Nonsingularity of Feedback Shift Registers of Degree at Most Three over a Finite Field

As a kind of generators of pseudo-random sequences, the Feedback shift register (FSR) is widely used in channel coding, cryptography and digital communication. A necessary and sufficient condition for the nonsingularity of a feedback shift register of degree at most three over a finite field is established. Using the above result, we can easily determine the nonsingularity of a feedback shift register from the algebraic normal form of the corresponding feedback function.