The Discrete Logarithm Problem in Bergman's non-representable ring

TL;DR

This paper demonstrates that the Discrete Logarithm Problem in Bergman's non-representable ring $E_p$ can be efficiently reduced to the classical problem in $ ext{Z}_p$, making it solvable in sub-exponential time with conventional computers.

Contribution

It provides a deterministic polynomial time reduction from the DLP in $E_p$ to the classical DLP in $ ext{Z}_p$, revealing the problem's computational tractability.

Findings

DLP in $E_p$ reduces to classical DLP in $ ext{Z}_p$

Discrete Logarithm Problem in $E_p$ is solvable in sub-exponential time

Bergman's ring $E_p$ may not be suitable for cryptographic hardness

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.