Cryptography, connections, cocycles and crystals: a p-adic exploration of the discrete logarithm problem

Authors:
H. Gopalkrishna Gadiyar;K. M. Sangeeta Maini;R. Padma
Affiliations:
AU-KBC Research Centre, M.I.T. Campus of Anna University, Chennai, India;AU-KBC Research Centre, M.I.T. Campus of Anna University, Chennai, India;AU-KBC Research Centre, M.I.T. Campus of Anna University, Chennai, India
Venue:
INDOCRYPT'04 Proceedings of the 5th international conference on Cryptology in India
Year:
2004

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Abstract

Applying Hensel’s lemma to the discrete logarithm problem over prime fields reveals the rich geometric and algebraic structure underlying the problem. It is shown that the problem has links to cocycles, connections, group extensions and crystalline cohomology. It is reminiscent of the recent use of Monsky-Washnitzer cohomology for counting points on hyperelliptic curves. Further some weak keys of the cryptosystems based on the hardness of the discrete logarithm problem over prime fields are discussed.