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.

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Keywords
Discrete Logarithm ProblemBergman RingLinear Representation
Contact author(s)
tsaban @ math biu ac il
History
2012-06-12: received
Short URL
https://ia.cr/2012/320
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2012/320,
      author = {Matan Banin and Boaz Tsaban},
      title = {The Discrete Logarithm Problem in non-representable rings},
      howpublished = {Cryptology ePrint Archive, Paper 2012/320},
      year = {2012},
      note = {\url{https://eprint.iacr.org/2012/320}},
      url = {https://eprint.iacr.org/2012/320}
}
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