Permutation Polynomials, de Bruijn Sequences, and Linear Complexity

Abstract

The paper establishes a connection between the theory of permutation polynomials and the question of whether a de Bruijn sequence over a general finite field of a given linear complexity exists. The connection is used both to construct span 1 de Bruijn sequences (permutations) of a range of linear complexities and to prove non-existence results for arbitrary spans. Upper and lower bounds for the linear complexity of a de Bruijn sequence of span n over a finite field are established. Constructions are given to show that the upper bound is always tight, and that the lower bound is also tight in many cases.

On the Minimum Linear Complexity of de Bruijn Sequences over Non-prime Finite Fields

“…The proof that, given a span n de Bruijn sequence over F p m of linear complexity d+1, there corresponds an orthogonal system of degree d of the form OS is given in Theorem 18 of [1]. We shall prove the converse.…”

“…The actual bound is 2p+1. When n=3 they found by computer search that in the case p=3 the bound 3 2 +3=12 is not realized (Table V of [1] shows that the minimum linear complexity is 17). It may be that, for odd prime fields, a better minimum is yet to be found.…”

“…In 1996 the lower bound over F 2 of 2 n&1 +n was generalised to any finite field F p m by Blackburn, Etzion and Paterson [1] who showed that the linear complexity was never less than p mn&1 +n. On the question of whether this bound is ever realised, and under what circumstances, they gave partial solutions.…”

“…However, this appeared to them not to be the case for non-prime fields. They showed that for m 2 the lower bound p nm&1 +n is realised for some n including n=2 (Theorems 26 and 29 of [1]) and they conjectured that over non-prime fields the lower bound is always realised.…”

“…It is known that the maximum possible linear complexity of a span n de Bruijn sequence over F p m is p nm &1, and that such a sequence may be constructed in a straightforward manner (a Linear Feedback Shift Register m-sequence with an extra zero added to the run of m&1 zeroes will produce such a sequence) [1,2].…”