GGH encryption scheme

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The Goldreich–Goldwasser–Halevi (GGH) lattice-based cryptosystem is an asymmetric cryptosystem based on lattices. There is also a GGH signature scheme.

The Goldreich–Goldwasser–Halevi (GGH) cryptosystem makes use of the fact that the closest vector problem can be a hard problem. It was published in 1997 by Oded Goldreich, Shafi Goldwasser, and Shai Halevi, and uses a trapdoor one-way function that is relying on the difficulty of lattice reduction. The idea included in this trapdoor function is that, given any basis for a lattice, it is easy to generate a vector which is close to a lattice point, for example taking a lattice point and adding a small error vector. But to return from this erroneous vector to the original lattice point a special basis is needed.

The GGH encryption scheme was cryptanalyzed in 1999 by Phong Q. Nguyen.


GGH involves a private key and a public key.

The private key is a basis of a lattice with good properties (such as short nearly orthogonal vectors) and a unimodular matrix .

The public key is another basis of the lattice of the form .

For some chosen M, the message space consists of the vector in the range .


Given a message , error , and a public key compute

In matrix notation this is


Remember consists of integer values, and is a lattice point, so v is also a lattice point. The ciphertext is then


To decrypt the cyphertext one computes

The Babai rounding technique will be used to remove the term as long as it is small enough. Finally compute

to get the messagetext.


Let be a lattice with the basis and its inverse




this gives

Let the message be and the error vector . Then the ciphertext is

To decrypt one must compute

This is rounded to and the message is recovered with

Security of the scheme[edit]

1999 Nguyen showed at the Crypto conference that the GGH encryption scheme has a flaw in the design of the schemes. He showed that every ciphertext reveals information about the plaintext and that the problem of decryption could be turned into a special closest vector problem much easier to solve than the general CVP.


  • Oded Goldreich, Shafi Goldwasser, and Shai Halevi. Public-key cryptosystems from lattice reduction problems. In CRYPTO ’97: Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology, pages 112–131, London, UK, 1997. Springer-Verlag.
  • Phong Q. Nguyen. Cryptanalysis of the Goldreich–Goldwasser–Halevi Cryptosystem from Crypto ’97. In CRYPTO ’99: Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology, pages 288–304, London, UK, 1999. Springer-Verlag.