# Linear complexity of Legendre-polynomial quotients

## TL;DR

This paper analyzes the linear complexity of a binary sequence defined via Legendre symbols, extending previous results to all w in the range 1 to p-1, under certain number-theoretic conditions.

## Contribution

It provides a comprehensive determination of the linear complexity for all relevant w values, generalizing earlier work limited to w=p-1.

## Key findings

- Linear complexity values are characterized for all 1 ≤ w < p-1.
- The case w ≥ p can be reduced to smaller w cases.
- Results depend on the condition 2^{p-1} ≠ 1 mod p^2.

## Abstract

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict $\left(\frac{0}{p}\right)=1$. In an earlier work, the linear complexity of $(f_u)$ was determined for $w=p-1$ under the assumption of $2^{p-1}\not\equiv 1 \pmod {p^2}$. In this work, we give possible values on the linear complexity of $(f_u)$ for all $1\le w<p-1$ under the same conditions. We also state that the case of larger $w(\geq p)$ can be reduced to that of $0\leq w\leq p-1$.