El Gamal

Warning

Even though ElGamal algorithms are in theory reasonably secure, in practice there are no real good reasons to prefer them to RSA instead.

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 \(y=g^x \text{ mod } p\)) 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 2017, a sufficient size for p is deemed to be 2048 bits. For more information, see the most recent ECRYPT_ report.

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 constructing them from known components.

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.Random import random
>>> from Crypto.PublicKey import ElGamal
>>> from Crypto.Util.number import GCD
>>> from Crypto.Hash import SHA
>>>
>>> message = "Hello"
>>> key = ElGamal.generate(1024, Random.new().read)
>>> h = SHA.new(message).digest()
>>> while 1:
>>>     k = random.StrongRandom().randint(1,key.p-1)
>>>     if GCD(k,key.p-1)==1: break
>>> sig = key.sign(h,k)
>>> ...
>>> if key.verify(h,sig):
>>>     print "OK"
>>> else:
>>>     print "Incorrect signature"
class Crypto.PublicKey.ElGamal.ElGamalobj

Class defining an ElGamal key.

Undocumented:

__getstate__, __setstate__, __repr__, __getattr__

decrypt(ciphertext)

Decrypt a piece of data with ElGamal.

Parameter ciphertext:

The piece of data to decrypt with ElGamal.

Type ciphertext:

byte string, long or a 2-item tuple as returned by encrypt

Return:

A byte string if ciphertext was a byte string or a tuple of byte strings. A long otherwise.

encrypt(plaintext, K)

Encrypt a piece of data with ElGamal.

Parameter plaintext:

The piece of data to encrypt with ElGamal. It must be numerically smaller than the module (p).

Type plaintext:

byte string or long

Parameter K:

A secret number, chosen randomly in the closed range [1,p-2].

Type K:

long (recommended) or byte string (not recommended)

Return:

A tuple with two items. Each item is of the same type as the plaintext (string or long).

Attention:

selection of K is crucial for security. Generating a random number larger than p-1 and taking the modulus by p-1 is not secure, since smaller values will occur more frequently. Generating a random number systematically smaller than p-1 (e.g. floor((p-1)/8) random bytes) is also not secure. In general, it shall not be possible for an attacker to know the value of any bit of K.

Attention:

The number K shall not be reused for any other operation and shall be discarded immediately.

has_private()

Tell if the key object contains private components.

Return:

bool

keydata = ['p', 'g', 'y', 'x']

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.

publickey()

Construct a new key carrying only the public information.

Return:

A new pubkey object.

sign(M, K)

Sign a piece of data with ElGamal.

Parameter M:

The piece of data to sign with ElGamal. It may not be longer in bit size than p-1.

Type M:

byte string or long

Parameter K:

A secret number, chosen randomly in the closed range [1,p-2] and such that gcd(k,p-1)=1.

Type K:

long (recommended) or byte string (not recommended)

Attention:

selection of K is crucial for security. Generating a random number larger than p-1 and taking the modulus by p-1 is not secure, since smaller values will occur more frequently. Generating a random number systematically smaller than p-1 (e.g. floor((p-1)/8) random bytes) is also not secure. In general, it shall not be possible for an attacker to know the value of any bit of K.

Attention:

The number K shall not be reused for any other operation and shall be discarded immediately.

Attention:

M must be be a cryptographic hash, otherwise an attacker may mount an existential forgery attack.

Return:

A tuple with 2 longs.

size()

Tell the maximum number of bits that can be handled by this key.

Return:

int

verify(M, signature)

Verify the validity of an ElGamal signature.

Parameter M:

The expected message.

Type M:

byte string or long

Parameter signature:

The ElGamal signature to verify.

Type signature:

A tuple with 2 longs as return by sign

Return:

True if the signature is correct, False otherwise.

Crypto.PublicKey.ElGamal.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:
tuptuple

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.

Return:

An ElGamal key object (ElGamalobj).

exception Crypto.PublicKey.ElGamal.error
Crypto.PublicKey.ElGamal.generate(bits, randfunc, progress_func=None)

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:
bitsint

Key length, or size (in bits) of the modulus p. Recommended value is 2048.

randfunccallable

Random number generation function; it should accept a single integer N and return a string of random data N bytes long.

progress_funccallable

Optional function that will be called with a short string containing the key parameter currently being generated; it’s useful for interactive applications where a user is waiting for a key to be generated.

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 (ElGamalobj).