Interpolation of the discrete logarithm in Fq by Boolean functions and by polynomials in several variables modulo a divisor of q - 1

Authors:
Tanja Lange;Arne Winterhof
Affiliations:
Institute of Experimental Mathematics, University of Essen, Ellernstr. 29, 5326 Essen, Germany;Institute of Discrete Mathematics, Austrian Academy of Sciences, Sonnenfelsgasse 19/2, 1010 Wien, Austria
Venue:
Discrete Applied Mathematics - Special issue: International workshop on coding and cryptography (WCC 2001)
Year:
2003

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Abstract

Recently, Shparlinski proved several results on the interpolation of the discrete logarithm in finite prime fields by Boolean functions. In the first part of the paper, these results are extended to arbitrary finite fields of odd characteristic. More precisely, we prove some complexity lower bounds for Boolean functions representing the least significant bit of the discrete logarithm in a finite field.In the second part of the paper we obtain lower bounds on the sparsity and the degree of polynomials over Fq in several variables computing the discrete logarithm modulo a prime divisor of q - 1. These results are valid for even characteristic, as well.