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 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 $B$ 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

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' = 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 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.

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