**The Discrete Logarithm Problem in non-representable rings**

*Matan Banin and Boaz Tsaban*

**Abstract: **Bergman's Ring $E_p$, parameterized by a prime number $p$,
is a ring with $p^5$ elements that cannot be embedded in a ring of matrices over any commutative ring.
This ring was discovered in 1974.
In 2011, Climent, Navarro and Tortosa described an efficient implementation of $E_p$
using simple modular arithmetic, and suggested that this ring may be a useful source
for intractable cryptographic problems.

We present a deterministic polynomial time reduction of the Discrete Logarithm Problem in $E_p$ to the classical Discrete Logarithm Problem in $\Zp$, the $p$-element field. In particular, the Discrete Logarithm Problem in $E_p$ can be solved, by conventional computers, in sub-exponential time.

Along the way, we collect a number of useful basic reductions for the toolbox of discrete logarithm solvers.

**Category / Keywords: **public-key cryptography / Discrete Logarithm Problem, Bergman Ring, Linear Representation

**Date: **received 5 Jun 2012

**Contact author: **tsaban at math biu ac il

**Available format(s): **PDF | BibTeX Citation

**Version: **20120612:035021 (All versions of this report)

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