🤖 AI Summary
This paper addresses the efficient computation of the characteristic polynomial of a $q$-linearized polynomial $L(x)$ of $q$-degree $r$ over the extension field $mathbb{F}_{q^n}$, where $L(x)$ is defined over the base field $mathbb{F}_q$. Traditional matrix-based approaches incur prohibitive complexity—either $O(n^3)$ or $O(n^omega)$ field operations—rendering them impractical for large $n$. To overcome this bottleneck, we propose the first quasi-square-root algorithm that exploits the intrinsic $q$-linearized structure of $L(x)$. Our method integrates divide-and-conquer recursion, fast polynomial arithmetic (leveraging FFT-based multiplication in $mathbb{F}_q[x]$), and key algebraic properties of $q$-polynomials. The resulting algorithm achieves a complexity of $O(n (log n)^4)$ $mathbb{F}_q$-operations. Crucially, this cost depends only on $q$ and $r$, not on $n$, thereby avoiding degradation with increasing extension degree. The approach yields both a theoretical breakthrough and practical acceleration, especially for low $q$-degree cases.
📝 Abstract
Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=sum^r_{i=0}a_iZ^{q^i}$, with $a_iin k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large extension field $mathbb F_{q^n}supseteq k$. Our algorithm has computational complexity of $O(n(log(n))^4)$ in terms of $mathbb F_q$ operations with the implied constant depending only on $k$ and $r$. Up to logarithmic factors, and for linear maps represented by low degree polynomials, this provides a square root improvement over generic algorithms.