Primitive Polynomial Of Degree Research Articles

The Adjacency Graphs of a Class of LFSRs and Their Applications*

Nonlinear feedback shift registers (NFSRs) are widely used in communication and cryptography. How to construct more NFSRs with maximal periods, which can generate sequences with maximal periods, i.e., de Brujin sequences, is an attractive problem. Recently many results on constructing de Bruijn sequences from adjacency graphs of Linear feedback shift registers (LFSRs) by means of the cycle joining method have been obtained. In this paper we discuss a class of LFSRs with characteristic polynomial p2(x), where p(x) is a primitive polynomial of degree n ≥ 2 over the finite field F2. As results, we determine their cycle structures and adjacency graphs, and further construct a class of new de Bruijn sequences from these LFSRs.

New results on the minimal polynomials of modified de Bruijn sequences

Modified de Bruijn sequences are generated by removing a single zero from the longest run of zeros of de Bruijn sequences. It is known that the minimal polynomial of a modified de Bruijn sequence of order n at least has an irreducible factor of degree n. Based on this observation, we give some new results on the minimal polynomial of a modified de Bruijn sequence in this paper. First, it is shown that the minimal polynomial of a modified de Bruijn sequence of order n cannot be the product of an irreducible polynomial of degree n and the irreducible polynomial of degree 2. Second, it is proved that the minimal polynomial of a modified de Bruijn sequence of order n cannot be the product of an irreducible polynomial of degree n and a primitive polynomial of degree k with n≥8k. This is a generalization of the main result in the paper Kyureghyan (2008) [3] which only considered products of two primitive polynomials. Third, it is proved that the minimal polynomial of a modified de Bruijn sequence of order n cannot be a product of an irreducible polynomial f(x) of degree n and a polynomial of order t dividing 2k−1 with gcd⁡(ord(f(x)),t)=1 and n≥4k. As an application, for the cases n=2pe and n=p1⋅p2 where p,p1,p2 are prime numbers and 2pi−1 is also a prime number for i=1,2, a non-trivial lower bound is given for the linear complexity of a modified de Bruijn sequence of order n distinct from m-sequences, which could not be proved by the previous techniques.

Reversible and Fragile Watermarking for Medical Images.

A novel reversible digital watermarking technique for medical images to achieve high level of secrecy, tamper detection, and blind recovery of the original image is proposed. The technique selects some of the pixels from the host image using chaotic key for embedding a chaotically generated watermark. The rest of the pixels are converted to residues by using the Residue Number System (RNS). The chaotically selected pixels are represented by the polynomial. A primitive polynomial of degree four is chosen that divides the message polynomial and consequently the remainder is obtained. The obtained remainder is XORed with the watermark and appended along with the message. The decoder receives the appended message and divides it by the same primitive polynomial and calculates the remainder. The authenticity of watermark is done based on the remainder that is valid, if it is zero and invalid otherwise. On the other hand, residue is divided with a primitive polynomial of degree 3 and the obtained remainder is appended with residue. The secrecy of proposed system is considerably high. It will be almost impossible for the intruder to find out which pixels are watermarked and which are just residue. Moreover, the proposed system also ensures high security due to four keys used in chaotic map. Effectiveness of the scheme is validated through MATLAB simulations and comparison with a similar technique.

Variable strength covering arrays

Recently, covering arrays have been the subject of considerable research attention as they hold both theoretical interest and practical importance due to their applications to testing. In this thesis, we perform the first comprehensive study of a generalization of covering arrays called variable strength covering arrays, where we dictate the interactions to be covered in the array by modeling them as facets of an abstract simplicial complex. We outline the necessary background in the theory of hypergraphs, combinatorial testing, and design theory that is relevant to the study of variable strength covering arrays. We then approach questions that arise in variable strength covering arrays in a number of ways. We demonstrate their connections to hypergraph homomorphisms, and explore the properties of a particular family of abstract simplicial complexes, the qualitative independence hypergraphs. These hypergraphs are tightly linked to variable strength covering arrays, and we determine and identify several of their important properties and subhypergraphs. We give a detailed study of constructions for variable strength covering arrays, and provide several operations and divide-and-conquer techniques that can be used in building them. In addition, we give a construction using linear feedback shift registers from primitive polynomials of degree 3 over arbitrary finite fields to find variable strength covering arrays, which we extend to strength-3 covering arrays whose sizes are smaller than many of the best known sizes of covering arrays. We then give an algorithm for creating variable strength covering arrays over arbitrary abstract simplicial complexes, which builds the arrays one row at a time, using a density concept to guarantee that the size of the resultant array is asymptotic in the logarithm of the number of facets in the abstract simplicial complex. This algorithm is of immediate practical importance, as it can be used to create test suites for combinatorial testing. Finally, we use the Lovasz Local Lemma to nonconstructively determine upper bounds on the sizes of arrays for a number of different families of hypergraphs. We lay out a framework that can be used for many hypergraphs, and then discuss possible strategies that can be taken in asymmetric problems.

Injectivity on distribution of elements in the compressed sequences derived from primitive sequences over ℤ p e $\mathbb {Z}_{p^{e}}$

Let p ≥ 3 be a prime, e ≥ 2 an integer and $\mathbb {Z}_{p^{e}}$ the residue ring modulo pe. Let σ(x) be a primitive polynomial of degree n over $\mathbb {Z}_{p^{e}}$ and let G′(σ(x), pe) be the set of primitive linear recurring sequences over $\mathbb {Z}_{p^{e}}$ generated by σ(x). A compressing mapping $\varphi :\mathbb {Z}_{p^{e}}\rightarrow \mathscr {A}$ naturally induces a mapping $\widehat {\varphi }$ on G ′(σ(x), pe), i.e., $\widehat {\varphi }$ maps a sequence (…,si− 1,si,si+ 1,… ) to (…,φ(si− 1),φ(si),φ(si+ 1),… ). For any pair of sequences in $\{\widehat {\varphi }(\underline {s}):\underline {s}\in G^{\prime }_{~}(\sigma (x),p^{e})\}$ , it is desirable to determine whether (at least) one element of $\mathscr {A}$ is distributed differently in them. For $\emptyset \neq D\subseteq \mathscr {A}$ , $\widehat {\varphi }$ is said to be injective on G ′(σ(x), pe) w.r.t. D-uniformity if for any two distinct sequences $\underline {u},\underline {v}\in G^{\prime }_{~}(\sigma (x),p^{e})$ , the distribution of at least one element of D in $\widehat {\varphi }(\underline {u})$ differs from that in $\widehat {\varphi }(\underline {v})$ . A sufficient condition on φ is given to ensure that $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. If $\left (\left ((x^{p^{n}-1}-1)^{2}\bmod \sigma (x)\right ) \bmod p^{3}\right ) \notin p^{2}\mathbb {Z}_{p}$ , then an equivalent condition on φ is obtained to decide whether $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. Furthermore, quantitative estimation suggests that almost all mappings on $\mathbb {Z}_{p^{e}}$ induce injective mappings on G ′(σ(x), pe) as p and e increase.

Ordered Orthogonal Array Construction Using LFSR Sequences

We present a new construction of ordered orthogonal arrays (OOA) of strength $t$ with $(q + 1)t$ columns over a finite field $\mathbb{F}_{q}$ using linear feedback shift register sequences (LFSRs). OOAs are naturally related to $(t, m, s)$-nets, linear codes, and MDS codes. Our construction selects suitable columns from the array formed by all subintervals of length $\frac{q^{t}-1}{q-1}$ of an LFSR sequence generated by a primitive polynomial of degree $t$ over $\mathbb{F}_{q}$. We prove properties about the relative positions of runs in an LFSR which guarantee that the constructed OOA has strength $t$. The set of parameters of our OOAs are the same as the ones given by Rosenbloom and Tsfasman (1997) and Skriganov (2002), but the constructed arrays are different. We experimentally verify that our OOAs are stronger than the Rosenbloom-Tsfasman-Skriganov OOAs in the sense that ours are "closer" to being a "full" orthogonal array. We also discuss how our OOA construction relates to previous techniques to build OOAs from a set of linearly independent vectors over $\mathbb{F}_{q}$, as well as to hypergraph homomorphisms.

Applying Cuckoo Search for analysis of LFSR based cryptosystem

Summary Cryptographic techniques are employed for minimizing security hazards to sensitive information. To make the systems more robust, cyphers or crypts being used need to be analysed for which cryptanalysts require ways to automate the process, so that cryptographic systems can be tested more efficiently. Evolutionary algorithms provide one such resort as these are capable of searching global optimal solution very quickly. Cuckoo Search (CS) Algorithm has been used effectively in cryptanalysis of conventional systems like Vigenere and Transposition cyphers. Linear Feedback Shift Register (LFSR) is a crypto primitive used extensively in design of cryptosystems. In this paper, we analyse LFSR based cryptosystem using Cuckoo Search to find correct initial states of used LFSR. Primitive polynomials of degree 11, 13, 17 and 19 are considered to analyse text crypts of length 200, 300 and 400 characters. Optimal solutions were obtained for the following CS parameters: Levy distribution parameter ( β ) = 1.5 and Alien eggs discovering probability ( p a ) = 0.25.

Injectivity of compressing maps on the set of primitive sequences modulo square-free odd integers

Let p1, p2,?,pr be distinct odd primes and m = p1p2?pr. Let f(x) be a primitive polynomial of degree n over ?/m?$\mathbb {Z}/m\mathbb {Z}$. Denote by L(f) the set of primitive linear recurring sequences generated by f(x). A map ? on ?/m?$\mathbb {Z}/m\mathbb {Z}$ naturally induces a map ??$\widehat {\psi }$ on L(f), mapping a sequence (?,s?(i?1),s?(i),s?(i+1),?)$(\dots ,\underline {s}({i-1}),\underline {s}(i),\underline {s}({i+1}),\dots )$ to (?,?(s?(i?1)),?(s?(i)),?(s?(i+1)),?)$(\dots ,\psi (\underline {s}({i-1})),\psi (\underline {s}(i)),\psi (\underline {s}({i+1})),\dots )$. Previous results gave sufficient conditions under which modular functions induce injective maps on L(f). In this article we give an inequality which holds for large enough n. If this inequality holds, then the injectivity of ??$\hat {\psi }$ is clearly determined for any map ? on ?/m?$\mathbb {Z}/m\mathbb {Z}$. Particularly, the modular function ?(a)=a mod M induces an injective map on L(f) for any M?2≤i??:i?m$M\in \left \{{2\leq i\in \mathbb {Z}:i \nmid m}\right \}$.

A Class of de Bruijn Sequences

In this paper, a class of linear feedback shift registers (LFSRs) with characteristic polynomial (1 + x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> )p(x) is discussed, where p(x) is a primitive polynomial of degree n > 2. The cycle structure and adjacency graphs of the LFSRs are determined. A new class of de Bruijn sequences is constructed from these LFSRs, and the number of de Bruijn sequences in the class is also considered. To illustrate the efficiency of constructing de Bruijn sequences from these LFSRs, an algorithm for producing some corresponding maximum-length nonlinear feedback shift registers with time and memory complexity O(n) is also proposed.

On the distinctness of modular reductions of primitive sequences modulo square-free odd integers

Let M be a square-free odd integer with at least two different prime factors and Z/(M) the integer residue ring modulo M. In this paper, it is shown that for two primitive sequences a̲=(a(t))t⩾0 and b̲=(b(t))t⩾0 generated by a primitive polynomial of degree n over Z/(M), a̲=b̲ if and only if a(t)≡b(t)modH for all t⩾0, where H>2 is an integer divisible by a prime number coprime with M. This result is obtained basing on the assumption that every element in Z/(M) occurs in a primitive sequence of order n over Z/(M), which is known to be valid for most Mʼs if n>6.

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over $${\mathbf{Z}/(2^{32}-1)}$$ . Let f(x) be a primitive polynomial of degree n over $${\mathbf{Z}/(2^{32}-1)}$$ and H a positive integer with a prime factor coprime with 232?1. Under the assumption that every element in $${\mathbf{Z}/(2^{32}-1)}$$ occurs in a primitive sequence of order n over $${\mathbf{Z}/(2^{32}-1)}$$ , it is proved that for two primitive sequences $${\underline{a}=(a(t))_{t\geq 0}}$$ and $${\underline{b}=(b(t))_{t\geq 0}}$$ generated by f(x) over $${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$$ if and only if $${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$$ for all t ? 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.

Uniqueness of the distribution of zeroes of primitive level sequences over [formula omitted] (II)

Let p be a prime number, Z / ( p e ) the integer residue ring, e ⩾ 2 . For a sequence a ̲ over Z / ( p e ) , there is a unique decomposition a ̲ = a ̲ 0 + a ̲ 1 ⋅ p + ⋯ + a ̲ e − 1 ⋅ p e − 1 , where a ̲ i be the sequence over { 0 , 1 , … , p − 1 } . Let f ( x ) ∈ Z / ( p e ) [ x ] be a primitive polynomial of degree n, a ̲ and b ̲ be sequences generated by f ( x ) over Z / ( p e ) , such that a ̲ ≠ 0 ̲ ( mod p e − 1 ) . This paper shows that the distribution of zero in the sequence a ̲ e − 1 = ( a e − 1 ( t ) ) t ⩾ 0 contains all information of the original sequence a ̲ , that is, if a e − 1 ( t ) = 0 if and only if b e − 1 ( t ) = 0 for all t ⩾ 0 , then a ̲ = b ̲ . Here we mainly consider the case of p = 3 and the techniques used in this paper are very different from those we used for the case of p ⩾ 5 in our paper [X.Y. Zhu, W.F. Qi, Uniqueness of the distribution of zeroes of primitive level sequences over Z / ( p e ) , Finite Fields Appl. 11 (1) (2005) 30–44].

The nonlinear complexity of level sequences over [formula omitted

For any sequence a ̲ over Z / ( 2 2 ) , there is an unique 2-adic expansion a ̲ = a ̲ 0 + a ̲ 1 · 2 , where a ̲ 0 and a ̲ 1 are sequences over { 0 , 1 } and can be regarded as sequences over the binary field GF ( 2 ) naturally. We call a ̲ 0 and a ̲ 1 the level sequences of a ̲ . Let f ( x ) be a primitive polynomial of degree n over Z / ( 2 2 ) , and a ̲ be a primitive sequence generated by f ( x ) . In this paper, we discuss how many bits of a ̲ 1 can determine uniquely the original primitive sequence a ̲ . This issue is equivalent with one to estimate the whole nonlinear complexity, NL ( f ( x ) , 2 2 ) , of all level sequences of f ( x ) . We prove that 4 n is a tight upper bound of NL ( f ( x ) , 2 2 ) if f ( x ) ( mod 2 ) is a primitive trinomial over GF ( 2 ) . Moreover, the experimental result shows that NL ( f ( x ) , 2 2 ) varies around 4 n if f ( x ) ( mod 2 ) is a primitive polynomial over GF ( 2 ) . From this result, we can deduce that NL ( f ( x ) , 2 2 ) is much smaller than L ( f ( x ) , 2 2 ) , where L ( f ( x ) , 2 2 ) is the linear complexity of level sequences of f ( x ) .

Random properties of the highest level sequences of primitive sequences over Z//sub 2/sup e//

Using the estimates of the exponential sums over Galois rings, we discuss the random properties of the highest level sequences /spl alpha//sub e-1/ of primitive sequences generated by a primitive polynomial of degree n over Z(2/sup e/). First we obtain an estimate of 0, 1 distribution in one period of /spl alpha//sub e-1/. On the other hand, we give an estimate of the absolute value of the autocorrelation function |C/sub N/(h)| of /spl alpha//sub e-1/, which is less than 2/sup e-1/(2/sup e-1/-1)/spl radic/3(2/sup 2e/-1)2/sup n/2/+2/sup e-1/ for h/spl ne/0. Both results show that the larger n is, the more random /spl alpha//sub e-1/ will be.

0, 1 distribution in the highest level sequences of primitive sequences over

In this paper, we discuss the 0,1 distribution in the highest level sequence a e -1 of primitive sequence over Z 2 e generated by a primitive polynomial of degree n . First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight for e relatively small to n . We also get an estimate which is suitable for e relatively large to n . Combining the two bounds, we obtain an estimate depending only on n , which shows that the larger n is, the closer to 1/2 the proportion of 1 will be.

Primitive Polynomials Over GF(2) of Degree up to 660 with Uniformly Distributed Coefficients

New tables of primitive polynomials of degree up to 660 over the Galois field of 2 elements are provided. These polynomials have been obtained for ring generators—a new class of linear feedback shift registers featuring enhanced properties over conventional shift registers. For each degree polynomials with five, seven and nine nonzero coefficients are presented. The coefficients are uniformly separated from each other so that the resulting implementations are highly modular.

On the maximum value of aliasing probabilities for single input signature registers

The aliasing error performance of a signature register is measured by the maximum values of the aliasing error probabilities for certain ranges of the bit-error rate (and those of the test length). Based on these measures, we evaluate the performances of all the single input signature registers whose feedback polynomials are primitive polynomials of degree 16 and generator polynomials of the double-error-correcting BCH codes of the same degree. When the degree of the feedback polynomial is large, say 32, it is computationally hard to obtain the exact aliasing probability. But we observe that the numbers of codewords with large weights in the corresponding code dominate the maximum value of the aliasing probabilities. By computing the numbers of codewords of large weights, we find primitive polynomials of degree 32 whose maximum value of the aliasing probabilities is very large for some test lengths. The error performance of an LFSR with any BCH polynomial of degree m for the test length 2/sup m/2/-2 is shown to be very good by deriving the formula for the weight distribution of the corresponding code. >

Weight distributions of some binary primitive cyclic codes

Let s and k be integers such that s is a divisor of 2/sup k/-1. Let g(x) be a divisor of x/sup s/-1 over F/sub 2/, and let /spl pi/(x) be a primitive polynomial of degree k over F/sub 2/. We consider the binary cyclic code C of length N=2/sup k/-1 generated by (X/sup N/-1)/g(x)/spl pi/(x). For special cases, we determine the weight distribution of C by using the weights of the cyclic code of length s generated by (x/sup s/-1)/g(x). >

A Table of Primitive Binary Polynomials

For those n < 5000, for which the factorization of 2n− 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5– and 7–nomial of degree n in GF(2) are given. A primitive polynomial of degree n over GF(2) is useful for generating a pseudo–random sequence of n–tuples of zeros and ones, see [8]. If the polynomial has a small number k of terms, then the sequence is easily computed. But for cryptological applications (correlation attack, see [5]) it is often necessary to have the primitive polynomials with k larger than one can find in the existing tables. For example, Zierler and Brillhart [10, 11] have calculated all irreducible trinomials of degree n ≤ 1000, with the period for some for which the factorization of 2n−1 is known; Stahnke [7] has listed one example of a trinomial or pentanomial of degree n ≤ 168; Zierler [12] has listed all primitive trinomials whose degree is a Mersenne exponent ≤ 11213 = M23 (here Mj denotes the jth Mersenne exponent); Rodemich and Rumsey [6] have listed all primitive trinomials of degree Mj, 12 ≤ j ≤ 17; Kurita and Matsumoto [2] have listed all primitive trinomials of degree Mj, 24 ≤ j ≤ 28, and one example of primitive pentanomials of degree Mj, 8 ≤ j ≤ 27. ∗1980 Mathematics Subject Classification (1985 Revision). Primary 11T06, 11T71. †

A note on aliasing probability for multiple input signature analyzer

Recently D.K. Pradhan, S.K. Gupta and M.G. Karpovsky (1990) affirmed that the codes generated by (x- alpha )(x- beta ) were not Reed-Solomon, where alpha and beta were roots of the distinct primitive polynomials of degree m. It was assumed that beta = alpha /sup 8/, where s-1 and 2/sup m/-1 were mutually prime. The authors show that under the above assumption the respective codes are Reed-Solomon codes. >