# 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.