Feedback linearly extended discrete functions

In this paper, we study a new flexible method to extend linearly the graph of nonlinear, and usually not bijective, function so that resulting extension is bijection. Our motivation comes from cryptography. Examples symmetric cryptography are given as how was used implicitly in construction some well-known block ciphers. The heavily relies on ideas brought linear coding theory secret sharing. We interested behavior composition many extensions, especially space parameters defines family equations based finite differences or forms. For any extension, characterize entirely for which such solvable terms render those corresponding nonlinear extended functions solvable. Conditions derived assess solvability kind number compositions iterations. prove relation between dimensions vector spaces appear our results. proofs properties rely mostly tools algebra.