The tap sequence of an LFSR can be represented as a polynomial mod 2. The sequence of numbers generated by an LFSR can be considered a binary numeral system just as valid as Gray code or the natural binary code. cycles through all possible 2^n-1 states within the shift register except the state where all bits are zero), unless it contains all zeros, in which case it will never change. * A maximal LFSR produces an m-sequence (i.e. * The outputs that influence the input are called 'taps' (white in the diagram). The taps are XOR'd sequentially with the output and then fed back into the leftmost bit. The list of the bits' positions that affect the next state is called the tap sequence. Both hardware and software implementations of LFSRs are common. However, an LFSR with a well-chosen feedback function can produce a sequence of bits which appears random and which has a very long cycle.Īpplications of LFSRs include generating pseudo-random numbers, pseudo-noise sequences, fast digital counters, and whitening sequences. Likewise, because the register has a finite number of possible states, it must eventually enter a repeating cycle. The initial value of the LFSR is called the seed, and because the operation of the register is deterministic, the sequence of values produced by the register is completely determined by its current (or previous) state. The only linear functions of single bits are xor and inverse-xor thus it is a shift register whose input bit is driven by the exclusive-or (xor) of some bits of the overall shift register value. The maximal sequence consists of every possible state except the '0000' state.]Ī linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. the content of stage 0 is output and forms part of the output sequence.Xor gate provides feedback to the register that shifts bits from left to right.During each unit of time the following operations are performed:

because of their structure, they can be readliy anallzed with algebraic techniquesĪ LFSR of length L consists of L stages (aka delay elements) numbered 0,1., L-1, each capable of storing one bit and having both one input and one output and a clock which controls the movement of data.They can produce sequences with very good statistical properties.They can produce sequences of large period.Well suited for hardware implementation.The LFSR, as these are known, is often a key design component in the keystream generator of a stream cipher. A linear feedback shift register is a feedback shift register.

Linear feedback shift register