floor(4^k / q)
// and let r = r - q * floor(m * r / 4^k). This will be the result
// or it will be at most one subtraction of q away from the result.
//
// Note: m is a 522-bit number, which fits in the same number of limbs
// as a 521-bit number assuming that limbs are 8 bits or more in size.
static limb_t const numM[NUM_LIMBS_521BIT] PROGMEM = {
LIMB_PAIR(0x6EC79BF7, 0x449048E1), LIMB_PAIR(0x7663B851, 0xC44A3647),
LIMB_PAIR(0x08F65A2F, 0x8033FEB7), LIMB_PAIR(0x40D06994, 0xAE79787C),
LIMB_PAIR(0x00000005, 0x00000000), LIMB_PAIR(0x00000000, 0x00000000),
LIMB_PAIR(0x00000000, 0x00000000), LIMB_PAIR(0x00000000, 0x00000000),
LIMB_PARTIAL(0x200)
};
limb_t temp[NUM_LIMBS_1042BIT + NUM_LIMBS_521BIT];
limb_t temp2[NUM_LIMBS_521BIT];
// Multiply r by m.
BigNumberUtil::mul_P(temp, r, NUM_LIMBS_1042BIT, numM, NUM_LIMBS_521BIT);
// Compute (m * r / 4^521) = (m * r / 2^1042).
#if BIGNUMBER_LIMB_8BIT || BIGNUMBER_LIMB_16BIT
dlimb_t carry = temp[NUM_LIMBS_BITS(1040)] >> 2;
for (uint8_t index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry += ((dlimb_t)(temp[NUM_LIMBS_BITS(1040) + index + 1])) << (LIMB_BITS - 2);
temp2[index] = (limb_t)carry;
carry >>= LIMB_BITS;
}
#elif BIGNUMBER_LIMB_32BIT || BIGNUMBER_LIMB_64BIT
dlimb_t carry = temp[NUM_LIMBS_BITS(1024)] >> 18;
for (uint8_t index = 0; index < NUM_LIMBS_521BIT; ++index) {
carry += ((dlimb_t)(temp[NUM_LIMBS_BITS(1024) + index + 1])) << (LIMB_BITS - 18);
temp2[index] = (limb_t)carry;
carry >>= LIMB_BITS;
}
#endif
// Multiply (m * r) / 2^1042 by q and subtract it from r.
// We can ignore the high words of the subtraction result
// because they will all turn into zero after the subtraction.
BigNumberUtil::mul_P(temp, temp2, NUM_LIMBS_521BIT,
P521_q, NUM_LIMBS_521BIT);
BigNumberUtil::sub(result, r, temp, NUM_LIMBS_521BIT);
// Perform a trial subtraction of q from the result to reduce it.
BigNumberUtil::reduceQuick_P(result, result, P521_q, NUM_LIMBS_521BIT);
// Clean up and exit.
clean(temp);
clean(temp2);
}
/**
* \brief Multiplies two values and then reduces the result modulo q.
*
* \param result The result, which must be NUM_LIMBS_521BIT limbs in size
* and can be the same array as \a x or \a y.
* \param x The first value to multiply, which must be NUM_LIMBS_521BIT limbs
* in size and less than q.
* \param y The second value to multiply, which must be NUM_LIMBS_521BIT limbs
* in size and less than q. This can be the same array as \a x.
*/
void P521::mulQ(limb_t *result, const limb_t *x, const limb_t *y)
{
limb_t temp[NUM_LIMBS_1042BIT];
mulNoReduce(temp, x, y);
reduceQ(result, temp);
strict_clean(temp);
}
/**
* \brief Computes the reciprocal of a number modulo q.
*
* \param result The result as a array of NUM_LIMBS_521BIT limbs in size.
* This cannot be the same array as \a x.
* \param x The number to compute the reciprocal for, also NUM_LIMBS_521BIT
* limbs in size.
*/
void P521::recipQ(limb_t *result, const limb_t *x)
{
// Bottom 265 bits of q - 2. The top 256 bits are all-1's.
static limb_t const P521_q_m2[] PROGMEM = {
LIMB_PAIR(0x91386407, 0xbb6fb71e), LIMB_PAIR(0x899c47ae, 0x3bb5c9b8),
LIMB_PAIR(0xf709a5d0, 0x7fcc0148), LIMB_PAIR(0xbf2f966b, 0x51868783),
LIMB_PARTIAL(0x1fa)
};
// Raise x to the power of q - 2, mod q. We start with the top
// 256 bits which are all-1's, using a similar technique to recip().
limb_t t1[NUM_LIMBS_521BIT];
mulQ(result, x, x);
mulQ(result, result, x);
mulQ(result, result, result);
mulQ(result, result, x);
mulQ(result, result, result);
mulQ(result, result, x);
for (size_t power = 4; power <= 128; power <<= 1) {
mulQ(t1, result, result);
for (size_t temp = 1; temp < power; ++temp)
mulQ(t1, t1, t1);
mulQ(result, result, t1);
}
clean(t1);
// Deal with the bottom 265 bits from highest to lowest. Square for
// each bit and multiply in x whenever there is a 1 bit. The timing
// is based on the publicly-known constant q - 2, not on the value of x.
size_t bit = 265;
while (bit > 0) {
--bit;
mulQ(result, result, result);
if (pgm_read_limb(&(P521_q_m2[bit / LIMB_BITS])) &
(((limb_t)1) << (bit % LIMB_BITS))) {
mulQ(result, result, x);
}
}
}
/**
* \brief Generates a k value using the algorithm from RFC 6979.
*
* \param k The value to generate.
* \param hm The hashed message formatted ready to be signed.
* \param x The private key to sign with.
* \param hash The hash algorithm to use.
* \param count Iteration counter for generating new values of k when the
* previous one is rejected.
*/
void P521::generateK(uint8_t k[66], const uint8_t hm[66],
const uint8_t x[66], Hash *hash, uint64_t count)
{
size_t hlen = hash->hashSize();
uint8_t V[64];
uint8_t K[64];
uint8_t marker;
// If for some reason a hash function was supplied with more than
// 512 bits of output, truncate hash values to the first 512 bits.
// We cannot support more than this yet.
if (hlen > 64)
hlen = 64;
// RFC 6979, Section 3.2, Step a. Hash the message, reduce modulo q,
// and produce an octet string the same length as q, bits2octets(H(m)).
// We support hashes up to 512 bits and q is a 521-bit number, so "hm"
// is already the bits2octets(H(m)) value that we need.
// Steps b and c. Set V to all-ones and K to all-zeroes.
memset(V, 0x01, hlen);
memset(K, 0x00, hlen);
// Step d. K = HMAC_K(V || 0x00 || x || hm). We make a small
// modification here to append the count value if it is non-zero.
// We use this to generate a new k if we have to re-enter this
// function because the previous one was rejected by sign().
// This is slightly different to RFC 6979 which says that the
// loop in step h below should be continued. That code path is
// difficult to access, so instead modify K and V in steps d and f.
// This alternative construction is compatible with the second
// variant described in section 3.6 of RFC 6979.
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
marker = 0x00;
hash->update(&marker, 1);
hash->update(x, 66);
hash->update(hm, 66);
if (count)
hash->update(&count, sizeof(count));
hash->finalizeHMAC(K, hlen, K, hlen);
// Step e. V = HMAC_K(V)
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
hash->finalizeHMAC(K, hlen, V, hlen);
// Step f. K = HMAC_K(V || 0x01 || x || hm)
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
marker = 0x01;
hash->update(&marker, 1);
hash->update(x, 66);
hash->update(hm, 66);
if (count)
hash->update(&count, sizeof(count));
hash->finalizeHMAC(K, hlen, K, hlen);
// Step g. V = HMAC_K(V)
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
hash->finalizeHMAC(K, hlen, V, hlen);
// Step h. Generate candidate k values until we find what we want.
for (;;) {
// Step h.1 and h.2. Generate a string of 66 bytes in length.
// T = empty
// while (len(T) < 66)
// V = HMAC_K(V)
// T = T || V
size_t posn = 0;
while (posn < 66) {
size_t temp = 66 - posn;
if (temp > hlen)
temp = hlen;
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
hash->finalizeHMAC(K, hlen, V, hlen);
memcpy(k + posn, V, temp);
posn += temp;
}
// Step h.3. k = bits2int(T) and exit the loop if k is not in
// the range 1 to q - 1. Note: We have to extract the 521 most
// significant bits of T, which means shifting it right by seven
// bits to put it into the correct form.
for (posn = 65; posn > 0; --posn)
k[posn] = (k[posn - 1] << 1) | (k[posn] >> 7);
k[0] >>= 7;
if (isValidPrivateKey(k))
break;
// Generate new K and V values and try again.
// K = HMAC_K(V || 0x00)
// V = HMAC_K(V)
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
marker = 0x00;
hash->update(&marker, 1);
hash->finalizeHMAC(K, hlen, K, hlen);
hash->resetHMAC(K, hlen);
hash->update(V, hlen);
hash->finalizeHMAC(K, hlen, V, hlen);
}
// Clean up.
clean(V);
clean(K);
}
/**
* \brief Generates a k value using the algorithm from RFC 6979.
*
* \param k The value to generate.
* \param hm The hashed message formatted ready to be signed.
* \param x The private key to sign with.
* \param count Iteration counter for generating new values of k when the
* previous one is rejected.
*
* This override uses SHA512 to generate k values. It is used when
* sign() was not passed an explicit hash object by the application.
*/
void P521::generateK(uint8_t k[66], const uint8_t hm[66],
const uint8_t x[66], uint64_t count)
{
SHA512 hash;
generateK(k, hm, x, &hash, count);
}
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