src/pkg/crypto/rsa/rsa.go - go - Git at Google

 // Copyright 2009 The Go Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.

 // This package implements RSA encryption as specified in PKCS#1.
 package rsa

 // TODO(agl): Add support for PSS padding.

 import (
 	"big"
 	"crypto/subtle"
 	"hash"
 	"io"
 	"os"
 	"sync"
 )

 var bigZero = big.NewInt(0)
 var bigOne = big.NewInt(1)

 // randomPrime returns a number, p, of the given size, such that p is prime
 // with high probability.
 func randomPrime(rand io.Reader, bits int) (p *big.Int, err os.Error) {
 	if bits < 1 {
 		err = os.EINVAL
 	}

 	bytes := make([]byte, (bits+7)/8)
 	p = new(big.Int)

 	for {
 		_, err = io.ReadFull(rand, bytes)
 		if err != nil {
 			return
 		}

 		// Don't let the value be too small.
 		bytes[0] |= 0x80
 		// Make the value odd since an even number this large certainly isn't prime.
 		bytes[len(bytes)-1] |= 1

 		p.SetBytes(bytes)
 		if big.ProbablyPrime(p, 20) {
 			return
 		}
 	}

 	return
 }

 // randomNumber returns a uniform random value in [0, max).
 func randomNumber(rand io.Reader, max *big.Int) (n *big.Int, err os.Error) {
 	k := (max.BitLen() + 7) / 8

 	// r is the number of bits in the used in the most significant byte of
 	// max.
 	r := uint(max.BitLen() % 8)
 	if r == 0 {
 		r = 8
 	}

 	bytes := make([]byte, k)
 	n = new(big.Int)

 	for {
 		_, err = io.ReadFull(rand, bytes)
 		if err != nil {
 			return
 		}

 		// Clear bits in the first byte to increase the probability
 		// that the candidate is < max.
 		bytes[0] &= uint8(int(1<<r) - 1)

 		n.SetBytes(bytes)
 		if n.Cmp(max) < 0 {
 			return
 		}
 	}

 	return
 }

 // A PublicKey represents the public part of an RSA key.
 type PublicKey struct {
 	N *big.Int // modulus
 	E int      // public exponent
 }

 // A PrivateKey represents an RSA key
 type PrivateKey struct {
 	PublicKey          // public part.
 	D         *big.Int // private exponent
 	P, Q      *big.Int // prime factors of N

 	rwMutex sync.RWMutex // protects the following
 	dP, dQ  *big.Int     // D mod (P-1) (or mod Q-1)
 	qInv    *big.Int     // q^-1 mod p
 }

 // Validate performs basic sanity checks on the key.
 // It returns nil if the key is valid, or else an os.Error describing a problem.

 func (priv *PrivateKey) Validate() os.Error {
 	// Check that p and q are prime. Note that this is just a sanity
 	// check. Since the random witnesses chosen by ProbablyPrime are
 	// deterministic, given the candidate number, it's easy for an attack
 	// to generate composites that pass this test.
 	if !big.ProbablyPrime(priv.P, 20) {
 		return os.ErrorString("P is composite")
 	}
 	if !big.ProbablyPrime(priv.Q, 20) {
 		return os.ErrorString("Q is composite")
 	}

 	// Check that p*q == n.
 	modulus := new(big.Int).Mul(priv.P, priv.Q)
 	if modulus.Cmp(priv.N) != 0 {
 		return os.ErrorString("invalid modulus")
 	}
 	// Check that e and totient(p, q) are coprime.
 	pminus1 := new(big.Int).Sub(priv.P, bigOne)
 	qminus1 := new(big.Int).Sub(priv.Q, bigOne)
 	totient := new(big.Int).Mul(pminus1, qminus1)
 	e := big.NewInt(int64(priv.E))
 	gcd := new(big.Int)
 	x := new(big.Int)
 	y := new(big.Int)
 	big.GcdInt(gcd, x, y, totient, e)
 	if gcd.Cmp(bigOne) != 0 {
 		return os.ErrorString("invalid public exponent E")
 	}
 	// Check that de ≡ 1 (mod totient(p, q))
 	de := new(big.Int).Mul(priv.D, e)
 	de.Mod(de, totient)
 	if de.Cmp(bigOne) != 0 {
 		return os.ErrorString("invalid private exponent D")
 	}
 	return nil
 }

 // GenerateKeyPair generates an RSA keypair of the given bit size.
 func GenerateKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
 	priv = new(PrivateKey)
 	// Smaller public exponents lead to faster public key
 	// operations. Since the exponent must be coprime to
 	// (p-1)(q-1), the smallest possible value is 3. Some have
 	// suggested that a larger exponent (often 2**16+1) be used
 	// since previous implementation bugs[1] were avoided when this
 	// was the case. However, there are no current reasons not to use
 	// small exponents.
 	// [1] http://marc.info/?l=cryptography&m=115694833312008&w=2
 	priv.E = 3

 	pminus1 := new(big.Int)
 	qminus1 := new(big.Int)
 	totient := new(big.Int)

 	for {
 		p, err := randomPrime(rand, bits/2)
 		if err != nil {
 			return nil, err
 		}

 		q, err := randomPrime(rand, bits/2)
 		if err != nil {
 			return nil, err
 		}

 		if p.Cmp(q) == 0 {
 			continue
 		}

 		n := new(big.Int).Mul(p, q)
 		pminus1.Sub(p, bigOne)
 		qminus1.Sub(q, bigOne)
 		totient.Mul(pminus1, qminus1)

 		g := new(big.Int)
 		priv.D = new(big.Int)
 		y := new(big.Int)
 		e := big.NewInt(int64(priv.E))
 		big.GcdInt(g, priv.D, y, e, totient)

 		if g.Cmp(bigOne) == 0 {
 			priv.D.Add(priv.D, totient)
 			priv.P = p
 			priv.Q = q
 			priv.N = n

 			break
 		}
 	}

 	return
 }

 // incCounter increments a four byte, big-endian counter.
 func incCounter(c *[4]byte) {
 	if c[3]++; c[3] != 0 {
 		return
 	}
 	if c[2]++; c[2] != 0 {
 		return
 	}
 	if c[1]++; c[1] != 0 {
 		return
 	}
 	c[0]++
 }

 // mgf1XOR XORs the bytes in out with a mask generated using the MGF1 function
 // specified in PKCS#1 v2.1.
 func mgf1XOR(out []byte, hash hash.Hash, seed []byte) {
 	var counter [4]byte

 	done := 0
 	for done < len(out) {
 		hash.Write(seed)
 		hash.Write(counter[0:4])
 		digest := hash.Sum()
 		hash.Reset()

 		for i := 0; i < len(digest) && done < len(out); i++ {
 			out[done] ^= digest[i]
 			done++
 		}
 		incCounter(&counter)
 	}
 }

 // MessageTooLongError is returned when attempting to encrypt a message which
 // is too large for the size of the public key.
 type MessageTooLongError struct{}

 func (MessageTooLongError) String() string {
 	return "message too long for RSA public key size"
 }

 func encrypt(c *big.Int, pub *PublicKey, m *big.Int) *big.Int {
 	e := big.NewInt(int64(pub.E))
 	c.Exp(m, e, pub.N)
 	return c
 }

 // EncryptOAEP encrypts the given message with RSA-OAEP.
 // The message must be no longer than the length of the public modulus less
 // twice the hash length plus 2.
 func EncryptOAEP(hash hash.Hash, rand io.Reader, pub *PublicKey, msg []byte, label []byte) (out []byte, err os.Error) {
 	hash.Reset()
 	k := (pub.N.BitLen() + 7) / 8
 	if len(msg) > k-2*hash.Size()-2 {
 		err = MessageTooLongError{}
 		return
 	}

 	hash.Write(label)
 	lHash := hash.Sum()
 	hash.Reset()

 	em := make([]byte, k)
 	seed := em[1 : 1+hash.Size()]
 	db := em[1+hash.Size():]

 	copy(db[0:hash.Size()], lHash)
 	db[len(db)-len(msg)-1] = 1
 	copy(db[len(db)-len(msg):], msg)

 	_, err = io.ReadFull(rand, seed)
 	if err != nil {
 		return
 	}

 	mgf1XOR(db, hash, seed)
 	mgf1XOR(seed, hash, db)

 	m := new(big.Int)
 	m.SetBytes(em)
 	c := encrypt(new(big.Int), pub, m)
 	out = c.Bytes()

 	if len(out) < k {
 		// If the output is too small, we need to left-pad with zeros.
 		t := make([]byte, k)
 		copy(t[k-len(out):], out)
 		out = t
 	}

 	return
 }

 // A DecryptionError represents a failure to decrypt a message.
 // It is deliberately vague to avoid adaptive attacks.
 type DecryptionError struct{}

 func (DecryptionError) String() string { return "RSA decryption error" }

 // A VerificationError represents a failure to verify a signature.
 // It is deliberately vague to avoid adaptive attacks.
 type VerificationError struct{}

 func (VerificationError) String() string { return "RSA verification error" }

 // modInverse returns ia, the inverse of a in the multiplicative group of prime
 // order n. It requires that a be a member of the group (i.e. less than n).
 func modInverse(a, n *big.Int) (ia *big.Int, ok bool) {
 	g := new(big.Int)
 	x := new(big.Int)
 	y := new(big.Int)
 	big.GcdInt(g, x, y, a, n)
 	if g.Cmp(bigOne) != 0 {
 		// In this case, a and n aren't coprime and we cannot calculate
 		// the inverse. This happens because the values of n are nearly
 		// prime (being the product of two primes) rather than truly
 		// prime.
 		return
 	}

 	if x.Cmp(bigOne) < 0 {
 		// 0 is not the multiplicative inverse of any element so, if x
 		// < 1, then x is negative.
 		x.Add(x, n)
 	}

 	return x, true
 }

 // precompute performs some calculations that speed up private key operations
 // in the future.
 func (priv *PrivateKey) precompute() {
 	priv.dP = new(big.Int).Sub(priv.P, bigOne)
 	priv.dP.Mod(priv.D, priv.dP)

 	priv.dQ = new(big.Int).Sub(priv.Q, bigOne)
 	priv.dQ.Mod(priv.D, priv.dQ)

 	priv.qInv = new(big.Int).ModInverse(priv.Q, priv.P)
 }

 // decrypt performs an RSA decryption, resulting in a plaintext integer. If a
 // random source is given, RSA blinding is used.
 func decrypt(rand io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err os.Error) {
 	// TODO(agl): can we get away with reusing blinds?
 	if c.Cmp(priv.N) > 0 {
 		err = DecryptionError{}
 		return
 	}

 	var ir *big.Int
 	if rand != nil {
 		// Blinding enabled. Blinding involves multiplying c by r^e.
 		// Then the decryption operation performs (m^e * r^e)^d mod n
 		// which equals mr mod n. The factor of r can then be removed
 		// by multipling by the multiplicative inverse of r.

 		var r *big.Int

 		for {
 			r, err = randomNumber(rand, priv.N)
 			if err != nil {
 				return
 			}
 			if r.Cmp(bigZero) == 0 {
 				r = bigOne
 			}
 			var ok bool
 			ir, ok = modInverse(r, priv.N)
 			if ok {
 				break
 			}
 		}
 		bigE := big.NewInt(int64(priv.E))
 		rpowe := new(big.Int).Exp(r, bigE, priv.N)
 		c.Mul(c, rpowe)
 		c.Mod(c, priv.N)
 	}

 	priv.rwMutex.RLock()

 	if priv.dP == nil && priv.P != nil {
 		priv.rwMutex.RUnlock()
 		priv.rwMutex.Lock()
 		if priv.dP == nil && priv.P != nil {
 			priv.precompute()
 		}
 		priv.rwMutex.Unlock()
 		priv.rwMutex.RLock()
 	}

 	if priv.dP == nil {
 		m = new(big.Int).Exp(c, priv.D, priv.N)
 	} else {
 		// We have the precalculated values needed for the CRT.
 		m = new(big.Int).Exp(c, priv.dP, priv.P)
 		m2 := new(big.Int).Exp(c, priv.dQ, priv.Q)
 		m.Sub(m, m2)
 		if m.Sign() < 0 {
 			m.Add(m, priv.P)
 		}
 		m.Mul(m, priv.qInv)
 		m.Mod(m, priv.P)
 		m.Mul(m, priv.Q)
 		m.Add(m, m2)
 	}

 	priv.rwMutex.RUnlock()

 	if ir != nil {
 		// Unblind.
 		m.Mul(m, ir)
 		m.Mod(m, priv.N)
 	}

 	return
 }

 // DecryptOAEP decrypts ciphertext using RSA-OAEP.
 // If rand != nil, DecryptOAEP uses RSA blinding to avoid timing side-channel attacks.
 func DecryptOAEP(hash hash.Hash, rand io.Reader, priv *PrivateKey, ciphertext []byte, label []byte) (msg []byte, err os.Error) {
 	k := (priv.N.BitLen() + 7) / 8
 	if len(ciphertext) > k ||
 		k < hash.Size()*2+2 {
 		err = DecryptionError{}
 		return
 	}

 	c := new(big.Int).SetBytes(ciphertext)

 	m, err := decrypt(rand, priv, c)
 	if err != nil {
 		return
 	}

 	hash.Write(label)
 	lHash := hash.Sum()
 	hash.Reset()

 	// Converting the plaintext number to bytes will strip any
 	// leading zeros so we may have to left pad. We do this unconditionally
 	// to avoid leaking timing information. (Although we still probably
 	// leak the number of leading zeros. It's not clear that we can do
 	// anything about this.)
 	em := leftPad(m.Bytes(), k)

 	firstByteIsZero := subtle.ConstantTimeByteEq(em[0], 0)

 	seed := em[1 : hash.Size()+1]
 	db := em[hash.Size()+1:]

 	mgf1XOR(seed, hash, db)
 	mgf1XOR(db, hash, seed)

 	lHash2 := db[0:hash.Size()]

 	// We have to validate the plaintext in constant time in order to avoid
 	// attacks like: J. Manger. A Chosen Ciphertext Attack on RSA Optimal
 	// Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1
 	// v2.0. In J. Kilian, editor, Advances in Cryptology.
 	lHash2Good := subtle.ConstantTimeCompare(lHash, lHash2)

 	// The remainder of the plaintext must be zero or more 0x00, followed
 	// by 0x01, followed by the message.
 	//   lookingForIndex: 1 iff we are still looking for the 0x01
 	//   index: the offset of the first 0x01 byte
 	//   invalid: 1 iff we saw a non-zero byte before the 0x01.
 	var lookingForIndex, index, invalid int
 	lookingForIndex = 1
 	rest := db[hash.Size():]

 	for i := 0; i < len(rest); i++ {
 		equals0 := subtle.ConstantTimeByteEq(rest[i], 0)
 		equals1 := subtle.ConstantTimeByteEq(rest[i], 1)
 		index = subtle.ConstantTimeSelect(lookingForIndex&equals1, i, index)
 		lookingForIndex = subtle.ConstantTimeSelect(equals1, 0, lookingForIndex)
 		invalid = subtle.ConstantTimeSelect(lookingForIndex&^equals0, 1, invalid)
 	}

 	if firstByteIsZero&lHash2Good&^invalid&^lookingForIndex != 1 {
 		err = DecryptionError{}
 		return
 	}

 	msg = rest[index+1:]
 	return
 }

 // leftPad returns a new slice of length size. The contents of input are right
 // aligned in the new slice.
 func leftPad(input []byte, size int) (out []byte) {
 	n := len(input)
 	if n > size {
 		n = size
 	}
 	out = make([]byte, size)
 	copy(out[len(out)-n:], input)
 	return
 }
	// Copyright 2009 The Go Authors. All rights reserved.
	// Use of this source code is governed by a BSD-style
	// license that can be found in the LICENSE file.

	// This package implements RSA encryption as specified in PKCS#1.
	package rsa

	// TODO(agl): Add support for PSS padding.

	import (
	"big"
	"crypto/subtle"
	"hash"
	"io"
	"os"
	"sync"
	)

	var bigZero = big.NewInt(0)
	var bigOne = big.NewInt(1)

	// randomPrime returns a number, p, of the given size, such that p is prime
	// with high probability.
	func randomPrime(rand io.Reader, bits int) (p *big.Int, err os.Error) {
	if bits < 1 {
	err = os.EINVAL
	}

	bytes := make([]byte, (bits+7)/8)
	p = new(big.Int)

	for {
	_, err = io.ReadFull(rand, bytes)
	if err != nil {
	return
	}

	// Don't let the value be too small.
	bytes[0] \|= 0x80
	// Make the value odd since an even number this large certainly isn't prime.
	bytes[len(bytes)-1] \|= 1

	p.SetBytes(bytes)
	if big.ProbablyPrime(p, 20) {
	return
	}
	}

	return
	}

	// randomNumber returns a uniform random value in [0, max).
	func randomNumber(rand io.Reader, max big.Int) (n big.Int, err os.Error) {
	k := (max.BitLen() + 7) / 8

	// r is the number of bits in the used in the most significant byte of
	// max.
	r := uint(max.BitLen() % 8)
	if r == 0 {
	r = 8
	}

	bytes := make([]byte, k)
	n = new(big.Int)

	for {
	_, err = io.ReadFull(rand, bytes)
	if err != nil {
	return
	}

	// Clear bits in the first byte to increase the probability
	// that the candidate is < max.
	bytes[0] &= uint8(int(1<<r) - 1)

	n.SetBytes(bytes)
	if n.Cmp(max) < 0 {
	return
	}
	}

	return
	}

	// A PublicKey represents the public part of an RSA key.
	type PublicKey struct {
	N *big.Int // modulus
	E int // public exponent
	}

	// A PrivateKey represents an RSA key
	type PrivateKey struct {
	PublicKey // public part.
	D *big.Int // private exponent
	P, Q *big.Int // prime factors of N

	rwMutex sync.RWMutex // protects the following
	dP, dQ *big.Int // D mod (P-1) (or mod Q-1)
	qInv *big.Int // q^-1 mod p
	}

	// Validate performs basic sanity checks on the key.
	// It returns nil if the key is valid, or else an os.Error describing a problem.

	func (priv *PrivateKey) Validate() os.Error {
	// Check that p and q are prime. Note that this is just a sanity
	// check. Since the random witnesses chosen by ProbablyPrime are
	// deterministic, given the candidate number, it's easy for an attack
	// to generate composites that pass this test.
	if !big.ProbablyPrime(priv.P, 20) {
	return os.ErrorString("P is composite")
	}
	if !big.ProbablyPrime(priv.Q, 20) {
	return os.ErrorString("Q is composite")
	}

	// Check that p*q == n.
	modulus := new(big.Int).Mul(priv.P, priv.Q)
	if modulus.Cmp(priv.N) != 0 {
	return os.ErrorString("invalid modulus")
	}
	// Check that e and totient(p, q) are coprime.
	pminus1 := new(big.Int).Sub(priv.P, bigOne)
	qminus1 := new(big.Int).Sub(priv.Q, bigOne)
	totient := new(big.Int).Mul(pminus1, qminus1)
	e := big.NewInt(int64(priv.E))
	gcd := new(big.Int)
	x := new(big.Int)
	y := new(big.Int)
	big.GcdInt(gcd, x, y, totient, e)
	if gcd.Cmp(bigOne) != 0 {
	return os.ErrorString("invalid public exponent E")
	}
	// Check that de ≡ 1 (mod totient(p, q))
	de := new(big.Int).Mul(priv.D, e)
	de.Mod(de, totient)
	if de.Cmp(bigOne) != 0 {
	return os.ErrorString("invalid private exponent D")
	}
	return nil
	}

	// GenerateKeyPair generates an RSA keypair of the given bit size.
	func GenerateKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
	priv = new(PrivateKey)
	// Smaller public exponents lead to faster public key
	// operations. Since the exponent must be coprime to
	// (p-1)(q-1), the smallest possible value is 3. Some have
	// suggested that a larger exponent (often 2**16+1) be used
	// since previous implementation bugs[1] were avoided when this
	// was the case. However, there are no current reasons not to use
	// small exponents.
	// [1] http://marc.info/?l=cryptography&m=115694833312008&w=2
	priv.E = 3

	pminus1 := new(big.Int)
	qminus1 := new(big.Int)
	totient := new(big.Int)

	for {
	p, err := randomPrime(rand, bits/2)
	if err != nil {
	return nil, err
	}

	q, err := randomPrime(rand, bits/2)
	if err != nil {
	return nil, err
	}

	if p.Cmp(q) == 0 {
	continue
	}

	n := new(big.Int).Mul(p, q)
	pminus1.Sub(p, bigOne)
	qminus1.Sub(q, bigOne)
	totient.Mul(pminus1, qminus1)

	g := new(big.Int)
	priv.D = new(big.Int)
	y := new(big.Int)
	e := big.NewInt(int64(priv.E))
	big.GcdInt(g, priv.D, y, e, totient)

	if g.Cmp(bigOne) == 0 {
	priv.D.Add(priv.D, totient)
	priv.P = p
	priv.Q = q
	priv.N = n

	break
	}
	}

	return
	}

	// incCounter increments a four byte, big-endian counter.
	func incCounter(c *[4]byte) {
	if c[3]++; c[3] != 0 {
	return
	}
	if c[2]++; c[2] != 0 {
	return
	}
	if c[1]++; c[1] != 0 {
	return
	}
	c[0]++
	}

	// mgf1XOR XORs the bytes in out with a mask generated using the MGF1 function
	// specified in PKCS#1 v2.1.
	func mgf1XOR(out []byte, hash hash.Hash, seed []byte) {
	var counter [4]byte

	done := 0
	for done < len(out) {
	hash.Write(seed)
	hash.Write(counter[0:4])
	digest := hash.Sum()
	hash.Reset()

	for i := 0; i < len(digest) && done < len(out); i++ {
	out[done] ^= digest[i]
	done++
	}
	incCounter(&counter)
	}
	}

	// MessageTooLongError is returned when attempting to encrypt a message which
	// is too large for the size of the public key.
	type MessageTooLongError struct{}

	func (MessageTooLongError) String() string {
	return "message too long for RSA public key size"
	}

	func encrypt(c big.Int, pub PublicKey, m big.Int) big.Int {
	e := big.NewInt(int64(pub.E))
	c.Exp(m, e, pub.N)
	return c
	}

	// EncryptOAEP encrypts the given message with RSA-OAEP.
	// The message must be no longer than the length of the public modulus less
	// twice the hash length plus 2.
	func EncryptOAEP(hash hash.Hash, rand io.Reader, pub *PublicKey, msg []byte, label []byte) (out []byte, err os.Error) {
	hash.Reset()
	k := (pub.N.BitLen() + 7) / 8
	if len(msg) > k-2*hash.Size()-2 {
	err = MessageTooLongError{}
	return
	}

	hash.Write(label)
	lHash := hash.Sum()
	hash.Reset()

	em := make([]byte, k)
	seed := em[1 : 1+hash.Size()]
	db := em[1+hash.Size():]

	copy(db[0:hash.Size()], lHash)
	db[len(db)-len(msg)-1] = 1
	copy(db[len(db)-len(msg):], msg)

	_, err = io.ReadFull(rand, seed)
	if err != nil {
	return
	}

	mgf1XOR(db, hash, seed)
	mgf1XOR(seed, hash, db)

	m := new(big.Int)
	m.SetBytes(em)
	c := encrypt(new(big.Int), pub, m)
	out = c.Bytes()

	if len(out) < k {
	// If the output is too small, we need to left-pad with zeros.
	t := make([]byte, k)
	copy(t[k-len(out):], out)
	out = t
	}

	return
	}

	// A DecryptionError represents a failure to decrypt a message.
	// It is deliberately vague to avoid adaptive attacks.
	type DecryptionError struct{}

	func (DecryptionError) String() string { return "RSA decryption error" }

	// A VerificationError represents a failure to verify a signature.
	// It is deliberately vague to avoid adaptive attacks.
	type VerificationError struct{}

	func (VerificationError) String() string { return "RSA verification error" }

	// modInverse returns ia, the inverse of a in the multiplicative group of prime
	// order n. It requires that a be a member of the group (i.e. less than n).
	func modInverse(a, n big.Int) (ia big.Int, ok bool) {
	g := new(big.Int)
	x := new(big.Int)
	y := new(big.Int)
	big.GcdInt(g, x, y, a, n)
	if g.Cmp(bigOne) != 0 {
	// In this case, a and n aren't coprime and we cannot calculate
	// the inverse. This happens because the values of n are nearly
	// prime (being the product of two primes) rather than truly
	// prime.
	return
	}

	if x.Cmp(bigOne) < 0 {
	// 0 is not the multiplicative inverse of any element so, if x
	// < 1, then x is negative.
	x.Add(x, n)
	}

	return x, true
	}

	// precompute performs some calculations that speed up private key operations
	// in the future.
	func (priv *PrivateKey) precompute() {
	priv.dP = new(big.Int).Sub(priv.P, bigOne)
	priv.dP.Mod(priv.D, priv.dP)

	priv.dQ = new(big.Int).Sub(priv.Q, bigOne)
	priv.dQ.Mod(priv.D, priv.dQ)

	priv.qInv = new(big.Int).ModInverse(priv.Q, priv.P)
	}

	// decrypt performs an RSA decryption, resulting in a plaintext integer. If a
	// random source is given, RSA blinding is used.
	func decrypt(rand io.Reader, priv PrivateKey, c big.Int) (m *big.Int, err os.Error) {
	// TODO(agl): can we get away with reusing blinds?
	if c.Cmp(priv.N) > 0 {
	err = DecryptionError{}
	return
	}

	var ir *big.Int
	if rand != nil {
	// Blinding enabled. Blinding involves multiplying c by r^e.
	// Then the decryption operation performs (m^e * r^e)^d mod n
	// which equals mr mod n. The factor of r can then be removed
	// by multipling by the multiplicative inverse of r.

	var r *big.Int

	for {
	r, err = randomNumber(rand, priv.N)
	if err != nil {
	return
	}
	if r.Cmp(bigZero) == 0 {
	r = bigOne
	}
	var ok bool
	ir, ok = modInverse(r, priv.N)
	if ok {
	break
	}
	}
	bigE := big.NewInt(int64(priv.E))
	rpowe := new(big.Int).Exp(r, bigE, priv.N)
	c.Mul(c, rpowe)
	c.Mod(c, priv.N)
	}

	priv.rwMutex.RLock()

	if priv.dP == nil && priv.P != nil {
	priv.rwMutex.RUnlock()
	priv.rwMutex.Lock()
	if priv.dP == nil && priv.P != nil {
	priv.precompute()
	}
	priv.rwMutex.Unlock()
	priv.rwMutex.RLock()
	}

	if priv.dP == nil {
	m = new(big.Int).Exp(c, priv.D, priv.N)
	} else {
	// We have the precalculated values needed for the CRT.
	m = new(big.Int).Exp(c, priv.dP, priv.P)
	m2 := new(big.Int).Exp(c, priv.dQ, priv.Q)
	m.Sub(m, m2)
	if m.Sign() < 0 {
	m.Add(m, priv.P)
	}
	m.Mul(m, priv.qInv)
	m.Mod(m, priv.P)
	m.Mul(m, priv.Q)
	m.Add(m, m2)
	}

	priv.rwMutex.RUnlock()

	if ir != nil {
	// Unblind.
	m.Mul(m, ir)
	m.Mod(m, priv.N)
	}

	return
	}

	// DecryptOAEP decrypts ciphertext using RSA-OAEP.
	// If rand != nil, DecryptOAEP uses RSA blinding to avoid timing side-channel attacks.
	func DecryptOAEP(hash hash.Hash, rand io.Reader, priv *PrivateKey, ciphertext []byte, label []byte) (msg []byte, err os.Error) {
	k := (priv.N.BitLen() + 7) / 8
	if len(ciphertext) > k \|\|
	k < hash.Size()*2+2 {
	err = DecryptionError{}
	return
	}

	c := new(big.Int).SetBytes(ciphertext)

	m, err := decrypt(rand, priv, c)
	if err != nil {
	return
	}

	hash.Write(label)
	lHash := hash.Sum()
	hash.Reset()

	// Converting the plaintext number to bytes will strip any
	// leading zeros so we may have to left pad. We do this unconditionally
	// to avoid leaking timing information. (Although we still probably
	// leak the number of leading zeros. It's not clear that we can do
	// anything about this.)
	em := leftPad(m.Bytes(), k)

	firstByteIsZero := subtle.ConstantTimeByteEq(em[0], 0)

	seed := em[1 : hash.Size()+1]
	db := em[hash.Size()+1:]

	mgf1XOR(seed, hash, db)
	mgf1XOR(db, hash, seed)

	lHash2 := db[0:hash.Size()]

	// We have to validate the plaintext in constant time in order to avoid
	// attacks like: J. Manger. A Chosen Ciphertext Attack on RSA Optimal
	// Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1
	// v2.0. In J. Kilian, editor, Advances in Cryptology.
	lHash2Good := subtle.ConstantTimeCompare(lHash, lHash2)

	// The remainder of the plaintext must be zero or more 0x00, followed
	// by 0x01, followed by the message.
	// lookingForIndex: 1 iff we are still looking for the 0x01
	// index: the offset of the first 0x01 byte
	// invalid: 1 iff we saw a non-zero byte before the 0x01.
	var lookingForIndex, index, invalid int
	lookingForIndex = 1
	rest := db[hash.Size():]

	for i := 0; i < len(rest); i++ {
	equals0 := subtle.ConstantTimeByteEq(rest[i], 0)
	equals1 := subtle.ConstantTimeByteEq(rest[i], 1)
	index = subtle.ConstantTimeSelect(lookingForIndex&equals1, i, index)
	lookingForIndex = subtle.ConstantTimeSelect(equals1, 0, lookingForIndex)
	invalid = subtle.ConstantTimeSelect(lookingForIndex&^equals0, 1, invalid)
	}

	if firstByteIsZero&lHash2Good&^invalid&^lookingForIndex != 1 {
	err = DecryptionError{}
	return
	}

	msg = rest[index+1:]
	return
	}

	// leftPad returns a new slice of length size. The contents of input are right
	// aligned in the new slice.
	func leftPad(input []byte, size int) (out []byte) {
	n := len(input)
	if n > size {
	n = size
	}
	out = make([]byte, size)
	copy(out[len(out)-n:], input)
	return
	}