# # ElGamal.py : ElGamal encryption/decryption and signatures # # Part of the Python Cryptography Toolkit # # Originally written by: A.M. Kuchling # # =================================================================== # The contents of this file are dedicated to the public domain. To # the extent that dedication to the public domain is not available, # everyone is granted a worldwide, perpetual, royalty-free, # non-exclusive license to exercise all rights associated with the # contents of this file for any purpose whatsoever. # No rights are reserved. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, # EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF # MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND # NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS # BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN # ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN # CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE # SOFTWARE. # =================================================================== """ElGamal public-key algorithm (randomized encryption and signature). Signature algorithm ------------------- The security of the ElGamal signature scheme is based (like DSA) on the discrete logarithm problem (DLP_). Given a cyclic group, a generator *g*, and an element *h*, it is hard to find an integer *x* such that *g^x = h*. The group is the largest multiplicative sub-group of the integers modulo *p*, with *p* prime. The signer holds a value *x* (*0<x<p-1*) as private key, and its public key (*y* where *y=g^x mod p*) is distributed. The ElGamal signature is twice as big as *p*. Encryption algorithm -------------------- The security of the ElGamal encryption scheme is based on the computational Diffie-Hellman problem (CDH_). Given a cyclic group, a generator *g*, and two integers *a* and *b*, it is difficult to find the element *g^{ab}* when only *g^a* and *g^b* are known, and not *a* and *b*. As before, the group is the largest multiplicative sub-group of the integers modulo *p*, with *p* prime. The receiver holds a value *a* (*0<a<p-1*) as private key, and its public key (*b* where *b*=g^a*) is given to the sender. The ElGamal ciphertext is twice as big as *p*. Domain parameters ----------------- For both signature and encryption schemes, the values *(p,g)* are called *domain parameters*. They are not sensitive but must be distributed to all parties (senders and receivers). Different signers can share the same domain parameters, as can different recipients of encrypted messages. Security -------- Both DLP and CDH problem are believed to be difficult, and they have been proved such (and therefore secure) for more than 30 years. The cryptographic strength is linked to the magnitude of *p*. In 2012, a sufficient size for *p* is deemed to be 2048 bits. For more information, see the most recent ECRYPT_ report. Even though ElGamal algorithms are in theory reasonably secure for new designs, in practice there are no real good reasons for using them. The signature is four times larger than the equivalent DSA, and the ciphertext is two times larger than the equivalent RSA. Functionality ------------- This module provides facilities for generating new ElGamal keys and for constructing them from known components. ElGamal keys allows you to perform basic signing, verification, encryption, and decryption. >>> from Crypto import Random >>> from Crypto.PublicKey import ElGamal >>> from Crypto.Hash import SHA >>> from Crypto.Math import Numbers >>> >>> message = "Hello" >>> key = ElGamal.generate(1024, Random.new().read) >>> h = SHA.new(message).digest() >>> while 1: >>> k = Numbers.random_range(min_inclusive=1, min_exclusive=key.p-1) >>> if k.gcd(key.p-1)==1: break >>> sig = key.sign(h,k) >>> ... >>> if key.verify(h,sig): >>> print "OK" >>> else: >>> print "Incorrect signature" .. _DLP: http://www.cosic.esat.kuleuven.be/publications/talk-78.pdf .. _CDH: http://en.wikipedia.org/wiki/Computational_Diffie%E2%80%93Hellman_assumption .. _ECRYPT: http://www.ecrypt.eu.org/documents/D.SPA.17.pdf """ __all__ = ['generate', 'construct', 'ElGamalKey'] from Crypto import Random from Crypto.Math.Primality import ( generate_probable_safe_prime, test_probable_prime, COMPOSITE ) from Crypto.Math.Numbers import Integer # Generate an ElGamal key with N bits def generate(bits, randfunc): """Randomly generate a fresh, new ElGamal key. The key will be safe for use for both encryption and signature (although it should be used for **only one** purpose). :Parameters: bits : int Key length, or size (in bits) of the modulus *p*. Recommended value is 2048. randfunc : callable Random number generation function; it should accept a single integer N and return a string of random data N bytes long. :attention: You should always use a cryptographically secure random number generator, such as the one defined in the ``Crypto.Random`` module; **don't** just use the current time and the ``random`` module. :Return: An ElGamal key object (`ElGamalKey`). """ obj=ElGamalKey() # Generate a safe prime p # See Algorithm 4.86 in Handbook of Applied Cryptography obj.p = generate_probable_safe_prime(exact_bits=bits, randfunc=randfunc) q = (obj.p - 1) >> 1 # Generate generator g # See Algorithm 4.80 in Handbook of Applied Cryptography # Note that the order of the group is n=p-1=2q, where q is prime while 1: # We must avoid g=2 because of Bleichenbacher's attack described # in "Generating ElGamal signatures without knowning the secret key", # 1996 # obj.g = Integer.random_range(min_inclusive=3, max_exclusive=obj.p, randfunc=randfunc) safe = 1 if pow(obj.g, 2, obj.p)==1: safe=0 if safe and pow(obj.g, q, obj.p)==1: safe=0 # Discard g if it divides p-1 because of the attack described # in Note 11.67 (iii) in HAC if safe and (obj.p-1) % obj.g == 0: safe=0 # g^{-1} must not divide p-1 because of Khadir's attack # described in "Conditions of the generator for forging ElGamal # signature", 2011 ginv = obj.g.inverse(obj.p) if safe and (obj.p-1) % ginv == 0: safe=0 if safe: break # Generate private key x obj.x = Integer.random_range(min_inclusive=2, max_exclusive=obj.p-1, randfunc=randfunc) # Generate public key y obj.y = pow(obj.g, obj.x, obj.p) return obj def construct(tup): """Construct an ElGamal key from a tuple of valid ElGamal components. The modulus *p* must be a prime. The following conditions must apply: - 1 < g < p-1 - g^{p-1} = 1 mod p - 1 < x < p-1 - g^x = y mod p :Parameters: tup : tuple A tuple of long integers, with 3 or 4 items in the following order: 1. Modulus (*p*). 2. Generator (*g*). 3. Public key (*y*). 4. Private key (*x*). Optional. :Raise PublicKey.ValueError: When the key being imported fails the most basic ElGamal validity checks. :Return: An ElGamal key object (`ElGamalKey`). """ obj=ElGamalKey() if len(tup) not in [3,4]: raise ValueError('argument for construct() wrong length') for i in range(len(tup)): field = obj._keydata[i] setattr(obj, field, Integer(tup[i])) fmt_error = test_probable_prime(obj.p) == COMPOSITE fmt_error |= obj.g<=1 or obj.g>=obj.p fmt_error |= pow(obj.g, obj.p-1, obj.p)!=1 fmt_error |= obj.y<1 or obj.y>=obj.p if len(tup)==4: fmt_error |= obj.x<=1 or obj.x>=obj.p fmt_error |= pow(obj.g, obj.x, obj.p)!=obj.y if fmt_error: raise ValueError("Invalid ElGamal key components") return obj class ElGamalKey(object): """Class defining an ElGamal key. :undocumented: __getstate__, __setstate__, __repr__, __getattr__ """ #: Dictionary of ElGamal parameters. #: #: A public key will only have the following entries: #: #: - **y**, the public key. #: - **g**, the generator. #: - **p**, the modulus. #: #: A private key will also have: #: #: - **x**, the private key. _keydata=['p', 'g', 'y', 'x'] def __init__(self, randfunc=None): if randfunc is None: randfunc = Random.new().read self._randfunc = randfunc def _encrypt(self, M, K): a=pow(self.g, K, self.p) b=( pow(self.y, K, self.p)*M ) % self.p return list(map(int, ( a,b ))) def _decrypt(self, M): if (not hasattr(self, 'x')): raise TypeError('Private key not available in this object') r = Integer.random_range(min_inclusive=2, max_exclusive=self.p-1, randfunc=self._randfunc) a_blind = (pow(self.g, r, self.p) * M[0]) % self.p ax=pow(a_blind, self.x, self.p) plaintext_blind = (ax.inverse(self.p) * M[1] ) % self.p plaintext = (plaintext_blind * pow(self.y, r, self.p)) % self.p return int(plaintext) def _sign(self, M, K): if (not hasattr(self, 'x')): raise TypeError('Private key not available in this object') p1=self.p-1 K = Integer(K) if (K.gcd(p1)!=1): raise ValueError('Bad K value: GCD(K,p-1)!=1') a=pow(self.g, K, self.p) t=(Integer(M)-self.x*a) % p1 while t<0: t=t+p1 b=(t*K.inverse(p1)) % p1 return list(map(int, (a, b))) def _verify(self, M, sig): sig = list(map(Integer, sig)) if sig[0]<1 or sig[0]>self.p-1: return 0 v1=pow(self.y, sig[0], self.p) v1=(v1*pow(sig[0], sig[1], self.p)) % self.p v2=pow(self.g, M, self.p) if v1==v2: return 1 return 0 def has_private(self): if hasattr(self, 'x'): return 1 else: return 0 def can_encrypt(self): return True def can_sign(self): return True def publickey(self): return construct((self.p, self.g, self.y)) def __eq__(self, other): if bool(self.has_private()) != bool(other.has_private()): return False result = True for comp in self._keydata: result = result and (getattr(self.key, comp, None) == getattr(other.key, comp, None)) return result def __ne__(self, other): return not self.__eq__(other) def __getstate__(self): # ElGamal key is not pickable from pickle import PicklingError raise PicklingError # Methods defined in PyCrypto that we don't support anymore def sign(self, M, K): raise NotImplementedError def verify(self, M, signature): raise NotImplementedError def encrypt(self, plaintext, K): raise NotImplementedError def decrypt(self, ciphertext): raise NotImplementedError def blind(self, M, B): raise NotImplementedError def unblind(self, M, B): raise NotImplementedError def size(): raise NotImplementedError