On the discrepancy and linear complexity of some counter-dependent recurrence sequences

Authors:
Igor E. Shparlinski;Arne Winterhof
Affiliations:
Department of Computing, Macquarie University, North Ryde, Australia;Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria
Venue:
SETA'06 Proceedings of the 4th international conference on Sequences and Their Applications
Year:
2006

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Abstract

We prove a discrepancy bound “on average” over all initial values aα(0)=α of congruential pseudorandom numbers obtained from the sequences aα(n) over a finite field of prime order defined by aα(n)=naα(n–1)+1, n=1,2,..., using new bounds on certain exponential sums. Moreover, we prove a lower bound on the linear complexity of this sequence showing that its structural properties are close to be best possible.