On the Optimality of the Binary Algorithm for the Jacobi Symbol

Authors:
J. Busch
Affiliations:
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA. E-mail: jbusch@math.ucla.edu
Venue:
Fundamenta Informaticae
Year:
2007

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Abstract

We establish lower bounds on the complexity of computing the following number-theoretic functions and relations from piecewise linear primitives: (i) the Legendre and Jacobi symbols, (ii) pseudoprimality, and (iii) modular exponentiation. As a corollary to the lower bound obtained for (i), an algorithm of Shallit and Sorenson is optimal (up to a multiplicative constant).