Generalization of Isomorphism of Polynomials with Two Secrets and Its Application to Public Key Encryption

Abstract

Most of the public key encryption (PKE) schemes based on multivariate quadratic polynomials rely on Hidden Field Equation (HFE) paradigm. However, most of HFE based schemes have been broken in only several years just after their introduction. In this paper, we propose an alternative paradigm for constructing PKE based on multivariate quadratic polynomials. At the heart of our proposal is a new family of computational problems based on the generalization of Isomorphism of Polynomials with Two Secrets (IP2S) problem. The main computational problem in the new family is proven as hard as the original IP2S problem and is more robust, in the sense that we can associate it with circulant matrices as solutions without degrading its computational hardness too much, in contrast to the original IP2S problem which immediately becomes easy as soon as it is associated with circulant matrices. By associating it to circulant matrices, we obtain a Diffie-Hellman like structure which allows us to have an El-Gamal like PKE scheme.