On the underlying hard lattice problems of GGH encryption scheme

Based on numerous experiments, the inventors of the Goldreich, Goldwasser and Halevi encryption scheme (GGH Scheme) conjectured that the Closest Vector Problem (CVP) instance which arose from the scheme was practically intractable in lattice with a dimension more than 300. However, some attacks had successfully broken the security of the scheme. Instead of solving the conjectured intractable CVP instance, these attacks managed to simplify the instance into its simpler form. Consequently, the security of the GGH Scheme is considered breached. In this paper, we address two most notable attacks on the GGH Scheme. On top of that, we propose a new attack on the GGH Scheme that manages to simplify the underlying CVP instance into a much simpler form. From that, we explicitly define the underlying CVP instance that arises from the GGH Scheme together with its corresponding simplified instances to give further illustration on the weakness points of the scheme. By identifying how these weaknesses are exploited by the attacks, further improvement on the GGH Scheme can be carried out in the future for making the scheme better and stronger.

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