Paper 2004/072 Original: eprint.iacr.org/2004/072 ↗

Asymmetric Cryptography: Hidden Field Equations

Abstract

The most popular public key cryptosystems rely on assumptions from algebraic number theory, e.g., the difficulty of factorisation or the discrete logarithm. The set of problems on which secure public key systems can be based is therefore very small: e.g., a breakthrough in factorisation would make RSA insecure and hence affect our digital economy quite dramatically. This would be the case if quantum-computer with a large number of qbits were available. Therefore, a wider range of candidate hard problems is needed. In 1996, Patarin proposed the ``Hidden Field Equations" (HFE) as a base for public key cryptosystems. In a nutshell, they use polynomials over finite fields of different size to disguise the relationship between the private key and the public key. In these systems, the public key consists of multivariate polynomials over finite fields with up to 256 elements for practical implementations. Over finite fields, solving these equations has been shown to be an NP-complete problem. In addition, empirical results show that this problem is also hard on average, i.e., it can be used for a secure public key signature or encryption scheme. In this article, we outline HFE, and its the variations HFE-, HFEv. Moreover, we describe the signature scheme Quartz, which is based on Hidden Field Equations. In addition, we describe the most recent attacks against HFE and sketch two versions of Quartz which are immune against these attacks.