# 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. 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.

## Operation

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 ${\displaystyle U}$ .

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

For some chosen M, the message space consists of the vector ${\displaystyle (m_{1},...,m_{n})}$  in the range ${\displaystyle -M .

### Encryption

Given a message ${\displaystyle m=(m_{1},...,m_{n})}$ , error ${\displaystyle e}$ , and a public key ${\displaystyle B'}$  compute

${\displaystyle v=\sum m_{i}b_{i}'}$

In matrix notation this is

${\displaystyle v=m\cdot B'}$ .

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

${\displaystyle c=v+e=m\cdot B'+e}$

### Decryption

To decrypt the cyphertext one computes

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

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

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

to get the messagetext.

## Example

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

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

With

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

this gives

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

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

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

To decrypt one must compute

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

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

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

## 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. Nguyen 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.