494 lines
13 KiB
Go
494 lines
13 KiB
Go
// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package rsa implements RSA encryption as specified in PKCS#1.
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package rsa
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// TODO(agl): Add support for PSS padding.
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import (
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"big"
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"crypto/rand"
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"crypto/subtle"
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"errors"
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"hash"
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"io"
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)
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var bigZero = big.NewInt(0)
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var bigOne = big.NewInt(1)
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// A PublicKey represents the public part of an RSA key.
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type PublicKey struct {
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N *big.Int // modulus
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E int // public exponent
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}
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// A PrivateKey represents an RSA key
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type PrivateKey struct {
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PublicKey // public part.
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D *big.Int // private exponent
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Primes []*big.Int // prime factors of N, has >= 2 elements.
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// Precomputed contains precomputed values that speed up private
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// operations, if available.
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Precomputed PrecomputedValues
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}
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type PrecomputedValues struct {
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Dp, Dq *big.Int // D mod (P-1) (or mod Q-1)
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Qinv *big.Int // Q^-1 mod Q
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// CRTValues is used for the 3rd and subsequent primes. Due to a
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// historical accident, the CRT for the first two primes is handled
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// differently in PKCS#1 and interoperability is sufficiently
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// important that we mirror this.
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CRTValues []CRTValue
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}
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// CRTValue contains the precomputed chinese remainder theorem values.
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type CRTValue struct {
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Exp *big.Int // D mod (prime-1).
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Coeff *big.Int // R·Coeff ≡ 1 mod Prime.
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R *big.Int // product of primes prior to this (inc p and q).
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}
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// Validate performs basic sanity checks on the key.
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// It returns nil if the key is valid, or else an error describing a problem.
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func (priv *PrivateKey) Validate() error {
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// Check that the prime factors are actually prime. Note that this is
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// just a sanity check. Since the random witnesses chosen by
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// ProbablyPrime are deterministic, given the candidate number, it's
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// easy for an attack to generate composites that pass this test.
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for _, prime := range priv.Primes {
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if !big.ProbablyPrime(prime, 20) {
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return errors.New("prime factor is composite")
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}
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}
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// Check that Πprimes == n.
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modulus := new(big.Int).Set(bigOne)
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for _, prime := range priv.Primes {
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modulus.Mul(modulus, prime)
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}
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if modulus.Cmp(priv.N) != 0 {
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return errors.New("invalid modulus")
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}
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// Check that e and totient(Πprimes) are coprime.
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totient := new(big.Int).Set(bigOne)
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for _, prime := range priv.Primes {
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pminus1 := new(big.Int).Sub(prime, bigOne)
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totient.Mul(totient, pminus1)
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}
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e := big.NewInt(int64(priv.E))
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gcd := new(big.Int)
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x := new(big.Int)
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y := new(big.Int)
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big.GcdInt(gcd, x, y, totient, e)
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if gcd.Cmp(bigOne) != 0 {
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return errors.New("invalid public exponent E")
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}
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// Check that de ≡ 1 (mod totient(Πprimes))
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de := new(big.Int).Mul(priv.D, e)
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de.Mod(de, totient)
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if de.Cmp(bigOne) != 0 {
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return errors.New("invalid private exponent D")
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}
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return nil
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}
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// GenerateKey generates an RSA keypair of the given bit size.
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func GenerateKey(random io.Reader, bits int) (priv *PrivateKey, err error) {
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return GenerateMultiPrimeKey(random, 2, bits)
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}
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// GenerateMultiPrimeKey generates a multi-prime RSA keypair of the given bit
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// size, as suggested in [1]. Although the public keys are compatible
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// (actually, indistinguishable) from the 2-prime case, the private keys are
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// not. Thus it may not be possible to export multi-prime private keys in
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// certain formats or to subsequently import them into other code.
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//
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// Table 1 in [2] suggests maximum numbers of primes for a given size.
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//
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// [1] US patent 4405829 (1972, expired)
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// [2] http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-16.pdf
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func GenerateMultiPrimeKey(random io.Reader, nprimes int, bits int) (priv *PrivateKey, err error) {
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priv = new(PrivateKey)
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priv.E = 65537
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if nprimes < 2 {
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return nil, errors.New("rsa.GenerateMultiPrimeKey: nprimes must be >= 2")
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}
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primes := make([]*big.Int, nprimes)
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NextSetOfPrimes:
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for {
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todo := bits
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for i := 0; i < nprimes; i++ {
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primes[i], err = rand.Prime(random, todo/(nprimes-i))
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if err != nil {
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return nil, err
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}
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todo -= primes[i].BitLen()
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}
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// Make sure that primes is pairwise unequal.
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for i, prime := range primes {
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for j := 0; j < i; j++ {
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if prime.Cmp(primes[j]) == 0 {
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continue NextSetOfPrimes
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}
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}
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}
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n := new(big.Int).Set(bigOne)
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totient := new(big.Int).Set(bigOne)
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pminus1 := new(big.Int)
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for _, prime := range primes {
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n.Mul(n, prime)
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pminus1.Sub(prime, bigOne)
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totient.Mul(totient, pminus1)
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}
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g := new(big.Int)
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priv.D = new(big.Int)
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y := new(big.Int)
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e := big.NewInt(int64(priv.E))
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big.GcdInt(g, priv.D, y, e, totient)
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if g.Cmp(bigOne) == 0 {
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priv.D.Add(priv.D, totient)
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priv.Primes = primes
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priv.N = n
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break
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}
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}
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priv.Precompute()
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return
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}
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// incCounter increments a four byte, big-endian counter.
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func incCounter(c *[4]byte) {
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if c[3]++; c[3] != 0 {
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return
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}
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if c[2]++; c[2] != 0 {
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return
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}
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if c[1]++; c[1] != 0 {
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return
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}
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c[0]++
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}
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// mgf1XOR XORs the bytes in out with a mask generated using the MGF1 function
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// specified in PKCS#1 v2.1.
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func mgf1XOR(out []byte, hash hash.Hash, seed []byte) {
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var counter [4]byte
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done := 0
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for done < len(out) {
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hash.Write(seed)
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hash.Write(counter[0:4])
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digest := hash.Sum()
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hash.Reset()
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for i := 0; i < len(digest) && done < len(out); i++ {
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out[done] ^= digest[i]
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done++
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}
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incCounter(&counter)
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}
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}
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// MessageTooLongError is returned when attempting to encrypt a message which
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// is too large for the size of the public key.
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type MessageTooLongError struct{}
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func (MessageTooLongError) Error() string {
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return "message too long for RSA public key size"
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}
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func encrypt(c *big.Int, pub *PublicKey, m *big.Int) *big.Int {
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e := big.NewInt(int64(pub.E))
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c.Exp(m, e, pub.N)
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return c
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}
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// EncryptOAEP encrypts the given message with RSA-OAEP.
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// The message must be no longer than the length of the public modulus less
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// twice the hash length plus 2.
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func EncryptOAEP(hash hash.Hash, random io.Reader, pub *PublicKey, msg []byte, label []byte) (out []byte, err error) {
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hash.Reset()
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k := (pub.N.BitLen() + 7) / 8
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if len(msg) > k-2*hash.Size()-2 {
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err = MessageTooLongError{}
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return
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}
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hash.Write(label)
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lHash := hash.Sum()
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hash.Reset()
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em := make([]byte, k)
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seed := em[1 : 1+hash.Size()]
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db := em[1+hash.Size():]
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copy(db[0:hash.Size()], lHash)
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db[len(db)-len(msg)-1] = 1
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copy(db[len(db)-len(msg):], msg)
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_, err = io.ReadFull(random, seed)
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if err != nil {
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return
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}
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mgf1XOR(db, hash, seed)
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mgf1XOR(seed, hash, db)
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m := new(big.Int)
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m.SetBytes(em)
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c := encrypt(new(big.Int), pub, m)
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out = c.Bytes()
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if len(out) < k {
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// If the output is too small, we need to left-pad with zeros.
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t := make([]byte, k)
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copy(t[k-len(out):], out)
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out = t
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}
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return
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}
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// A DecryptionError represents a failure to decrypt a message.
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// It is deliberately vague to avoid adaptive attacks.
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type DecryptionError struct{}
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func (DecryptionError) Error() string { return "RSA decryption error" }
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// A VerificationError represents a failure to verify a signature.
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// It is deliberately vague to avoid adaptive attacks.
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type VerificationError struct{}
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func (VerificationError) Error() string { return "RSA verification error" }
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// modInverse returns ia, the inverse of a in the multiplicative group of prime
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// order n. It requires that a be a member of the group (i.e. less than n).
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func modInverse(a, n *big.Int) (ia *big.Int, ok bool) {
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g := new(big.Int)
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x := new(big.Int)
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y := new(big.Int)
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big.GcdInt(g, x, y, a, n)
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if g.Cmp(bigOne) != 0 {
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// In this case, a and n aren't coprime and we cannot calculate
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// the inverse. This happens because the values of n are nearly
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// prime (being the product of two primes) rather than truly
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// prime.
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return
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}
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if x.Cmp(bigOne) < 0 {
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// 0 is not the multiplicative inverse of any element so, if x
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// < 1, then x is negative.
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x.Add(x, n)
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}
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return x, true
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}
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// Precompute performs some calculations that speed up private key operations
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// in the future.
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func (priv *PrivateKey) Precompute() {
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if priv.Precomputed.Dp != nil {
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return
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}
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priv.Precomputed.Dp = new(big.Int).Sub(priv.Primes[0], bigOne)
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priv.Precomputed.Dp.Mod(priv.D, priv.Precomputed.Dp)
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priv.Precomputed.Dq = new(big.Int).Sub(priv.Primes[1], bigOne)
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priv.Precomputed.Dq.Mod(priv.D, priv.Precomputed.Dq)
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priv.Precomputed.Qinv = new(big.Int).ModInverse(priv.Primes[1], priv.Primes[0])
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r := new(big.Int).Mul(priv.Primes[0], priv.Primes[1])
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priv.Precomputed.CRTValues = make([]CRTValue, len(priv.Primes)-2)
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for i := 2; i < len(priv.Primes); i++ {
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prime := priv.Primes[i]
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values := &priv.Precomputed.CRTValues[i-2]
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values.Exp = new(big.Int).Sub(prime, bigOne)
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values.Exp.Mod(priv.D, values.Exp)
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values.R = new(big.Int).Set(r)
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values.Coeff = new(big.Int).ModInverse(r, prime)
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r.Mul(r, prime)
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}
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}
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// decrypt performs an RSA decryption, resulting in a plaintext integer. If a
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// random source is given, RSA blinding is used.
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func decrypt(random io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err error) {
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// TODO(agl): can we get away with reusing blinds?
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if c.Cmp(priv.N) > 0 {
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err = DecryptionError{}
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return
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}
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var ir *big.Int
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if random != nil {
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// Blinding enabled. Blinding involves multiplying c by r^e.
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// Then the decryption operation performs (m^e * r^e)^d mod n
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// which equals mr mod n. The factor of r can then be removed
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// by multiplying by the multiplicative inverse of r.
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var r *big.Int
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for {
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r, err = rand.Int(random, priv.N)
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if err != nil {
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return
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}
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if r.Cmp(bigZero) == 0 {
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r = bigOne
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}
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var ok bool
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ir, ok = modInverse(r, priv.N)
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if ok {
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break
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}
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}
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bigE := big.NewInt(int64(priv.E))
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rpowe := new(big.Int).Exp(r, bigE, priv.N)
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cCopy := new(big.Int).Set(c)
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cCopy.Mul(cCopy, rpowe)
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cCopy.Mod(cCopy, priv.N)
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c = cCopy
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}
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if priv.Precomputed.Dp == nil {
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m = new(big.Int).Exp(c, priv.D, priv.N)
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} else {
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// We have the precalculated values needed for the CRT.
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m = new(big.Int).Exp(c, priv.Precomputed.Dp, priv.Primes[0])
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m2 := new(big.Int).Exp(c, priv.Precomputed.Dq, priv.Primes[1])
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m.Sub(m, m2)
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if m.Sign() < 0 {
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m.Add(m, priv.Primes[0])
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}
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m.Mul(m, priv.Precomputed.Qinv)
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m.Mod(m, priv.Primes[0])
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m.Mul(m, priv.Primes[1])
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m.Add(m, m2)
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for i, values := range priv.Precomputed.CRTValues {
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prime := priv.Primes[2+i]
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m2.Exp(c, values.Exp, prime)
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m2.Sub(m2, m)
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m2.Mul(m2, values.Coeff)
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m2.Mod(m2, prime)
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if m2.Sign() < 0 {
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m2.Add(m2, prime)
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}
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m2.Mul(m2, values.R)
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m.Add(m, m2)
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}
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}
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if ir != nil {
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// Unblind.
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m.Mul(m, ir)
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m.Mod(m, priv.N)
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}
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return
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}
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// DecryptOAEP decrypts ciphertext using RSA-OAEP.
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// If rand != nil, DecryptOAEP uses RSA blinding to avoid timing side-channel attacks.
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func DecryptOAEP(hash hash.Hash, random io.Reader, priv *PrivateKey, ciphertext []byte, label []byte) (msg []byte, err error) {
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k := (priv.N.BitLen() + 7) / 8
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if len(ciphertext) > k ||
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k < hash.Size()*2+2 {
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err = DecryptionError{}
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return
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}
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c := new(big.Int).SetBytes(ciphertext)
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m, err := decrypt(random, priv, c)
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if err != nil {
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return
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}
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hash.Write(label)
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lHash := hash.Sum()
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hash.Reset()
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// Converting the plaintext number to bytes will strip any
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// leading zeros so we may have to left pad. We do this unconditionally
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// to avoid leaking timing information. (Although we still probably
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// leak the number of leading zeros. It's not clear that we can do
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// anything about this.)
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em := leftPad(m.Bytes(), k)
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firstByteIsZero := subtle.ConstantTimeByteEq(em[0], 0)
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seed := em[1 : hash.Size()+1]
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db := em[hash.Size()+1:]
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mgf1XOR(seed, hash, db)
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mgf1XOR(db, hash, seed)
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lHash2 := db[0:hash.Size()]
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// We have to validate the plaintext in constant time in order to avoid
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// attacks like: J. Manger. A Chosen Ciphertext Attack on RSA Optimal
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// Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1
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// v2.0. In J. Kilian, editor, Advances in Cryptology.
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lHash2Good := subtle.ConstantTimeCompare(lHash, lHash2)
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// The remainder of the plaintext must be zero or more 0x00, followed
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// by 0x01, followed by the message.
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// lookingForIndex: 1 iff we are still looking for the 0x01
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// index: the offset of the first 0x01 byte
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// invalid: 1 iff we saw a non-zero byte before the 0x01.
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var lookingForIndex, index, invalid int
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lookingForIndex = 1
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rest := db[hash.Size():]
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for i := 0; i < len(rest); i++ {
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equals0 := subtle.ConstantTimeByteEq(rest[i], 0)
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equals1 := subtle.ConstantTimeByteEq(rest[i], 1)
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index = subtle.ConstantTimeSelect(lookingForIndex&equals1, i, index)
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lookingForIndex = subtle.ConstantTimeSelect(equals1, 0, lookingForIndex)
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invalid = subtle.ConstantTimeSelect(lookingForIndex&^equals0, 1, invalid)
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}
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if firstByteIsZero&lHash2Good&^invalid&^lookingForIndex != 1 {
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err = DecryptionError{}
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return
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}
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msg = rest[index+1:]
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return
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}
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// leftPad returns a new slice of length size. The contents of input are right
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// aligned in the new slice.
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func leftPad(input []byte, size int) (out []byte) {
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n := len(input)
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if n > size {
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n = size
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}
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out = make([]byte, size)
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copy(out[len(out)-n:], input)
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return
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}
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