In this paper, we study the complexity of solving generic over-determined bilinear systems over a finite field $\mathbb{F}$. Given a generic bilinear sequence $B \in \mathbb{F}[\mathbf{x},\mathbf{y}]$, with respect to a partition of variables $\mathbf{x}$, $\mathbf{y}$, we show that, the solutions of the system $B= \mathbf{0}$ can be efficiently found on the $\mathbb{F}[\mathbf{y}]$-module generated by $B$. Following this observation, we propose three variations of Gr\"obner basis algorithms, that only involve multiplication by monomials in they-variables, namely, $\mathbf{y}$-XL, based on the XL algorithm, $\mathbf{y}$-MLX, based on the mutant XL algorithm, and $\mathbf{y}$-HXL, basedon a hybrid approach. We define notions of regularity for over-determined bilinear systems,that capture the idea of genericity, and we develop the necessary theoretical tools to estimate the complexity of the algorithms for such sequences. We also present extensive experimental results, testing our conjecture, verifying our results, and comparing the complexity of the various methods.

中文翻译：

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关于求解泛型超定双线性系统的复杂性

在本文中，我们研究了在有限域 $\mathbb{F}$ 上求解泛型超定双线性系统的复杂性。给定一个泛型双线性序列 $B \in \mathbb{F}[\mathbf{x},\mathbf{y}]$，关于变量 $\mathbf{x}$, $\mathbf{y} $，我们表明，系统 $B= \mathbf{0}$ 的解可以在 $B$ 生成的 $\mathbb{F}[\mathbf{y}]$-module 上有效地找到。根据这一观察，我们提出了 Gr\"obner 基算法的三种变体，它们只涉及与变量中的单项式相乘，即 $\mathbf{y}$-XL，基于 XL 算法，$\mathbf{y }$-MLX，基于突变 XL 算法，和 $\mathbf{y}$-HXL，基于混合方法。我们定义了超定双线性系统的正则性概念，它捕捉了通用性的概念，我们开发了必要的理论工具来估计此类序列算法的复杂性。我们还展示了大量的实验结果，检验了我们的猜想，验证了我们的结果，并比较了各种方法的复杂性。