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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. The system provides an additional layer of security by asymmetrically encrypting keys previously used for symmetric message encryption. 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.

ElGamal encryption can be defined over any cyclic group . Its security depends upon the difficulty of a certain problem in related to computing discrete logarithms (see below).


The algorithmEdit

ElGamal encryption consists of three components: the key generator, the encryption algorithm, and the decryption algorithm.

Key generationEdit

The key generator works as follows:

  • Alice generates an efficient description of a cyclic group   of order   with generator  . See below for a discussion on the required properties of this group.
  • Alice chooses an   randomly from  .
  • Alice computes  .
  • Alice publishes  , along with the description of  , as her public key. Alice retains   as her private key, which must be kept secret.


The encryption algorithm works as follows: to encrypt a message   to Alice under her public key  ,

  • Bob chooses a random   from  , then calculates  .
  • Bob calculates the shared secret   .
  • Bob maps his message   onto an element   of  .
  • Bob calculates  .
  • Bob sends the ciphertext   to Alice.

Note that one can easily find   if one knows  . Therefore, a new   is generated for every message to improve security. For this reason,   is also called an ephemeral key.


The decryption algorithm works as follows: to decrypt a ciphertext   with her private key  ,

  • Alice calculates the shared secret  
  • and then computes   which she then converts back into the plaintext message  , where   is the inverse of   in the group  . (E.g. modular multiplicative inverse if   is a subgroup of a multiplicative group of integers modulo n).
The decryption algorithm produces the intended message, since

Practical useEdit

The ElGamal cryptosystem is usually used in a hybrid cryptosystem. I.e., the message itself is encrypted using a symmetric cryptosystem and ElGamal is then used to encrypt the key used for the symmetric cryptosystem. 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 symmetric key (which most of the time is quite small if compared to the size of the message) with Elgamal and the message (which can be arbitrarily large) with a symmetric cipher.


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  , then ElGamal achieves semantic security.[2] Semantic security is not implied by the computational Diffie–Hellman assumption alone.[3] 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 DHAES,[3] whose proof requires an assumption that is weaker than the DDH assumption.


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 2:1 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 need be. Decryption only requires one exponentiation:


The division by   can be avoided by using an alternative method for decryption. To decrypt a ciphertext   with Alice's private key  ,

  • Alice calculates  .

  is the inverse of  . This is a consequence of Lagrange's theorem, because

where   is the identity element of  .
  • Alice then computes  , which she then converts back into the plaintext message  .
The decryption algorithm produces the intended message, since

See alsoEdit


  1. ^ Taher ElGamal (1985). "A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms" (PDF). IEEE Transactions on Information Theory. 31 (4): 469–472. doi:10.1109/TIT.1985.1057074.  (conference version appeared in CRYPTO'84, pp. 10–18)
  2. ^ a b CRYPTUTOR, "Elgamal encryption scheme Archived 2009-04-21 at the Wayback Machine."
  3. ^ a b M. Abdalla, M. Bellare, P. Rogaway, "DHAES, An encryption scheme based on the Diffie–Hellman Problem" (Appendix A)