Generating all linear orthomorphisms without repetition Original Research Article

A permutation σ on F2n is an orthomorphism iff the mapping x↦x+σ(x) is also a permutation on F2n, as x takes all values in F2n. It is a linear orthomorphism iff σ is a linear transformation on F2n. (Here F2=GF(2).) This paper presents a non-redundant construction technique to generate all linear orthomorphisms on F2n. The computational complexity of this construction technique is determined. Implicitly, this solves the problem for linear orthomorphisms on F2n, which have important applications in the design of block ciphers, and a strong relationship to the design of hashing functions and pseudo-random sequence generators.