Constructions of Orthomorphisms of ℤ ⁿ₂

Abstract

A permutation σ on ℤ ⁿ₂ , the linear space over ℤ₂ of dimension n, is an orthormorphism iff the mapping x ↦ σ(x)+x is also a permutation on ℤ ⁿ₂ , as x takes all values in ℤ ⁿ₂ . It is a linear orthomorphism iff σ is a linear transformation on ℤ ⁿ₂ . This paper contains two parts. In the first part, in terms of the isomorphism between the linear space ℤ ⁿ₂ and the finite field GF(2ⁿ), an algebraic method of constructing linear orthomorphisms with maximal length cycles is provided. Then two algorithms to implement these linear orthomorphisms are presented. In the second part, by using this type of linear orthomorphisms, special types of Latin squares, called shift Latin squares are constructed and nonlinear orthomorphisms, which can be represented as transversals of such Latin squares, are obtained. Some discussion on nonlinearity of the resulting nonlinear orthomorphisms and a construction of arbitrary nonlinear orthormorphismsare also included in this part. A motivation is to use such mappings for encryption of digital data.