ElGamal encryption explained
In cryptography, the ElGamal encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie–Hellman key exchange. It was described by Taher Elgamal in 1985.[1] ElGamal encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption.
, like
multiplicative group of integers modulo n if and only if
n is 1, 2, 4,
pk or 2
pk, where
p is an odd prime and . Its security depends upon the difficulty of a certain problem in
related to computing
discrete logarithms.
The algorithm
The algorithm can be described as first performing a Diffie–Hellman key exchange to establish a shared secret
, then using this as a
one-time pad for encrypting the message. ElGamal encryption is performed in three phases: the key generation, the encryption, and the decryption. The first is purely key exchange, whereas the latter two mix key exchange computations with message computations.
Key generation
The first party, Alice, generates a key pair as follows:
of
order
with
generator
. Let
represent the identity element of
.
It is not necessary to come up with a group and generator for each new key. Indeed, one may expect a specific implementation of ElGamal to be hardcoded to use a specific group, or a group from a specific suite. The choice of group is mostly about how large keys you want to use.
randomly from
.
.
. Alice publishes this public key and retains
as her
private key, which must be kept secret.
Encryption
A second party, Bob, encrypts a message
to Alice under her public key
as follows:
to an element
of
using a reversible mapping function.
randomly from
.
. This is called the
shared secret.
.
.
to Alice.
Note that if one knows both the ciphertext
and the plaintext
, one can easily find the shared secret
, since
. Therefore, a new
and hence a new
is generated for every message to improve security. For this reason,
is also called an
ephemeral key.
Decryption
Alice decrypts a ciphertext
with her private key
as follows:
. Since
,
, and thus it is the same shared secret that was used by Bob in encryption.
, the inverse of
in the group
. This can be computed in one of several ways. If
is a subgroup of a multiplicative group of integers modulo
, where
is prime, the
modular multiplicative inverse can be computed using the
extended Euclidean algorithm. An alternative is to compute
as
. This is the inverse of
because of
Lagrange's theorem, since
s ⋅
=gxy ⋅ g(q-x)y=(gq)y=ey=e
.
. This calculation produces the original message
, because
; hence
c2 ⋅ s-1=(m ⋅ s) ⋅ s-1=m ⋅ e=m
.
back to the plaintext message
.
Practical use
Like most public key systems, the ElGamal cryptosystem is usually used as part of a hybrid cryptosystem, where the message itself is encrypted using a symmetric cryptosystem, and ElGamal is then used to encrypt only the symmetric key. This is because asymmetric cryptosystems like ElGamal are usually slower than symmetric ones for the same level of security, so it is faster to encrypt the message, which can be arbitrarily large, with a symmetric cipher, and then use ElGamal only to encrypt the symmetric key, which usually is quite small compared to the size of the message.
Security
The security of the ElGamal scheme depends on the properties of the underlying group
as well as any padding scheme used on the messages. If the
computational Diffie–Hellman assumption (CDH) holds in the underlying cyclic group
, then the encryption function is
one-way.
[2] If the decisional Diffie–Hellman assumption (DDH) holds in
, thenElGamal achieves
semantic security.
[2] [3] Semantic security is not implied by the computational Diffie–Hellman assumption alone. See
Decisional Diffie–Hellman assumption for a discussion of groups where the assumption is believed to hold.
ElGamal encryption is unconditionally malleable, and therefore is not secure under chosen ciphertext attack. For example, given an encryption
of some (possibly unknown) message
, one can easily construct a valid encryption
of the message
.
To achieve chosen-ciphertext security, the scheme must be further modified, or an appropriate padding scheme must be used. Depending on the modification, the DDH assumption may or may not be necessary.
Other schemes related to ElGamal which achieve security against chosen ciphertext attacks have also been proposed. The Cramer–Shoup cryptosystem is secure under chosen ciphertext attack assuming DDH holds for
. Its proof does not use the
random oracle model. Another proposed scheme is
DHIES,
[4] whose proof requires an assumption that is stronger than the DDH assumption.
Efficiency
ElGamal encryption is probabilistic, meaning that a single plaintext can be encrypted to many possible ciphertexts, with the consequence that a general ElGamal encryption produces a 1:2 expansion in size from plaintext to ciphertext.
Encryption under ElGamal requires two exponentiations; however, these exponentiations are independent of the message and can be computed ahead of time if needed. Decryption requires one exponentiation and one computation of a group inverse, which can, however, be easily combined into just one exponentiation.
See also
Further reading
- Book: A. J. Menezes . P. C. van Oorschot . S. A. Vanstone . CRC Press . https://cacr.uwaterloo.ca/hac/about/chap8.pdf . Handbook of Applied Cryptography . Chapter 8.4 ElGamal public-key encryption.
- Book: Dan Boneh . Algorithmic Number Theory . The Decision Diffie-Hellman problem . Lecture Notes in Computer Science . 1998 . 1423 . 48–63 . 10.1007/BFb0054851 . 978-3-540-64657-0 . 10.1.1.461.9971 . Dan Boneh.
Notes and References
- Taher ElGamal . A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms . IEEE Transactions on Information Theory . 31 . 4 . 1985 . 469–472 . 10.1109/TIT.1985.1057074 . 10.1.1.476.4791 . 2973271 . (conference version appeared in CRYPTO'84, pp. 10–18)
- Web site: Elgamal encryption scheme. Mike Rosulek. 2008-12-13. University of Illinois at Urbana-Champaign. https://web.archive.org/web/20160722005050/https://crypto.cs.uiuc.edu/wiki/index.php/Elgamal_encryption_scheme. 2016-07-22. dead.
- Book: Yiannis. Tsiounis. Moti. Yung. Public Key Cryptography. On the security of ElGamal based encryption. 2006-05-24. 117–134. 1431. 10.1007/BFb0054019. 978-3-540-69105-1. Lecture Notes in Computer Science.
- Book: Michel. Abdalla. Mihir. Bellare. Phillip. Rogaway. Topics in Cryptology — CT-RSA 2001. The Oracle Diffie-Hellman Assumptions and an Analysis of DHIES. 2001-01-01. Lecture Notes in Computer Science. 2020. 143–158. 10.1007/3-540-45353-9_12. 978-3-540-41898-6. https://link.springer.com/chapter/10.1007/3-540-45353-9_12.