用于密码学的并元MDS矩阵的构造
Construction of dyadic MDS matrices for cryptographic applications
摘要：
许多最近的分组密码在其扩散层中使用最大距离可分离（MDS）矩阵。该操作的主要目标是尽可能地扩展非线性Sbox的输出之间的差异。因此，它们通常在半字节或字节级别上起作用。MDS矩阵与比率为1/2的MDS码相关联。最著名的例子是AES分组密码的MixColumns操作。在这个例子中，MDS矩阵被仔细选择以获得紧凑和高效的实现。然而，该MDS矩阵专用于8位字，并且并不总是适于轻量级应用。最近，一些研究已经致力于递归扩散层的建设。这种方法允许使用迭代过程来应用MDS矩阵，该迭代过程看起来像具有线性函数而不是非线性函数的Feistel网络。我们的方法完全不同。我们提出了一个通用的经典MDS矩阵，不递归计算，但强对称的建设，以加速其评估与最小数量的查找表，或执行此评估与电路中的门的最小数量。我们称这种特殊的矩阵为“并元矩阵”，因为它们与并元码有关。我们研究了这类矩阵的一些基本性质。我们介绍了一个通用的构造对合并元MDS矩阵的里德所罗门码。最后，我们讨论了这些二元MDS矩阵的实现方面，以建立高效的分组密码。
英文摘要：
Many recent block ciphers use Maximum Distance Separable (MDS) matrices in their diffusion layer. The main objective of this operation is to spread as much as possible the differences between the outputs of nonlinear Sboxes. So they generally act at nibble or at byte level. The MDS matrices are associated to MDS codes of ratio 1/2. The most famous example is the MixColumns operation of the AES block cipher. In this example, the MDS matrix was carefully chosen to obtain compact and efficient implementations. However, this MDS matrix is dedicated to 8-bit words, and is not always adapted to lightweight applications. Recently, several studies have been devoted to the construction of recursive diffusion layers. Such a method allows to apply an MDS matrix using an iterative process which looks like a Feistel network with linear functions instead of nonlinear. Our approach is quite different. We present a generic construction of classical MDS matrices that are not recursively computed, but that are strong symmetric in order to either accelerate their evaluation with a minimal number of look-up tables, or to perform this evaluation with a minimal number of gates in a circuit. We call this particular kind of matrices "dyadic matrices", since they are related to dyadic codes. We study some basic properties of such matrices. We introduce a generic construction of involutive dyadic MDS matrices from Reed Solomon codes. Finally, we discuss the implementation aspects of these dyadic MDS matrices in order to build efficient block ciphers.
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。
