Cryptology ePrint Archive: Report 2008/543

Odd-Char Multivariate Hidden Field Equations

Chia-Hsin Owen Chen and Ming-Shing Chen and Jintai Ding and Fabian Werner and Bo-Yin Yang

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 Gr\"obner 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.

Category / Keywords: public-key cryptography / HFE, Gr\"{o}bner basis, multivariate public key cryptosystem

Date: received 28 Dec 2008

Contact author: by at crypto tw

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Version: 20081229:161921 (All versions of this report)

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