The Discrete Logarithm Problem in non-representable rings

Bergman’s Ring Ep, 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 Ep 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 Ep to the classical Discrete Logarithm Problem in Zp, the p-element field. In particular, the Discrete Logarithm Problem in Ep can be solved, by conventional computers, in subexponential time. Along the way, we collect a number of useful basic reductions for the toolbox of discrete logarithm solvers.