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Black-Box Extension Fields and the Inexistence of Field-Homomorphic One-Way Permutations

Ueli Maurer and Dominik Raub

The black-box field (BBF) extraction problem is, for a given field $\F$, to determine a secret field element hidden in a black-box which allows to add and multiply values in $\F$ in the box and which reports only equalities of elements in the box. This problem is of cryptographic interest for two reasons. First, for $\F=\F_p$ it corresponds to the generic reduction of the discrete logarithm problem to the computational Diffie-Hellman problem in a group of prime order $p$. Second, an efficient solution to the BBF extraction problem proves the inexistence of field-homomorphic one-way permutations whose realization is an interesting open problem in algebra-based cryptography. BBFs are also of independent interest in computational algebra.

In the previous literature BBFs had only been considered for the prime field case. In this paper we consider a generalization of the extraction problem to BBFs that are extension fields. More precisely we discuss the representation problem defined as follows: For given generators $g_1,\ldots,g_d$ algebraically generating a BBF and an additional element $x$, all hidden in a black-box, express $x$ algebraically in terms of $g_1,\ldots,g_d$. We give an efficient algorithm for this representation problem and related problems for fields with small characteristic (e.g. $\F=\F_{2^n}$ for some $n$). We also consider extension fields of large characteristic and show how to reduce the representation problem to the extraction problem for the underlying prime field.

These results imply the inexistence of field-homomorphic (as opposed to only group-homomorphic, like RSA) one-way permutations for fields of small characteristic.