Paper 2012/320

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.