The one use case where I see ElGamal being used over RSA is when a multiplicatively homomorphic cryptosystem is needed (noe that both ElGamal and RSA are multiplicatively homomorphic). ElGamal encryption is unconditionally malleable, and therefore is not secure under chosen ciphertext attack. It is mainly used to establish common keys and not to encrypt messages. • We can use the above fact to come up with two message where one of them is a quadratic residue and the other one is a quadratic non-residue so that The ElGamal cryptographic algorithm is a public key system like the Diffie-Hellman system. The ElGamal cryptographic algorithm is comparable to the Diffie-Hellman system. Nowadays, RSA is more convenient but not necessarily safer than ElGamal. The reason why GnuPG used to defaul to ElGamal was probably related to patents. Introduction. Notably, ElGamal is homomorphic (given the encryption of x and the encryption of y, one can from the outside compute the encryption of the product xy), which is a nice property in some cases, but can be bothersome in other conditions. As for ElGamal versus RSA: An encrypted ElGamal message is simply about twice the size of an encrypted RSA message for the same security level. The cryptosystem takes its name from its founder the Egyptian cryptographer Taher Elgamal who introduced the system in his 1985 paper entitled " A Public Key Cryptosystem and A Signature Scheme Based on Discrete Logarithms ". As this title suggests the security of this cryptosystem is based on the notion of discrete logarit It can be considered as the asymmetric algorithm where the encryption and decryption happen by the use of public and private keys. For example, given an encryption of some (possibly unknown) message , one can easily construct a valid encryption of the message . Diffie-Hellman enables two parties to agree a common shared secret that can be used subsequently in a symmetric algorithm like AES. Diffie-Hellman (DH) is a key agreement algorithm, ElGamal an asymmetric encryption algorithm. The Diffie-Hellman key exchange provides a method of sharing a secret key between Alice and Bob, but does not allow Alice and Bob to otherwise communicate securely. ElGamal cryptosystem can be defined as the cryptography algorithm that uses the public and private key concept to secure the communication occurring between two systems. While ElGamal over appropriate prime order subgroups is semantically secure and ElGamal over ∗ is not we should not conclude that any ElGamal implementation using the group ∗ is immediately insecure and any system using a prime order subgroup is secure. Stick to RSA. ElGamal encryption is an example of public-key or asymmetric cryptography. ElGamal is a public key encryption algorithm that was described by an Egyptian cryptographer Taher Elgamal in 1985. Erik-Oliver Blass and I found that the implementations of ElGamal encryption in libgcrypt, PyCrypto, PyCryptodome, and CryptoPP are not secure. It all depends on what padding scheme is used. Now, RSA patents have expired. • We can infer whether a ciphertext is quadratic residue or not. The ElGamal cryptosystem was first described by Taher Elgamal in 1985 and is closely related to the Diffie-Hellman key exchange. DSA and Elgamal; RSA (Sign only) DSA (Sign only) I found this Superuser question, but it may be outdated. Semantic Security of ElGamal • Note that the generic ElGamal encryption scheme is not semantically secure. RSA keys seem to be less secure since it's known that the NSA infiltrated RSA and made their key generation algorithm weaker. I don't know if this has a significant influence for a 4096 bit RSA key. Be cyber secure. 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