ElGamal encryption

"ElGamal" redirects here. For signature algorithm, seeElGamal signature scheme.

Incryptography,theElGamal encryption systemis anasymmetric key encryption algorithmforpublic-key cryptographywhich is based on theDiffie–Hellman key exchange.It was described byTaher Elgamalin 1985.ElGamal encryption is used in the freeGNU Privacy Guardsoftware, recent versions ofPGP,and othercryptosystems.TheDigital Signature Algorithm(DSA) is a variant of theElGamal signature scheme,which should not be confused with ElGamal encryption.

ElGamal encryption can be defined over anycyclic group$G$,likemultiplicative group of integers modulonif and only ifnis 1, 2, 4,p^kor 2p^k,wherepis an odd prime andk> 0.Its security depends upon the difficulty of a certain problem in$G$related to computingdiscrete logarithms.

The algorithm

The algorithm can be described as first performing a Diffie–Hellman key exchange to establish a shared secret$s$,then using this as aone-time padfor 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:

• Generate an efficient description of acyclic group$G\,$oforder$q\,$withgenerator$g$.Let$e$represent the identity element of$G$.

  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.

• Choose an integer$x$randomly from$\{1,\ldots,q-1\}$.
• Compute$h:=g^{x}$.
• Thepublic keyconsists of the values$(G,q,g,h)$.Alice publishes this public key and retains$x$as herprivate key,which must be kept secret.

Encryption

A second party, Bob, encrypts a message$M$to Alice under her public key$(G,q,g,h)$as follows:

• Map the message$M$to an element$m$of$G$using a reversible mapping function.
• Choose an integer$y$randomly from$\{1,\ldots,q-1\}$.
• Compute$s:=h^{y}$.This is called theshared secret.
• Compute$c_{1}:=g^{y}$.
• Compute$c_{2}:=m\cdot s$.
• Bob sends the ciphertext$(c_{1},c_{2})$to Alice.

Note that if one knows both the ciphertext$(c_{1},c_{2})$and the plaintext$m$,one can easily find the shared secret$s$,since$c_{2}\cdot m^{-1}=s$.Therefore, a new$y$and hence a new$s$is generated for every message to improve security. For this reason,$y$is also called anephemeral key.

Decryption

Alice decrypts a ciphertext$(c_{1},c_{2})$with her private key$x$as follows:

• Compute$s:=c_{1}^{x}$.Since$c_{1}=g^{y}$,$c_{1}^{x}=g^{xy}=h^{y}$,and thus it is the same shared secret that was used by Bob in encryption.
• Compute$s^{-1}$,the inverse of$s$in the group$G$.This can be computed in one of several ways. If$G$is a subgroup of a multiplicative group of integers modulo$n$,where$n$is prime, themodular multiplicative inversecan be computed using theextended Euclidean algorithm.An alternative is to compute$s^{-1}$as$c_{1}^{q-x}$.This is the inverse of$s$because ofLagrange's theorem,since$s\cdot c_{1}^{q-x}=g^{xy}\cdot g^{(q-x)y}=(g^{q})^{y}=e^{y}=e$.
• Compute$m:=c_{2}\cdot s^{-1}$.This calculation produces the original message$m$,because$c_{2}=m\cdot s$;hence$c_{2}\cdot s^{-1}=(m\cdot s)\cdot s^{-1}=m\cdot e=m$.
• Map$m$back to the plaintext message$M$.

Practical use

Like most public key systems, the ElGamal cryptosystem is usually used as part of ahybrid 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 samelevel 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$G$as well as any padding scheme used on the messages. If thecomputational Diffie–Hellman assumption(CDH) holds in the underlying cyclic group$G$,then the encryption function isone-way.

If thedecisional Diffie–Hellman assumption(DDH) holds in$G$,then ElGamal achievessemantic security.Semantic security is not implied by the computational Diffie–Hellman assumption alone. SeeDecisional Diffie–Hellman assumptionfor a discussion of groups where the assumption is believed to hold.

ElGamal encryption is unconditionallymalleable,and therefore is not secure underchosen ciphertext attack.For example, given an encryption$(c_{1},c_{2})$of some (possibly unknown) message$m$,one can easily construct a valid encryption$(c_{1},2c_{2})$of the message$2m$.

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. TheCramer–Shoup cryptosystemis secure under chosen ciphertext attack assuming DDH holds for$G$.Its proof does not use therandom oracle model.Another proposed scheme isDHIES,whose proof requires an assumption that is stronger than the DDH assumption.

Efficiency

ElGamal encryption isprobabilistic,meaning that a singleplaintextcan 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 twoexponentiations;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

• Taher Elgamal,designer of this and other cryptosystems
• ElGamal signature scheme
• Homomorphic encryption

Further reading

• A. J. Menezes; P. C. van Oorschot; S. A. Vanstone."Chapter 8.4 ElGamal public-key encryption"(PDF).Handbook of Applied Cryptography.CRC Press.
• Dan Boneh(1998). "The Decision Diffie-Hellman problem".Algorithmic Number Theory.Lecture Notes in Computer Science. Vol. 1423. pp. 48–63.CiteSeerX10.1.1.461.9971.doi:10.1007/BFb0054851.ISBN978-3-540-64657-0.