Solving quadratic equations over finite fields is a fundamental task in algebraic coding theory and serves as a key subroutine for computing the roots of cubic and quartic polynomials. Notably, any quadratic polynomial over binary extension fields can be transformed into the reduced form $x^2+x+c\in \mathbb{F}_{2^m}[x]$, for which existing formula-based methods rely on heavy exponentiation or case distinctions on $m$ (odd/even or powers of two), limiting uniformity and efficiency. This paper presents a unified, formula-based solution for all positive integers $m$ that uses only exclusive-OR operations (XORs). The approach leverages a Reed-Muller matrix characterization of evaluations and transforms the problem into computing a binary matrix-vector multiplication. The total cost is at most $m^2-2m+1$ XORs, and under parallelism, the latency is $\lceil \log_2 m\rceil$ XORs, making the method attractive for low-power, low-latency applications.

翻译：有限域上的二次方程求解是代数编码理论中的基础任务，也是计算三次和四次多项式根的关键子程序。特别地，二元扩域上的任意二次多项式可化为简化形式$x^2+x+c\in \mathbb{F}_{2^m}[x]$，现有基于公式的方法依赖繁重的指数运算或对$m$的奇偶性/2的幂次分类处理，限制了统一性与效率。本文提出一种适用于所有正整数$m$的统一公式解法，仅需使用异或运算。该方法利用Reed-Muller矩阵表征评估过程，将问题转化为二进制矩阵-向量乘法计算。总代价至多为$m^2-2m+1$次异或运算，并行化时延迟为$\lceil \log_2 m\rceil$次异或运算，使该方法在低功耗、低延迟应用中具有吸引力。