An extension of binary threshold sequences from Fermat quotients

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