# GGH encryption scheme

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 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.

## Operation

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}.

### Encryption

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

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.

## Example

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

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

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

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

To decrypt one must compute

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

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

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

## Security of the scheme

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.

## Bibliography

• 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.