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. For the reduced quadratic polynomial $x^2+x+c\in \mathbb{F}_{2^m}[x]$, existing formula-based methods rely on heavy exponentiation or case distinctions on $m$ (odd/even or powers of two), which limits 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 reduces the problem to solving a binary linear system. 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：在代数编码理论中，求解有限域上的二次方程是一项基本任务，也是计算三次和四次多项式根的关键子程序。对于约化二次多项式 $x^2+x+c\in \mathbb{F}_{2^m}[x]$，现有的基于公式的方法依赖于繁重的指数运算或对 $m$（奇数/偶数或2的幂）进行情况区分，这限制了方法的统一性和效率。本文提出了一种适用于所有正整数 $m$ 的统一、基于公式的解法，该方法仅使用异或（XOR）运算。该方案利用Reed-Muller矩阵对求值过程进行刻画，并将问题简化为求解一个二元线性系统。总计算成本至多为 $m^2-2m+1$ 次异或运算，在并行条件下，延迟仅为 $\lceil \log_2 m\rceil$ 次异或运算，这使得该方法在低功耗、低延迟应用中具有吸引力。