Homomorphisms of binary shift registers

We consider homomorphisms of binary shift registers with linear in the input variable feedback function, that is, homomorphisms of automata generating binary linearly controlled complicated recurring sequences. It is proved that any homomorphism of such register can be represented as a composition of a homomorphism into a register and a homomorphism close to an isomorphism. In the description of this decomposition, the main role is played by some operation which extends the operation of multiplication of polynomials to the set of all binary functions. The problem on homomorphism of a shift register is reduced to the problem of searching common divisors of the feedback function and the output function.