Cryptology ePrint Archive: Report 2010/577

Discrete Logarithms, Diffie-Hellman, and Reductions

Neal Koblitz and Alfred Menezes and Igor Shparlinski

Abstract: We consider the One-Prime-Not-p and All-Primes-But-p variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the Diffie-Hellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show that the One-Prime-Not-p-DL problem reduces to DH in time roughly L_p(1/2); the All-Primes-But-p-DL problem reduces to DH in time roughly L_p(2/5); and the All-Primes-But-p-DL problem reduces to the DH plus Integer Factorization problems in polynomial time. We also prove that under the Riemann Hypothesis, with e*log p queries to a yes-or-no oracle one can reduce DL to DH in time roughly L_p(1/2); and under a conjecture about smooth numbers, with e*log p queries to a yes-or-no oracle one can reduce DL to DH in polynomial time.

Category / Keywords: public-key cryptography /

Publication Info: Also available at http://anotherlook.ca

Date: received 13 Nov 2010, last revised 15 Aug 2011

Contact author: ajmeneze at uwaterloo ca

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Version: 20110815:114941 (All versions of this report)

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