On the $k$-error linear complexity of binary sequences derived from polynomial quotients
This work addresses a theoretical problem in cryptography and coding theory by analyzing the robustness of sequences against errors, but it is incremental as it builds on known sequences and methods.
The paper tackles the problem of determining the k-error linear complexity of binary sequences derived from polynomial quotients, such as Fermat quotients, and obtains exact values for these complexities under specific conditions, indicating that the sequences have good error linear complexity.
We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0, $$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field $\F_2$ for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field $\F_p$ for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.