FCSR connection integers: efficient generation, testing and counting

Abstract The period of a binary sequence generated by feedback with carry shift register (FCSR) depends on the connection integer (q) of the register. We define FCSR primes as primes generating maximum possible period sequences when used as connection integer of an FCSR. Hence, to design good FCSR pseudorandom generators, one needs suitable feedback primes. The authors of this article study some properties about these primes and give algorithms to generate some of them. The first algorithm is somewhat straightforward and is based on the criterion given in Arnault and Berger. We propose two new algorithms for the efficient generation of FCSR primes. Comparison between these algorithms, supported by experimental and analytical results is presented. Further, we address the question of quantification of such primes and derive some explicit bounds.