Abstract : In this report, a matrix model is used to discover some of the properties of the linear feedback shift register (LFSR) and to consider its application to security systems. First the hardware and operation of the LFSR is briefly discussed. Then a representation of the LFSR as a finite state device is used to obtain the matrix model for the LFSR. The matrix model is employed to derive a number of known results about the period of an LFSR as well as some new results concerning subperiods of an LFSR. Cryptographic applications are suggested by the randomness properties of the LFSR bit stream output. The matrix model provides a concise treatment of the cryptanalysis of the simple LFSR system. Some suggestions are made to improve the security of LFSR secrecy systems.