Algebraic attack on NTRU using Witt vectors and Grobner bases

摘要

We present an algebraic attack on NTRU (restricted to the case where the parameter q is a power of two) using the method of the Witt vectors proposed by Silverman, Smart and Vercauteren [17]; the latter considered only the first two bits of a Witt vector attached to the recovering of the secret key in order to reduce the problem to the resolution of an algebraic system over F_2. The theoretical complexity of this resolution was not studied by the authors. In this paper, we use the first three bits of the Witt vectors to obtain supplementary equations which allow us to reduce the complexity of the attack. Using Grobner basis complexity results of overdetermined systems, we have been able to provide a theoretical complexity analysis. Additionally we provide experimental results illustrating the efficiency of this approach. Moreover, we prove that the use of the fourth bit does not improve the complexity, what is surprising. Unfortunately, for standard values of the NTRU parameters, the proven complexity is around 2~(246) and this attack does not make it possible to find the private key.