Structure of Pseudorandom Numbers Derived from Fermat Quotients Zhixiong Chen 1 , Alina Ostafe 2 , and Arne Winterhof 3 1 Department of Mathematics, Putian University, Fujian 351100, China ptczx@126.com 2 Institut f¨ ur Mathematik, Universit¨at Z¨ urich, Winterthurerstr. 190, CH-8057 Z¨ urich, Switzerland alina.ostafe@math.uzh.ch 3 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria arne.winterhof@oeaw.ac.at Abstract. We 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 uncondi- tional 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 thresh- old 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 S´ ark¨ozy.We also prove a lower boundon 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 S´ark¨ozy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags. Keywords: Fermat quotients, finite fields, pseudorandom sequences, ex- ponential sums, discrepancy, well-distribution measure, correlation mea- sure, linear complexity, lattice test. MSC(2010): Primary 11T23; Secondary 65C10, 94A55, 94A60. 1 Introduction For a prime p and an integer u with gcd(u, p) = 1 the Fermat quotient q p (u) modulo p is defined as the unique integer with q p (u) ≡ u p-1 - 1 p (mod p), 0 ≤ q p (u) ≤ p - 1, and we also define q p (kp)=0, k ∈ Z. We note that (q p (u)) is a p 2 -periodic sequence modulo p for u ≥ 1. There are several results which involve the distribution and structure of Fermat quotients M.A. Hasan and T. Helleseth (Eds.): WAIFI 2010, LNCS 6087, pp. 73–85, 2010. c  Springer-Verlag Berlin Heidelberg 2010