P521.cpp

/*
 * 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);
}