15 research outputs found
On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials
We propose a construction of de Bruijn sequences by the cycle joining method
from linear feedback shift registers (LFSRs) with arbitrary characteristic
polynomial . We study in detail the cycle structure of the set
that contains all sequences produced by a specific LFSR on
distinct inputs and provide a fast way to find a state of each cycle. This
leads to an efficient algorithm to find all conjugate pairs between any two
cycles, yielding the adjacency graph. The approach is practical to generate a
large class of de Bruijn sequences up to order . Many previously
proposed constructions of de Bruijn sequences are shown to be special cases of
our construction
The Cycle Structure of LFSR with Arbitrary Characteristic Polynomial over Finite Fields
We determine the cycle structure of linear feedback shift register with
arbitrary monic characteristic polynomial over any finite field. For each
cycle, a method to find a state and a new way to represent the state are
proposed.Comment: An extended abstract containing preliminary results was presented at
SETA 201
De Bruijn Sequences, Adjacency Graphs and Cyclotomy
We study the problem of constructing De Bruijn sequences by joining cycles of linear feedback shift registers (LFSRs) with reducible characteristic polynomials. The main difficulty for joining cycles is to find the location of conjugate pairs between cycles, and the distribution of conjugate pairs in cycles is defined to be adjacency graphs. Let l(x) be a characteristic polynomial, and l(x)=l_1(x)l_2(x)\cdots l_r(x) be a decomposition of l(x) into pairwise co-prime factors. Firstly, we show a connection between the adjacency graph of FSR(l(x)) and the association graphs of FSR(l_i(x)), 1\leq i\leq r. By this connection, the problem of determining the adjacency graph of FSR(l(x)) is decomposed to the problem of determining the association graphs of FSR(l_i(x)), 1\leq i\leq r, which is much easier to handle. Then, we study the association graphs of LFSRs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields. At last, as an application of these results, we explicitly determine the adjacency graphs of some LFSRs, and show that our results cover the previous ones
On the Maximum Nonlinearity of De Bruijn Sequence Feedback Function
The nonlinearity of Boolean function is an important cryptographic criteria in the Best Affine Attack approach. In this paper, based on the definition of nonlinearity, we propose a new design index of nonlinear feedback shift registers. Using the index and the correlative necessary conditions of de Bruijn sequence feedback function, we prove that when , the maximum nonlinearity of arbitrary order de Bruijn sequence feedback function satisfies and the nonlinearity of de Bruijn sequence feedback function, based on the spanning tree of adjacency graph of affine shift registers, has a fixed value. At the same time, this paper gives the correlation analysis and practical application of the index
The Adjacency Graphs of Linear Feedback Shift Registers with Primitive-like Characteristic Polynomials
We consider the adjacency graphs of the linear feedback shift registers (LFSRs) with characteristic polynomials of the form l(x)p(x), where l(x) is a polynomial of small degree and p(x) is a primitive polynomial. It is shown that, their adjacency graphs are closely related to the association graph of l(x) and the cyclotomic numbers over finite fields. By using this connection, we give a unified method to determine their adjacency graphs. As an application of this method, we explicitly calculate the adjacency graphs of LFSRs with characteristic polynomials of the form (1+x+x^3+x^4)p(x), and construct a large class of De Bruijn sequences from them
Complexity measures for classes of sequences and cryptographic apllications
Pseudo-random sequences are a crucial component of cryptography, particularly
in stream cipher design. In this thesis we will investigate several measures of
randomness for certain classes of finitely generated sequences.
We will present a heuristic algorithm for calculating the k-error linear complexity
of a general sequence, of either finite or infinite length, and results on the
closeness of the approximation generated.
We will present an linear time algorithm for determining the linear complexity
of a sequence whose characteristic polynomial is a power of an irreducible element,
again presenting variations for both finite and infinite sequences. This algorithm
allows the linear complexity of such sequences to be determined faster than was
previously possible.
Finally we investigate the stability of m-sequences, in terms of both k-error
linear complexity and k-error period. We show that such sequences are inherently
stable, but show that some are more stable than others