Abstract: Orthomorphic permutations play important roles in the design of cryptosystem. Studying the properties, enumeration and counting of orthomorphic permutations is heavily significant to design and analysis of cryptosystem. The theory research of orthomorphic permutations has become a focal issue. Counting and enumeration of orthomorphic permutations are discussed in this paper. An enumeration algorithm is given using sum matrix. and all of n-tuples orthomorphic permutations can be listed with this enumeration method. This is the first enumeration method of orthomorphisms. The lower bound and upper bound of the n-tuples orthomorphic permutation's number is given with this enumeration method. They are the best bounds until now.