Algorithm for strengthening some cryptographic systems

In the present work a field is built on the set of permutations, using the construction of a bijection, I_m, which goes from the set N_m = {n ∈ N{double-struck}|0 ≤ n ≤ m! - 1} to the set IIm = {π|π is a permutation of array 0, 1, 2,...,m -1}. During this work, it is shown that this bijection defines an isomorphism I_m: N_m → Π_m; also, two binary operations -⊙ and -⊕ on the set II_m are defined, such that a field can be built on a subset of IIm. On the other side, it is also possible to define a unary operation on II_m whose result is the inverse of a permutation, denoted as ¬π. Finally, an application of this tool to cryptography is presented, specifically to symmetric cryptosystems DES and Triple DES [1], with a modification: the inverse permutation at the end of the ciphering process is not applied; only the initial permutation is applied, but using a variable permutation. It can be seen that, as a result, both cryptosystems are strengthened against brute-force attacks, as well as against differential and linear cryptanalysis in the case of DES [2],[3].