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// 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.
// Package rsa implements RSA encryption as specified in PKCS#1.
package rsa
// TODO(agl): Add support for PSS padding.
import (
"big"
"crypto/rand"
"crypto/subtle"
"hash"
"io"
"os"
)
var bigZero = big.NewInt(0)
var bigOne = big.NewInt(1)
// 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
Primes []*big.Int // prime factors of N, has >= 2 elements.
// Precomputed contains precomputed values that speed up private
// operations, if available.
Precomputed PrecomputedValues
}
type PrecomputedValues struct {
Dp, Dq *big.Int // D mod (P-1) (or mod Q-1)
Qinv *big.Int // Q^-1 mod Q
// CRTValues is used for the 3rd and subsequent primes. Due to a
// historical accident, the CRT for the first two primes is handled
// differently in PKCS#1 and interoperability is sufficiently
// important that we mirror this.
CRTValues []CRTValue
}
// CRTValue contains the precomputed chinese remainder theorem values.
type CRTValue struct {
Exp *big.Int // D mod (prime-1).
Coeff *big.Int // R·Coeff ≡ 1 mod Prime.
R *big.Int // product of primes prior to this (inc p and q).
}
// 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 the prime factors are actually 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.
for _, prime := range priv.Primes {
if !big.ProbablyPrime(prime, 20) {
return os.NewError("prime factor is composite")
}
}
// Check that Πprimes == n.
modulus := new(big.Int).Set(bigOne)
for _, prime := range priv.Primes {
modulus.Mul(modulus, prime)
}
if modulus.Cmp(priv.N) != 0 {
return os.NewError("invalid modulus")
}
// Check that e and totient(Πprimes) are coprime.
totient := new(big.Int).Set(bigOne)
for _, prime := range priv.Primes {
pminus1 := new(big.Int).Sub(prime, bigOne)
totient.Mul(totient, pminus1)
}
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.NewError("invalid public exponent E")
}
// Check that de ≡ 1 (mod totient(Πprimes))
de := new(big.Int).Mul(priv.D, e)
de.Mod(de, totient)
if de.Cmp(bigOne) != 0 {
return os.NewError("invalid private exponent D")
}
return nil
}
// GenerateKey generates an RSA keypair of the given bit size.
func GenerateKey(random io.Reader, bits int) (priv *PrivateKey, err os.Error) {
return GenerateMultiPrimeKey(random, 2, bits)
}
// GenerateMultiPrimeKey generates a multi-prime RSA keypair of the given bit
// size, as suggested in [1]. Although the public keys are compatible
// (actually, indistinguishable) from the 2-prime case, the private keys are
// not. Thus it may not be possible to export multi-prime private keys in
// certain formats or to subsequently import them into other code.
//
// Table 1 in [2] suggests maximum numbers of primes for a given size.
//
// [1] US patent 4405829 (1972, expired)
// [2] http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-16.pdf
func GenerateMultiPrimeKey(random io.Reader, nprimes int, 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
if nprimes < 2 {
return nil, os.NewError("rsa.GenerateMultiPrimeKey: nprimes must be >= 2")
}
primes := make([]*big.Int, nprimes)
NextSetOfPrimes:
for {
todo := bits
for i := 0; i < nprimes; i++ {
primes[i], err = rand.Prime(random, todo/(nprimes-i))
if err != nil {
return nil, err
}
todo -= primes[i].BitLen()
}
// Make sure that primes is pairwise unequal.
for i, prime := range primes {
for j := 0; j < i; j++ {
if prime.Cmp(primes[j]) == 0 {
continue NextSetOfPrimes
}
}
}
n := new(big.Int).Set(bigOne)
totient := new(big.Int).Set(bigOne)
pminus1 := new(big.Int)
for _, prime := range primes {
n.Mul(n, prime)
pminus1.Sub(prime, bigOne)
totient.Mul(totient, pminus1)
}
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.Primes = primes
priv.N = n
break
}
}
priv.Precompute()
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, random 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(random, 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() {
if priv.Precomputed.Dp != nil {
return
}
priv.Precomputed.Dp = new(big.Int).Sub(priv.Primes[0], bigOne)
priv.Precomputed.Dp.Mod(priv.D, priv.Precomputed.Dp)
priv.Precomputed.Dq = new(big.Int).Sub(priv.Primes[1], bigOne)
priv.Precomputed.Dq.Mod(priv.D, priv.Precomputed.Dq)
priv.Precomputed.Qinv = new(big.Int).ModInverse(priv.Primes[1], priv.Primes[0])
r := new(big.Int).Mul(priv.Primes[0], priv.Primes[1])
priv.Precomputed.CRTValues = make([]CRTValue, len(priv.Primes)-2)
for i := 2; i < len(priv.Primes); i++ {
prime := priv.Primes[i]
values := &priv.Precomputed.CRTValues[i-2]
values.Exp = new(big.Int).Sub(prime, bigOne)
values.Exp.Mod(priv.D, values.Exp)
values.R = new(big.Int).Set(r)
values.Coeff = new(big.Int).ModInverse(r, prime)
r.Mul(r, prime)
}
}
// decrypt performs an RSA decryption, resulting in a plaintext integer. If a
// random source is given, RSA blinding is used.
func decrypt(random 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 random != 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 multiplying by the multiplicative inverse of r.
var r *big.Int
for {
r, err = rand.Int(random, 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)
cCopy := new(big.Int).Set(c)
cCopy.Mul(cCopy, rpowe)
cCopy.Mod(cCopy, priv.N)
c = cCopy
}
if priv.Precomputed.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.Precomputed.Dp, priv.Primes[0])
m2 := new(big.Int).Exp(c, priv.Precomputed.Dq, priv.Primes[1])
m.Sub(m, m2)
if m.Sign() < 0 {
m.Add(m, priv.Primes[0])
}
m.Mul(m, priv.Precomputed.Qinv)
m.Mod(m, priv.Primes[0])
m.Mul(m, priv.Primes[1])
m.Add(m, m2)
for i, values := range priv.Precomputed.CRTValues {
prime := priv.Primes[2+i]
m2.Exp(c, values.Exp, prime)
m2.Sub(m2, m)
m2.Mul(m2, values.Coeff)
m2.Mod(m2, prime)
if m2.Sign() < 0 {
m2.Add(m2, prime)
}
m2.Mul(m2, values.R)
m.Add(m, m2)
}
}
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, random 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(random, 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
}