Cryptanalysis of HFE with Internal Perturbation

Multivariate Cryptography has been an active line of research for almost twenty years. While most multivariate cryptosystems have been under attack, variations of the basic schemes came up as potential repairs. In this paper, we study the Internal Perturbation variation of HFE recently proposed by Ding and Schmidt. Although several results indicate that HFE is vulnerable against algebraic attacks for moderate size parameters, Ding and Schmidt claim that the cryptosystem with internal perturbation should be immune against them. However in this paper, we apply the recently discovered method of differential analysis to the Internal Perturbation of HFE and we find a subtle property which allows to disclose the kernel of the perturbation. Once this has been achieved, the public key can be inverted by attacking the underlying HFE provided the parameters were taken low enough to make the perturbed scheme of competitive performance.