Linear Feedback Shift Register - Fibonacci LFSRs

Fibonacci LFSRs

The bit positions that affect the next state are called the taps. In the diagram the taps are . The rightmost bit of the LFSR is called the output bit. The taps are XOR'd sequentially with the output bit and then fed back into the leftmost bit. The sequence of bits in the rightmost position is called the output stream.

• The bits in the LFSR state which influence the input are called taps (white in the diagram).
• A maximum-length LFSR produces an m-sequence (i.e. it cycles through all possible 2n − 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.
• As an alternative to the XOR based feedback in an LFSR, one can also use XNOR. This function is an affine map, not strictly a linear map, but it results in an equivalent polynomial counter whose state of this counter is the complement of the state of an LFSR. A state with all ones is illegal when using an XNOR feedback, in the same way as a state with all zeroes is illegal when using XOR. This state is considered illegal because the counter would remain "locked-up" in this state.

The sequence of numbers generated by an LFSR or its XNOR counterpart can be considered a binary numeral system just as valid as Gray code or the natural binary code.

The arrangement of taps for feedback in an LFSR can be expressed in finite field arithmetic as a polynomial mod 2. This means that the coefficients of the polynomial must be 1's or 0's. This is called the feedback polynomial or reciprocal characteristic polynomial. For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is

The 'one' in the polynomial does not correspond to a tap — it corresponds to the input to the first bit (i.e. x0, which is equivalent to 1). The powers of the terms represent the tapped bits, counting from the left. The first and last bits are always connected as an input and output tap respectively.

Tables of primitive polynomials from which maximum-length LFSRs can be constructed are given below and in the references.

• The LFSR will only be maximum-length if the number of taps is even; just 2 or 4 taps can suffice even for extremely long sequences.
• The set of taps — taken all together, not pairwise (i.e. as pairs of elements) — must be relatively prime. In other words, there must be no common divisor to all taps.
• There can be more than one maximum-length tap sequence for a given LFSR length
• Once one maximum-length tap sequence has been found, another automatically follows. If the tap sequence, in an n-bit LFSR, is, where the 0 corresponds to the x0 = 1 term, then the corresponding 'mirror' sequence is . So the tap sequence has as its counterpart . Both give a maximum-length sequence.

Some example C code is below: