The Discrete Logarithm Problem in Bergman's non-representable ring
This work addresses cryptographic security concerns by demonstrating that a previously suggested intractable problem in a non-representable ring is not secure, which is incremental as it builds on prior implementations and reductions.
The authors tackled the discrete logarithm problem in Bergman's non-representable ring $E_p$ by providing a deterministic polynomial time reduction to the classical discrete logarithm problem in the finite field $\\mathbb{Z}_p$, showing it can be solved in sub-exponential time by conventional computers.
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.