#
# 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