374 lines
13 KiB
Python
374 lines
13 KiB
Python
#
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# ElGamal.py : ElGamal encryption/decryption and signatures
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#
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# Part of the Python Cryptography Toolkit
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#
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# Originally written by: A.M. Kuchling
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#
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# ===================================================================
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# The contents of this file are dedicated to the public domain. To
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# the extent that dedication to the public domain is not available,
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# everyone is granted a worldwide, perpetual, royalty-free,
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# non-exclusive license to exercise all rights associated with the
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# contents of this file for any purpose whatsoever.
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# No rights are reserved.
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#
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# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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# NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
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# BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
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# ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
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# CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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# SOFTWARE.
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# ===================================================================
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"""ElGamal public-key algorithm (randomized encryption and signature).
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Signature algorithm
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-------------------
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The security of the ElGamal signature scheme is based (like DSA) on the discrete
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logarithm problem (DLP_). Given a cyclic group, a generator *g*,
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and an element *h*, it is hard to find an integer *x* such that *g^x = h*.
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The group is the largest multiplicative sub-group of the integers modulo *p*,
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with *p* prime.
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The signer holds a value *x* (*0<x<p-1*) as private key, and its public
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key (*y* where *y=g^x mod p*) is distributed.
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The ElGamal signature is twice as big as *p*.
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Encryption algorithm
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--------------------
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The security of the ElGamal encryption scheme is based on the computational
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Diffie-Hellman problem (CDH_). Given a cyclic group, a generator *g*,
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and two integers *a* and *b*, it is difficult to find
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the element *g^{ab}* when only *g^a* and *g^b* are known, and not *a* and *b*.
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As before, the group is the largest multiplicative sub-group of the integers
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modulo *p*, with *p* prime.
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The receiver holds a value *a* (*0<a<p-1*) as private key, and its public key
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(*b* where *b*=g^a*) is given to the sender.
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The ElGamal ciphertext is twice as big as *p*.
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Domain parameters
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-----------------
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For both signature and encryption schemes, the values *(p,g)* are called
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*domain parameters*.
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They are not sensitive but must be distributed to all parties (senders and
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receivers).
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Different signers can share the same domain parameters, as can
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different recipients of encrypted messages.
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Security
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--------
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Both DLP and CDH problem are believed to be difficult, and they have been proved
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such (and therefore secure) for more than 30 years.
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The cryptographic strength is linked to the magnitude of *p*.
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In 2012, a sufficient size for *p* is deemed to be 2048 bits.
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For more information, see the most recent ECRYPT_ report.
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Even though ElGamal algorithms are in theory reasonably secure for new designs,
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in practice there are no real good reasons for using them.
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The signature is four times larger than the equivalent DSA, and the ciphertext
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is two times larger than the equivalent RSA.
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Functionality
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-------------
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This module provides facilities for generating new ElGamal keys and for constructing
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them from known components. ElGamal keys allows you to perform basic signing,
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verification, encryption, and decryption.
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>>> from Crypto import Random
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>>> from Crypto.Random import random
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>>> from Crypto.PublicKey import ElGamal
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>>> from Crypto.Util.number import GCD
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>>> from Crypto.Hash import SHA
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>>>
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>>> message = "Hello"
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>>> key = ElGamal.generate(1024, Random.new().read)
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>>> h = SHA.new(message).digest()
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>>> while 1:
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>>> k = random.StrongRandom().randint(1,key.p-1)
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>>> if GCD(k,key.p-1)==1: break
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>>> sig = key.sign(h,k)
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>>> ...
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>>> if key.verify(h,sig):
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>>> print "OK"
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>>> else:
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>>> print "Incorrect signature"
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.. _DLP: http://www.cosic.esat.kuleuven.be/publications/talk-78.pdf
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.. _CDH: http://en.wikipedia.org/wiki/Computational_Diffie%E2%80%93Hellman_assumption
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.. _ECRYPT: http://www.ecrypt.eu.org/documents/D.SPA.17.pdf
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"""
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__revision__ = "$Id$"
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__all__ = ['generate', 'construct', 'error', 'ElGamalobj']
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from Crypto.PublicKey.pubkey import *
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from Crypto.Util import number
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class error (Exception):
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pass
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# Generate an ElGamal key with N bits
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def generate(bits, randfunc, progress_func=None):
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"""Randomly generate a fresh, new ElGamal key.
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The key will be safe for use for both encryption and signature
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(although it should be used for **only one** purpose).
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:Parameters:
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bits : int
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Key length, or size (in bits) of the modulus *p*.
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Recommended value is 2048.
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randfunc : callable
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Random number generation function; it should accept
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a single integer N and return a string of random data
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N bytes long.
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progress_func : callable
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Optional function that will be called with a short string
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containing the key parameter currently being generated;
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it's useful for interactive applications where a user is
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waiting for a key to be generated.
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:attention: You should always use a cryptographically secure random number generator,
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such as the one defined in the ``Crypto.Random`` module; **don't** just use the
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current time and the ``random`` module.
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:Return: An ElGamal key object (`ElGamalobj`).
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"""
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obj=ElGamalobj()
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# Generate a safe prime p
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# See Algorithm 4.86 in Handbook of Applied Cryptography
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if progress_func:
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progress_func('p\n')
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while 1:
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q = bignum(getPrime(bits-1, randfunc))
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obj.p = 2*q+1
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if number.isPrime(obj.p, randfunc=randfunc):
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break
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# Generate generator g
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# See Algorithm 4.80 in Handbook of Applied Cryptography
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# Note that the order of the group is n=p-1=2q, where q is prime
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if progress_func:
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progress_func('g\n')
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while 1:
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# We must avoid g=2 because of Bleichenbacher's attack described
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# in "Generating ElGamal signatures without knowning the secret key",
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# 1996
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#
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obj.g = number.getRandomRange(3, obj.p, randfunc)
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safe = 1
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if pow(obj.g, 2, obj.p)==1:
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safe=0
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if safe and pow(obj.g, q, obj.p)==1:
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safe=0
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# Discard g if it divides p-1 because of the attack described
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# in Note 11.67 (iii) in HAC
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if safe and divmod(obj.p-1, obj.g)[1]==0:
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safe=0
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# g^{-1} must not divide p-1 because of Khadir's attack
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# described in "Conditions of the generator for forging ElGamal
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# signature", 2011
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ginv = number.inverse(obj.g, obj.p)
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if safe and divmod(obj.p-1, ginv)[1]==0:
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safe=0
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if safe:
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break
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# Generate private key x
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if progress_func:
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progress_func('x\n')
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obj.x=number.getRandomRange(2, obj.p-1, randfunc)
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# Generate public key y
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if progress_func:
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progress_func('y\n')
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obj.y = pow(obj.g, obj.x, obj.p)
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return obj
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def construct(tup):
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"""Construct an ElGamal key from a tuple of valid ElGamal components.
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The modulus *p* must be a prime.
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The following conditions must apply:
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- 1 < g < p-1
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- g^{p-1} = 1 mod p
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- 1 < x < p-1
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- g^x = y mod p
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:Parameters:
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tup : tuple
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A tuple of long integers, with 3 or 4 items
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in the following order:
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1. Modulus (*p*).
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2. Generator (*g*).
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3. Public key (*y*).
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4. Private key (*x*). Optional.
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:Return: An ElGamal key object (`ElGamalobj`).
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"""
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obj=ElGamalobj()
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if len(tup) not in [3,4]:
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raise ValueError('argument for construct() wrong length')
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for i in range(len(tup)):
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field = obj.keydata[i]
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setattr(obj, field, tup[i])
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return obj
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class ElGamalobj(pubkey):
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"""Class defining an ElGamal key.
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:undocumented: __getstate__, __setstate__, __repr__, __getattr__
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"""
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#: Dictionary of ElGamal parameters.
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#:
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#: A public key will only have the following entries:
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#:
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#: - **y**, the public key.
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#: - **g**, the generator.
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#: - **p**, the modulus.
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#:
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#: A private key will also have:
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#:
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#: - **x**, the private key.
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keydata=['p', 'g', 'y', 'x']
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def encrypt(self, plaintext, K):
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"""Encrypt a piece of data with ElGamal.
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:Parameter plaintext: The piece of data to encrypt with ElGamal.
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It must be numerically smaller than the module (*p*).
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:Type plaintext: byte string or long
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:Parameter K: A secret number, chosen randomly in the closed
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range *[1,p-2]*.
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:Type K: long (recommended) or byte string (not recommended)
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:Return: A tuple with two items. Each item is of the same type as the
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plaintext (string or long).
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:attention: selection of *K* is crucial for security. Generating a
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random number larger than *p-1* and taking the modulus by *p-1* is
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**not** secure, since smaller values will occur more frequently.
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Generating a random number systematically smaller than *p-1*
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(e.g. *floor((p-1)/8)* random bytes) is also **not** secure.
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In general, it shall not be possible for an attacker to know
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the value of any bit of K.
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:attention: The number *K* shall not be reused for any other
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operation and shall be discarded immediately.
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"""
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return pubkey.encrypt(self, plaintext, K)
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def decrypt(self, ciphertext):
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"""Decrypt a piece of data with ElGamal.
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:Parameter ciphertext: The piece of data to decrypt with ElGamal.
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:Type ciphertext: byte string, long or a 2-item tuple as returned
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by `encrypt`
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:Return: A byte string if ciphertext was a byte string or a tuple
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of byte strings. A long otherwise.
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"""
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return pubkey.decrypt(self, ciphertext)
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def sign(self, M, K):
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"""Sign a piece of data with ElGamal.
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:Parameter M: The piece of data to sign with ElGamal. It may
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not be longer in bit size than *p-1*.
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:Type M: byte string or long
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:Parameter K: A secret number, chosen randomly in the closed
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range *[1,p-2]* and such that *gcd(k,p-1)=1*.
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:Type K: long (recommended) or byte string (not recommended)
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:attention: selection of *K* is crucial for security. Generating a
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random number larger than *p-1* and taking the modulus by *p-1* is
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**not** secure, since smaller values will occur more frequently.
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Generating a random number systematically smaller than *p-1*
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(e.g. *floor((p-1)/8)* random bytes) is also **not** secure.
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In general, it shall not be possible for an attacker to know
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the value of any bit of K.
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:attention: The number *K* shall not be reused for any other
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operation and shall be discarded immediately.
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:attention: M must be be a cryptographic hash, otherwise an
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attacker may mount an existential forgery attack.
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:Return: A tuple with 2 longs.
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"""
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return pubkey.sign(self, M, K)
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def verify(self, M, signature):
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"""Verify the validity of an ElGamal signature.
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:Parameter M: The expected message.
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:Type M: byte string or long
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:Parameter signature: The ElGamal signature to verify.
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:Type signature: A tuple with 2 longs as return by `sign`
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:Return: True if the signature is correct, False otherwise.
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"""
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return pubkey.verify(self, M, signature)
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def _encrypt(self, M, K):
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a=pow(self.g, K, self.p)
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b=( M*pow(self.y, K, self.p) ) % self.p
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return ( a,b )
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def _decrypt(self, M):
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if (not hasattr(self, 'x')):
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raise TypeError('Private key not available in this object')
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ax=pow(M[0], self.x, self.p)
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plaintext=(M[1] * inverse(ax, self.p ) ) % self.p
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return plaintext
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def _sign(self, M, K):
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if (not hasattr(self, 'x')):
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raise TypeError('Private key not available in this object')
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p1=self.p-1
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if (GCD(K, p1)!=1):
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raise ValueError('Bad K value: GCD(K,p-1)!=1')
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a=pow(self.g, K, self.p)
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t=(M-self.x*a) % p1
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while t<0: t=t+p1
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b=(t*inverse(K, p1)) % p1
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return (a, b)
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def _verify(self, M, sig):
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if sig[0]<1 or sig[0]>self.p-1:
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return 0
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v1=pow(self.y, sig[0], self.p)
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v1=(v1*pow(sig[0], sig[1], self.p)) % self.p
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v2=pow(self.g, M, self.p)
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if v1==v2:
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return 1
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return 0
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def size(self):
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return number.size(self.p) - 1
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def has_private(self):
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if hasattr(self, 'x'):
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return 1
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else:
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return 0
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def publickey(self):
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return construct((self.p, self.g, self.y))
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object=ElGamalobj
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