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:
Compute. Since, and thus it is the same shared secret that was used by Bob in encryption.
Compute, 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 n, 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.
Compute. This calculation produces the original message, because ; hence.
Map 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 holds in the underlying cyclic group, then the encryption function is one-way. If the decisional Diffie–Hellman assumption holds in, then ElGamal achieves semantic security;. 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 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, whose proof requires an assumption that is weaker 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 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 requires one exponentiation and one computation of a group inverse which can however be easily combined into just one exponentiation.
