A New Method for Construction of Orthomorphic Permutations with the Highest Degree

Orthomorphic permutation is proposed in 1942, which is a special permutation with many good cryptographic properties. Orthomorphic permutation has similar structure with a one-way function used in hash functions called Davies-Meyer construction. However, the algebraic degrees of previous constructions of orthomorphic permutations by Boolean functions are not the highest, which can be attacked by higher order differential attack. In this paper, we develop a new method for construction of orthomorphic permutations with the highest degree so that they are resistant to higher order differential analysis. We also generalize the construction from four aspects to generate more orthomorphic permutations with the highest degree. Moreover, we give the total number of orthomorphic permutations constructed after applying these generalizations. Our improvement on the degree of orthomorphic permutation can be used as a reference of designing S-box with great cryptographic properties in new algorithms.