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**2012/320**

**Title:**

The Discrete Logarithm Problem in non-representable rings

**Authors:**
Matan Banin, 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.

**ePrint:**
https://eprint.iacr.org/2012/320

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