Construction of a cryptosystem using the AES box and a bijective function from the natural numbers to the set of permutations

Abstract

Given a positive integer n, an algorithm is constructed that associates to each positive integer m, with 0 ≤ m ≤ n!-1, a permutation of n different elements in n-1 steps. In fact, the algorithm defines a bijective function, that is, one-to-one and onto, from the set of natural numbers to the set of permutations. Furthermore, for any permutation πL defi ned in the set of numbers {0,1, …,L-1}, with L a multiple of 3, this permutation may be constructed by means of 3 permutations defined on the set of numbers {0,1,…,2/3L-1}. The former allows to defi ne a cryptosystem on blocks of chains of 96 bits in length where one operates on numbers of 64! – 1 ≈ 1090 instead of 96! – 1 ≈ 10150, which reduces time and computational resources. It is also shown that the set of keys grows factorially in such a way that the amount of elements of the set is of the order of 10150 ≈ 2500 when working with chains of 96 bits. An example is given using the box of the Advanced Encryption Standard (AES) and an encryption procedure for blocks of 96 bits of clear text. The AES box is proposed because it is highly non-linear [1]. A hardware design for this cryptosystem is given to be implemented. Finally, we mention that by associating a permutation to an integer the permutations may be variable, that is, the permutations may be considered to be keys.