黑盒线性代数：扩展Wiedemann对小域上稀疏矩阵预处理器的分析
Black Box Linear Algebra: Extending Wiedemann's Analysis of a Sparse Matrix Preconditioner for Computations over Small Fields
AI总结：
本文扩展Wiedemann对小域上稀疏矩阵预处理器的分析，旨在明确矩阵非零元素数量的期望及最大秩的概率界，从而为预处理矩阵的Jordan标准形中非平凡幂零块数量提供界，进而实现均匀采样原矩阵的核空间。
摘要：
Wiedemann的论文介绍了其用于任意域上稀疏和结构化矩阵计算的算法，同时也提出了用于小域上计算的两个矩阵预处理器。本文扩展了对第二个预处理器的分析，以提供所获得矩阵中非零元素数量的预期陈述以及此类矩阵具有最大秩的概率界。这属于持续工作的一部分，旨在证明该矩阵预处理器也能用于对预处理矩阵的Jordan标准形中的非平凡幂零块数量进行界估计，从而能够从原矩阵的核空间中均匀采样。如果成功，这将导致一种用于整数因数分解中所需矩阵计算的黑盒算法，其可靠性已得到证明，并且通过一个小因子，渐近效率高于其他使用其他矩阵预处理器或需要在域扩展上进行计算的技术。
英文摘要：
Wiedemann's paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is extended in order to provide more explicit statements of the expected number of nonzero entries in the matrices obtained as well as bounds on the probability that such matrices have maximal rank. This is part of ongoing work to establish that this matrix preconditioner can also be used to bound the number of nontrivial nilpotent blocks in the Jordan normal form of a preconditioned matrix, in such a way that one can also sample uniformly from the null space of the originally given matrix. If successful this will result in a black box algorithm for the type of matrix computation required when using the number field sieve for integer factorization that is provably reliable and - by a small factor - asymptotically more efficient than alternative techniques that make use of other matrix preconditioners or require computations over field extensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
