Nowadays, Big Data security processes require mining large amounts of content that was traditionally not typically used for security analysis in the past. The RSA algorithm has become the de facto standard for encryption, especially for data sent over the internet. RSA takes its security from the hardness of the Integer Factorisation Problem. As the size of the modulus of an RSA key grows with the number of bytes to be encrypted, the corresponding linear system to be solved in the adversary integer factorisation algorithm also grows. In the age of big data this makes it compelling to redesign linear solvers over finite fields so that they exploit the memory hierarchy. To this end, we examine several matrix layouts based on space-filling curves that allow for a cache-oblivious adaptation of parallel TU decomposition for rectangular matrices over finite fields. The TU algorithm of Dumas and Roche (2002) requires index conversion routines for which the cost to encode and decode the chosen curve is significant. Using a detailed analysis of the number of bit operations required for the encoding and decoding procedures, and filtering the cost of lookup tables that represent the recursive decomposition of the Hilbert curve, we show that the Morton-hybrid order incurs the least cost for index conversion routines that are required throughout the matrix decomposition as compared to the Hilbert, Peano, or Morton orders. The motivation lies in that cache efficient parallel adaptations for which the natural sequential evaluation order demonstrates lower cache miss rate result in overall faster performance on parallel machines with private or shared caches and on GPU’s.

如今，大数据安全流程需要挖掘大量的内容，这些内容过去通常不用于安全分析。RSA算法已成为事实上的加密标准，尤其是对于通过Internet发送的数据。RSA从整数分解问题的难度中获得了安全性。随着RSA密钥模数的大小随要加密的字节数的增长而增长，在对抗整数分解算法中要解决的相应线性系统也随之增长。在大数据时代，这迫使人们必须重新设计有限域上的线性求解器，以便它们利用内存层次结构。为此，我们研究了基于空间填充曲线的几种矩阵布局，这些布局允许对有限域上的矩形矩阵进行并行TU分解的高速缓存忽略式适配。Dumas和Roche（2002）的TU算法需要索引转换例程，对于该例程而言，编码和解码所选曲线的成本很高。通过对编码和解码过程所需的位数运算的数量进行详细分析，并过滤表示希尔伯特曲线递归分解的查找表的成本，我们证明了莫顿混合阶数的索引转换成本最低与希尔伯特（Hilbert），皮亚诺（Peano）或莫顿（Morton）阶相比，整个矩阵分解所需的例程。