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 $f(x)$. We study in detail the cycle structure of the set $\Omega(f(x))$ 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 $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction

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

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

An internet-based IP protection scheme for circuit designs using linear feedback shift register (LFSR)-based locking

Abstract—Due to emerging trend of design reuse in VLSI circuits, the intellectual property (IP) of design faces serious challenges like forgery, theft, misappropriation etc. These in-creasing risks of design IP stored in design repositories, or the threat of hacking the same during its Internet-based trans-mission, mandates design file encryption and its appropriate watermarking. In this paper, we propose a novel Internet-based scheme to tackle this problem. Input to the proposed scheme is a generic graph corresponding to a digital system design. Watermarking of the graph and its encryption are achieved using a new linear feedback shift register(LFSR)-based locking scheme. The proposed scheme makes unauthorized disclosure of valuable designs almost infeasible, and can easily detect any alteration of the design file during transmission. It ensures authentication of the original designer as well as non-repudiation between the seller and the buyer. Empirical evidences on several well-known benchmark problem sets are encouraging. Index Terms—Intellectual property protection (IPP), Water-marking, Encryption, Decryption

De Bruijn Sequences from Joining Cycles of Nonlinear Feedback Shift Registers

De Bruijn sequences are a class of nonlinear recurring sequences that have wide applications in cryptography and modern communication systems. One main method for constructing them is to join the cycles of a feedback shift register (FSR) into a full cycle, which is called the cycle joining method. Jansen et al. (IEEE Trans on Information Theory 1991) proposed an algorithm for joining cycles of an arbitrary FSR. This classical algorithm is further studied in this paper. Motivated by their work, we propose a new algorithm for joining cycles, which doubles the efficiency of the classical cycle joining algorithm. Since both algorithms need FSRs that only generate short cycles, we also propose efficient ways to construct short-cycle FSRs. These FSRs are nonlinear and are easy to obtain. As a result, a large number of de Bruijn sequences are constructed from them. We explicitly determine the size of these de Bruijn sequences. Besides, we show a property of the pure circulating register, which is important for searching for short-cycle FSRs

Codes and Sequences for Information Retrieval and Stream Ciphers

Given a self-similar structure in codes and de Bruijn sequences, recursive techniques may be used to analyze and construct them. Batch codes partition the indices of code words into m buckets, where recovery of t symbols is accomplished by accessing at most tau in each bucket. This finds use in the retrieval of information spread over several devices. We introduce the concept of optimal batch codes, showing that binary Hamming codes and first order Reed-Muller codes are optimal. Then we study batch properties of binary Reed-Muller codes which have order less than half their length. Cartesian codes are defined by the evaluation of polynomials at a subset of points in F_q. We partition F_q into buckets defined by the quotient with a subspace V. Several properties equivalent to (V intersect ) = {0} for all i,j between 1 and mu are explored. With this framework, a code in F_q^(mu-1) capable of reconstructing mu indices is expanded to one in F_q^(mu) capable of reconstructing mu+1 indices. Using a base case in F_q^3, we are able to prove batch properties for codes in F_q. We generalize this to Cartesian Codes with a limit on the degree mu of the polynomials. De Bruijn sequences are cyclic sequences of length q^n that contain every q-ary word of length n exactly once. The pseudorandom properties of such sequences make them useful for stream ciphers. Under a particular homomorphism, the preimages of a binary de Bruijn sequence form two cycles. We examine a method for identifying points where these sequences may be joined to make a de Bruijn sequence of order n. Using the recursive structure of this construction, we are able to calculate sums of subsequences in O(n^4 log(n)) time, and the location of a word in O(n^5 log(n)) time. Together, these functions allow us to check the validity of any potential toggle point, which provides a method for efficiently generating a recursive specification. Each successful step takes O(k^5 log(k)), for k from 3 to n

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 $n \ge 9$, the maximum nonlinearity $Nl{(f)_{\max }}$ of arbitrary $n -$order de Bruijn sequence feedback function $f$ satisfies $3 \cdot {2^{n - 3}} - ({Z_n} + 1) < Nl{(f)_{\max }} \le {2^{n - 1}} - {2^{\frac{{n - 1}}{2}}}$ 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

Analyse et Conception d'Algorithmes de Chiffrement Légers

The work presented in this thesis has been completed as part of the FUI Paclido project, whose aim is to provide new security protocols and algorithms for the Internet of Things, and more specifically wireless sensor networks. As a result, this thesis investigates so-called lightweight authenticated encryption algorithms, which are designed to fit into the limited resources of constrained environments. The first main contribution focuses on the design of a lightweight cipher called Lilliput-AE, which is based on the extended generalized Feistel network (EGFN) structure and was submitted to the Lightweight Cryptography (LWC) standardization project initiated by NIST (National Institute of Standards and Technology). Another part of the work concerns theoretical attacks against existing solutions, including some candidates of the nist lwc standardization process. Therefore, some specific analyses of the Skinny and Spook algorithms are presented, along with a more general study of boomerang attacks against ciphers following a Feistel construction.Les travaux présentés dans cette thèse s’inscrivent dans le cadre du projet FUI Paclido, qui a pour but de définir de nouveaux protocoles et algorithmes de sécurité pour l’Internet des Objets, et plus particulièrement les réseaux de capteurs sans fil. Cette thèse s’intéresse donc aux algorithmes de chiffrements authentifiés dits à bas coût ou également, légers, pouvant être implémentés sur des systèmes très limités en ressources. Une première partie des contributions porte sur la conception de l’algorithme léger Lilliput-AE, basé sur un schéma de Feistel généralisé étendu (EGFN) et soumis au projet de standardisation international Lightweight Cryptography (LWC) organisé par le NIST (National Institute of Standards and Technology). Une autre partie des travaux se concentre sur des attaques théoriques menées contre des solutions déjà existantes, notamment un certain nombre de candidats à la compétition LWC du NIST. Elle présente donc des analyses spécifiques des algorithmes Skinny et Spook ainsi qu’une étude plus générale des attaques de type boomerang contre les schémas de Feistel