Paper 2026/821

A spectral approach to arithmetic correlations for binary FCSR sequences with prime connection integers

Abstract

Arithmetic correlation is an important metric for measuring feedback with carry shift register (FCSR) sequences, and its value should be as small as possible. For binary FCSR sequences with a prime connection integer $p$ and for which $\operatorname{ord}_p(2)$ is odd, where $\operatorname{ord}_p(2)$ is the order of $2$ modulo $p$, the arithmetic correlation can be expressed as the difference between the number of even representatives and the number of odd representatives within the subgroup generated by $2$ and all its cosets. From this perspective, we develop a unified spectral method for arithmetic correlation, derive an upper bound on it, and establish conditions for its with small values. We also analyze cases with a prime connection integer $p$ where the number of cosets is $2$, $4$, or $6$, and characterize when the arithmetic correlation takes small values.