Paper 2008/543 Original: eprint.iacr.org/2008/543 ↗

Odd-Char Multivariate Hidden Field Equations

Abstract

We present a multivariate version of Hidden Field Equations (HFE) over a finite field of odd characteristic, with an extra ``embedding'' modifier. Combining these known ideas makes our new MPKC (multivariate public key cryptosystem) more efficient and scalable than any other extant multivariate encryption scheme. Switching to odd characteristics in HFE-like schemes affects how an attacker can make use of field equations. Extensive empirical tests (using MAGMA-2.14, the best commercially available \mathbold{F_4} implementation) suggests that our new construction is indeed secure against algebraic attacks using Gröbner Basis algorithms. The ``embedding'' serves both to narrow down choices of pre-images and to guard against a possible Kipnis-Shamir type (rank-based) attack. We may hence reasonably argue that for practical sizes, prior attacks take exponential time. We demonstrate that our construction is in fact efficient by implementing practical-sized examples of our ``odd-char HFE'' with 3 variables (``THFE'') over $\mathrm{GF}(31)$. To be precise, our preliminary THFE implementation is $15\times$--$20\times$ the speed of RSA-1024.