GGH encryption algorithm

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. This system 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 ${\displaystyle B}$ of a lattice ${\displaystyle L}$ with good properties (such as short nearly orthogonal vectors) and a unimodular matrix ${\displaystyle U}$.

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