Usually the naming of the shift register follows a type of convention shown normally in digital logic, with the least significant bit on the left. Considering that a shift register has n stages, the waveform is delayed by n discrete clock times. Always the shift registers produce a discrete delay of a digital signal or waveform. Shift registers are a form of sequential logic like counters. In digital circuits a shift register is formed by flip-flops and EXOR gates chained together with a synchronous clock. A LFSR is composed of memory cells connected together as a shift register with linear feedback. Every LFSR works by taking the XOR of the selected bits in its internal state and any LFSR containing all zero bits will never move to any other state, so one possible state must be excluded from any cycle. An n bits register will always have n + 1 signals. For an n bits LFSR, all the registers will be configured as shift registers, but only the last significant register will determine the feedback. The difference of status is, of course, that the equivalent status will be 1, where it was 0. LFSR can be built based on XOR (exclusive OR) circuits or XNOR (exclusive denied OR). A primitive polynomial satisfies some additional mathematical conditions and determines for the LFSR to have its maximum possible period, meaning (2n-1), where n is the number of cells of the shift register or the length.

The basis of every LFSR is developed with a polynomial, which can be irreducible or primitive (Angheloiu et al., 1986 Schneier, 1996). For each clock pulse the new bit in the string is produced using the XOR of certain positions. A string of memory cells that stored a string of bits and a clock pulse can advance the bits with one position in that string. The LFSR (Linear Feedback Shift Register) is the basis of the stream ciphers and it is the most often used one in hardware designs. Many different implementation forms were developed along the years. It was a five-stage device built of vacuum tubes and thyratrons. Introduction A code-breaking machine appeared as one of the first forms of shift register early in the 40's, in Colossus. The analysis of functioning for Primitive Polynomials of 16th degree shows that almost all the obtained results are in the same time distribution.

Storing data in Galois Fields allows effective and manageable manipulation, mainly in computer cryptographic applications. Usually LFSR functions in a Galois Field GF(2 n), meaning that all the operations are done with arithmetic modulo n degree Irreducible and especially Primitive Polynomials. Almost all of the major applications in the specific Fields of Communication used a well-known device called Linear Feedback Shift Register.

Linear feedback shift register pdf