一类传递置换群阶的下界估计与实例

The Lower Bound Estimation of Order of a Class of Transitive Permutation Groups and Instantiation

基于非交换群的抗量子密码体制是密码学的一个研究热点,其群的阶在一定程度上保证了求逆运算的困难性.本文对二元生成的传递置换群<g1,g2>的阶这一代数命题进行了研究,给出了传递置换群的充分必要条件,以及二元生成的传递置换群阶的下界估计式.在实例化生成g1,g2使传递置换群<g1,g2>的阶满足相应下界值的过程中,给出了一类特殊n阶轮换表成两个n元置换g1,g2乘积的方法,以及相应的二元生成的传递置换群<g1,g2>的设计算法.最后,阐述了传递置换群在对称密码体制中的应用.

Post quantum cryptography based on non-commutative group is a hot topic in cryptography.The order of the group ensures the difficulty of inverse operation to some extent.We mainly study the algebraic proposition of order of transi⁃tive permutation groups<g1,g2>generated by two elements g1,g2,give a necessary and sufficient conditions of transitive permutation group,and get a lower bound estimation of order of transitive permutation groups generated by two elements.In the process of the instantiation for generating g1,g2 which enables the order of transitive permutation groups<g1,g2>to satis⁃fy the corresponding lower bound value,we give a method expressing a class of special n⁃order cycles as the product of two n⁃ary permutations and a corresponding design algorithm on transitive permutation groups<g1,g2>generated by two ele⁃ments.In the end,this paper describes the application of transitive permutation group in symmetric cryptography.