// Generate3PrimeKey generates a 3-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 3-prime private keys in certain formats or to
// subsequently import them into other code.
//
// Table 1 in [2] suggests that size should be >= 1024 when using 3 primes.
//
// [1] US patent 4405829 (1972, expired)
// [2] http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-16.pdf
func Generate3PrimeKey(rand io.Reader, bits int) (priv *PrivateKey, err os.Error) {
priv = new(PrivateKey)
priv.E = 3
pminus1 := new(big.Int)
qminus1 := new(big.Int)
rminus1 := new(big.Int)
totient := new(big.Int)
for {
p, err := randomPrime(rand, bits/3)
if err != nil {
return nil, err
}
todo := bits - p.BitLen()
q, err := randomPrime(rand, todo/2)
if err != nil {
return nil, err
}
todo -= q.BitLen()
r, err := randomPrime(rand, todo)
if err != nil {
return nil, err
}
if p.Cmp(q) == 0 ||
q.Cmp(r) == 0 ||
r.Cmp(p) == 0 {
continue
}
n := new(big.Int).Mul(p, q)
n.Mul(n, r)
pminus1.Sub(p, bigOne)
qminus1.Sub(q, bigOne)
rminus1.Sub(r, bigOne)
totient.Mul(pminus1, qminus1)
totient.Mul(totient, rminus1)
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.R = r
priv.N = n
break
}
}
return
}
// 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))
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
}
// 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
}
// 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) {
g := new(big.Int)
x := new(big.Int)
y := new(big.Int)
big.GcdInt(g, x, y, a, n)
if big.CmpInt(x, 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
}
func (priv *PrivateKey) Validate() os.Error {
// Check that p, q and, maybe, r 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")
}
if priv.R != nil && !big.ProbablyPrime(priv.R, 20) {
return os.ErrorString("R is composite")
}
// Check that p*q*r == n.
modulus := new(big.Int).Mul(priv.P, priv.Q)
if priv.R != nil {
modulus.Mul(modulus, priv.R)
}
if modulus.Cmp(priv.N) != 0 {
return os.ErrorString("invalid modulus")
}
// Check that e and totient(p, q, r) 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)
if priv.R != nil {
rminus1 := new(big.Int).Sub(priv.R, bigOne)
totient.Mul(totient, rminus1)
}
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, r))
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
}
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.ErrorString("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.ErrorString("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.ErrorString("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.ErrorString("invalid private exponent D")
}
return nil
}
// 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
}
func (priv PrivateKey) Validate() os.Error {
/*
TODO(agl): Enable once big implements ProbablyPrime.
// Check that p and q are prime.
if !priv.P.ProbablyPrime(20) {
return os.ErrorString("P is composite");
}
if !priv.Q.ProbablyPrime(20) {
return os.ErrorString("Q is composite");
}
*/
// Check that p*q == n.
modulus := new(big.Int).Mul(priv.P, priv.Q)
if big.CmpInt(modulus, 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 big.CmpInt(gcd, 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 big.CmpInt(de, bigOne) != 0 {
return os.ErrorString("invalid private exponent D")
}
return nil
}
// 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.ErrorString("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
}
