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 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 it is not known how to simply return from this erroneous vector to the original lattice point.

## OperationEdit

GGH involves a private key and a public key.

The private key is a basis $ B $ of a lattice $ L $ with good properties, such as short nearly orthogonal vectors and a unimodular matrix $ U $.

The public key is another basis of the lattice $ L $ of the form $ B'=UB $.

For some chosen M, the message space consists of the vector $ (\lambda_1,..., \lambda_n) $ in the range $ -M <\lambda_i < M $.

### Encryption Edit

Given a message $ m = (\lambda_1,..., \lambda_n) $, error $ e $, and a public key $ B' $ compute

- $ v = \sum \lambda_i b_i' $

In matrix notation this is

- $ v=m\cdot B' $.

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

- $ c=v+e=m \cdot B' + e $

### Decryption Edit

To decrypt the cyphertext one computes

- $ c \cdot B^{-1} = (m\cdot B^\prime +e)B^{-1} = m\cdot U\cdot B\cdot B^{-1} + e\cdot B^{-1} = m\cdot U + e\cdot B^{-1} $

The Babai rounding technique will be used to remove the term $ e \cdot B^{-1} $ as long as it is small enough. Finally compute

- $ m = m \cdot U \cdot U^{-1} $

to get the messagetext.

## ExampleEdit

Let $ L \subset \mathbb{R}^2 $ be a lattice with the basis $ B $ and its inverse $ B^{-1} $

- $ B= \begin{pmatrix} 7 & 0 \\ 0 & 3 \\ \end{pmatrix} $ and $ B^{-1}= \begin{pmatrix} \frac{1}{7} & 0 \\ 0 & \frac{1}{3} \\ \end{pmatrix} $

With

- $ U = \begin{pmatrix} 2 & 3 \\ 3 &5\\ \end{pmatrix} $ and
- $ U^{-1} = \begin{pmatrix} 5 & -3 \\ -3 &2\\ \end{pmatrix} $

this gives

- $ B' = U B = \begin{pmatrix} 14 & 9 \\ 21 & 15 \\ \end{pmatrix} $

Let the message be $ m = (3, -7) $ and the error vector $ e = (1, -1) $. Then the ciphertext is

- $ c = m B'+e=(-104, -79).\, $

To decrypt one must compute

- $ c B^{-1} = \left( \frac{-104}{7}, \frac{-79}{3}\right). $

This is rounded to $ (-15, -26) $ and the message is recovered with

- $ m= (-15, -26) U^{-1} = (3, -7).\, $

## Security of the schemeEdit

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.

## BibliographyEdit

- 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.