Generalisation of Kipnis and Shamir Cryptanalysis of the HFE public key cryptosystem

Abstract — In [4], Kipnis and Shamir have cryptanaliseda version of HFE of degree 2. In this paper, we describe thegeneralization of this attack of HFE of degree more than 2.We are based on Fourier Transformation to acheive partiallythis attack. Keywords — Public, cryptosystem, cryptanalisis, HFE.I. INTRODUCTION P UBLIC key cryptography depends on a handful of alge-braic problems which try to achieve security. The originalRSA problem requires large blocs sizes. Other altenatives withsmall size have been proposed: Elliptic curves and recently thefamily of quadratic multivariate schemes such as HFE (HiddenField Equations)[3][1].The security of this system is based on the difficulty ofsolving large systems of quadratic multivariate polynomialequations[2]. This problem is NP-hard over any field. The mostefficient attack is the one of Kipnis and Shamir that consist indetermining the secret key from the public key.This attack is based on a non standard representation of theHFE. In this paper, we generalise the idea of Kipnis andShamir to attack partially the HFE cryptosystem of degree3.II. HFE