Classification of Exponent Permutations over finite fields GF($2^n$) and its applications

In this paper, we define an equivalence relation on the group of all permutations over the finite field GF() and show each equivalence class has common cryptographic properties. And, we classify all exponent permutations over GF() and GF(). Then, three applications of our results are described. We suggest a method for designing S(ubstitution)-boxes by the concatenation of two exponent permutations over GF() and study the differential and linear resistance of them. And we can easily indicate that the conjecture of Beth in Eurocrypt '93 is wrong, and discuss the security of S-box in LOKI encryption algorithm.