An extension of binary threshold sequences from Fermat quotients

Abstract

We extend the construction of $p^2$-periodic binary thresholdsequences derived from Fermat quotients to the $d$-ary case where $d$is an odd prime divisor of $p-1$, and then by defining cyclotomic classesmodulo $p^{2}$, we present exact values of the linear complexityunder the condition of $d^{p-1}\not \equiv 1 \pmod {p^2}$. Also, weextend the results to the Euler quotients modulo $p^{r}$ with oddprime $p$ and $r \geq 2$. The linear complexity is very close to theperiod and is of desired value for cryptographic purpose. Theresults extend the linear complexity of the corresponding $d$-arysequences when $d$ is a primitive root modulo $p^2$ in earlier work.Finally, partial results for the linear complexity of the sequenceswhen $d^{p-1} \equiv 1 \pmod {p^2}$ is given.