Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for solving the MQ problem with linearization over a coefficient field. Furthermore, the hybrid approach with XL (h-XL) is a variant of XL guessing some variables beforehand. In this paper, we present a variant of h-XL, which we call the \textit{polynomial XL (PXL)}. In PXL, the whole $n$ variables are divided into $k$ variables to be fixed and the remaining $n-k$ variables as ``main variables'', and we generate a Macaulay matrix with respect to the $n-k$ main variables over a polynomial ring of the $k$ (sub-)variables. By eliminating some columns of the Macaulay matrix over the polynomial ring before guessing $k$ variables, the amount of operations required for each guessed value can be reduced compared with h-XL. Our complexity analysis of PXL (under some practical assumptions and heuristics) gives a new theoretical bound, and it indicates that PXL could be more efficient than other algorithms in theory on the random system with $n=m$, which is the case of general multivariate signatures. For example, on systems over the finite field with ${2^8}$ elements with $n=m=80$, the numbers of operations deduced from the theoretical bounds of the hybrid approaches with XL and Wiedemann XL, Crossbred, and PXL with optimal $k$ are estimated as $2^{252}$, $2^{234}$, $2^{237}$, and $2^{220}$, respectively.

翻译：求解有限域上 $n$ 个变量 $m$ 个多元二次方程组成的系统（MQ问题）是计算机科学理论中的关键问题之一。XL算法（简称XL）是一种通过在系数域上线性化来求解MQ问题的主要方法。此外，XL混合方法（h-XL）是预先猜测部分变量的XL变体。本文提出一种h-XL的变体，称为多项式XL（PXL）。在PXL中，全部 $n$ 个变量被分为 $k$ 个待固定变量和 $n-k$ 个"主变量"，我们在 $k$ 个（子）变量的多项式环上，针对 $n-k$ 个主变量生成Macaulay矩阵。通过预先猜测 $k$ 个变量前消去多项式环上Macaulay矩阵的某些列，相比于h-XL，每个猜测值所需的运算量得以降低。在若干实用假设和启发式条件下，PXL的复杂度分析给出了新的理论界，表明对于 $n=m$（即一般多元签名场景）的随机系统，PXL理论上可能比其他算法更高效。例如，在元素数为 $2^8$ 的有限域上，当 $n=m=80$ 时，基于XL混合方法、Wiedemann XL混合方法、Crossbred算法以及最优 $k$ 下PXL的理论界推导出的运算次数分别估计为 $2^{252}$、$2^{234}$、$2^{237}$ 和 $2^{220}$。