Measures of pseudorandomness: Arithmetic autocorrelation and correlation measure Richard Hofer, L´aszl´ o M´ erai, Arne Winterhof Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstr. 69, 4040 Linz, Austria e-mail: {richard.hofer,laszlo.merai,arne.winterhof}@oeaw.ac.at Dedicated to Robert F. Tichy on the occasion of his 60th birthday. Abstract We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials. 1 Introduction Pseudorandom numbers are generated by deterministic algorithms and are not random at all. However, in contrast to truly random numbers they guarantee certain randomness properties. Their desirable features depend on the application area. For example, unpredictable sequences are needed for cryptography and uncorrelated sequences for wireless communication or radar. Some corresponding quality measures are linear complexity and ex- pansion complexity for unpredictability and autocorrelation or more general correlation measure of order k. Finding relations between different measures of pseudorandomness is an important goal. For example, the linear complexity provides essentially the same quality measure as certain lattice tests coming from the area of Monte Carlo methods, see [4, 20]. The correlation measure of order k is a rather general measure of pseudorandomness introduced by Mauduit and S´ ark¨ozy [16]. A relation between linear complexity and the correlation measure of 1