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

*Ueli Maurer and Dominik Raub*

**Abstract: **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 problem proves the
inexistence of certain field-homomorphic encryption schemes whose
realization is an interesting open problems 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.

**Category / Keywords: **foundations / black-box fields, generic algorithms,

**Date: **received 8 Mar 2007

**Contact author: **d raub at inf ethz ch

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**Version: **20070308:140502 (All versions of this report)

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