Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences

Vivek, Srinivas and Madhavan, Veni CE (2014) Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences. In: JOURNAL OF SYMBOLIC COMPUTATION, 64 (SI). pp. 22-34.

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Abstract

The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) x(3) equivalent to y(2)z (mod p), where p is a given prime number. A nontrivial solution must also satisfy x(3) not equal y(2)z and 1 <= x, y, z < p(alpha), where alpha is a given real number such that 1/3 < alpha <= 1/2. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as x equivalent to v(2)z (mod p) and y equivalent to v(3)z (mod p). In this paper, we give a deterministic polynomial-time (O(ln(3) p) bit-operations) algorithm to determine, for a given v, a nontrivial solution to CSC, if one exists. Previously it took (O) over tilde (p(alpha)) time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers N satisfying the fractional part inequality {theta N} < phi (theta and phi are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter z in the former problem corresponds to determining N in the latter problem. We also show in the alpha = 1/2 case of CSC that for a certain class of primes the CSC problem can be solved deterministically in <(O)over tilde>(p(1/3)) time compared to the previous best of (O) over tilde (p(1/2)). It is empirically observed that about one out of three primes is covered by the above class. (C) 2013 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	JOURNAL OF SYMBOLIC COMPUTATION
Publisher:	ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Additional Information:	Copyright for this article belongs to the ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD, ENGLAND
Keywords:	Computational number theory; Cryptanalysis; Diophantine equation; Discrete Logarithm Problem; Fractional part sequence
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	22 May 2014 08:04
Last Modified:	22 May 2014 08:05
URI:	http://eprints.iisc.ac.in/id/eprint/49009

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