Linear decomposition of discrete functions in terms of shift-composition operation

We investigate the shift-composition operation on discrete functions that arise in connection with homomorphisms of shift registers. For an arbitrary function over a finite field all possible representations in the form of shift-compositions of two functions (where the right function is linear) are described. Besides, the possibility to represent an arbitrary function as a shift-composition of three functions such that both left and right functions are linear is studied. It is proved that in the case of a simple field for linear functions and quadratic functions that are linear in the extreme variable the concepts of reducibility and linear reducibility coincide.