| // 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 |
| } |