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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 (
        "crypto/rand"
        "crypto/subtle"
        "errors"
        "hash"
        "io"
        "math/big"
)

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 error describing a problem.
func (priv *PrivateKey) Validate() 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 !prime.ProbablyPrime(20) {
                        return errors.New("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 errors.New("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)
        gcd.GCD(x, y, totient, e)
        if gcd.Cmp(bigOne) != 0 {
                return errors.New("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 errors.New("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 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 error) {
        priv = new(PrivateKey)
        priv.E = 65537

        if nprimes < 2 {
                return nil, errors.New("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))
                g.GCD(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
        var digest []byte

        done := 0
        for done < len(out) {
                hash.Write(seed)
                hash.Write(counter[0:4])
                digest = hash.Sum(digest[:0])
                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) Error() 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 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(nil)
        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) Error() 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) Error() 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)
        g.GCD(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 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 random != 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 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(nil)
        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
}