Constructions of complete sets of orthogonal diagonal Sudoku squares

We prove that complete sets of orthogonal diagonal Sudoku latin squares (sometimes called Sudoku frames) exist of all orders p, where p is a prime. We also show that complete sets of orthogonal Sudoku frames which are left semi-diagonal exist of all orders p, s > 1. We conjecture that these may be right semi-diagonal also but we do not have a general proof. We show how these complete sets may be constructed.