Structure of Pseudorandom Numbers Derived from Fermat Quotients

Abstract

AbstractWe study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2. We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sárközy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sárközy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.KeywordsFermat quotientsfinite fieldspseudorandom sequencesexponential sumsdiscrepancywell-distribution measurecorrelation measurelinear complexitylattice testMSC(2010)Primary 11T23Secondary 65C1094A5594A60