Orthomorphism plays an important role in the design of symmetric cryptography. To analyze its construction, counting and properties further, a problem in a conclusion about linear orthomorphism was pointed out and corrected. Then, with the corrected conclusion, a non-redundant construction method to generate all maximal linear orthomorphisms was presented, while the previous method would produce repeatable results. The number of orthomorphism was proved to be a multiple of 2 to the power (n+1) based on the relationship between affine orthomorphism and complementary permutation. At last, a definition of algebraic immunity was proposed and proved to be CCZ-equivalence-invariant. The algebraic immunity of a non-affine orthomorphism was also proved to be equal to that of complementary permutation of this orthomorphism.Same is the case with some other cryptographic properties,such as difference uniformity, nonlinearity and algebraic degree.
