Searcharxiv⌕ Search

arXiv subjects

Jianqin Zhou

Publications and source records attributed to Jianqin Zhou.

17 recordsLinked to original sources
arXiv2022

Granule Description based on Compound Concepts

Concise granule descriptions for definable granules and approaching descriptions for indefinable granules are challenging and important issues in granular computing. The concept with only common attributes has been intensively studied. To investigate the granules with some special needs, we propose a novel type of compound concepts in this paper, i.e., common-and-necessary concept. Based on the definitions of concept-forming operations, the logical formulas are derived for each of the following types of concepts: formal concept, object-induced three-way concept, object oriented concept and common-and-necessary concept. Furthermore, by utilizing the logical relationship among various concepts, we have derived concise and unified equivalent conditions for definable granules and approaching descriptions for indefinable granules for all four kinds of concepts.

cs.AI↗
arXiv2021

On the prime distribution

In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots, A_i. \ldots $, such that the number of prime pairs ($p_i$ and $q_i $, $p_i-q_i = p-q $) in $A_i $ increases gradually, where $i>0$. So twin prime conjecture is true. We also prove that for any even integer $m>2700$, there exist more than 10 prime pairs $(p,q)$, such that $p+q=m$. Thus Goldbach conjecture is true.

math.GM↗
arXiv2021

A New Algorithm based on Extent Bit-array for Computing Formal Concepts

The emergence of Formal Concept Analysis (FCA) as a data analysis technique has increased the need for developing algorithms which can compute formal concepts quickly. The current efficient algorithms for FCA are variants of the Close-By-One (CbO) algorithm, such as In-Close2, In-Close3 and In-Close4, which are all based on horizontal storage of contexts. In this paper, based on algorithm In-Close4, a new algorithm based on the vertical storage of contexts, called In-Close5, is proposed, which can significantly reduce both the time complexity and space complexity of algorithm In-Close4. Technically, the new algorithm stores both context and extent of a concept as a vertical bit-array, while within In-Close4 algorithm the context is stored only as a horizontal bit-array, which is very slow in finding the intersection of two extent sets. Experimental results demonstrate that the proposed algorithm is much more effective than In-Close4 algorithm, and it also has a broader scope of applicability in computing formal concept in which one can solve the problems that cannot be solved by the In-Close4 algorithm.

cs.AI↗
arXiv2021

Concept and Attribute Reduction Based on Rectangle Theory of Formal Concept

Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative necessary concepts and unnecessary concepts. First, we present the new judgment results for relative necessary concepts and unnecessary concepts. Second, we derive the bounds for both the maximum number of relative necessary concepts and the maximum number of unnecessary concepts and it is a difficult problem as either in concept reduction preserving binary relations or attribute reduction of decision formal contexts, the computation of formal contexts from formal concepts is a challenging problem. Third, based on rectangle theory of formal concept, a fast algorithm for reducing attributes while preserving the extensions for a set of formal concepts is proposed using the extension bit-array technique, which allows multiple context cells to be processed by a single 32-bit or 64-bit operator. Technically, the new algorithm could store both formal context and extent of a concept as bit-arrays, and we can use bit-operations to process set operations "or" as well as "and". One more merit is that the new algorithm does not need to consider other concepts in the concept lattice, thus the algorithm is explicit to understand and fast. Experiments demonstrate that the new algorithm is effective in the computation of attribute reductions.

cs.AI↗
arXiv2014

On the $k$-error linear complexity for $p^n$-periodic binary sequences via hypercube theory

The linear complexity and the $k$-error linear complexity of a binary sequence are important security measures for key stream strength. By studying binary sequences with the minimum Hamming weight, a new tool named as hypercube theory is developed for $p^n$-periodic binary sequences. In fact, hypercube theory is based on a typical sequence decomposition and it is a very important tool in investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a $p^n$-periodic binary sequence $s$. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the $m$-hypercubes with the same linear complexity.

cs.CR↗
arXiv2014

Structure Analysis on the $k$-error Linear Complexity for $2^n$-periodic Binary Sequences

In this paper, in order to characterize the critical error linear complexity spectrum (CELCS) for $2^n$-periodic binary sequences, we first propose a decomposition based on the cube theory. Based on the proposed $k$-error cube decomposition, and the famous inclusion-exclusion principle, we obtain the complete characterization of $i$th descent point (critical point) of the k-error linear complexity for $i=2,3$. Second, by using the sieve method and Games-Chan algorithm, we characterize the second descent point (critical point) distribution of the $k$-error linear complexity for $2^n$-periodic binary sequences. As a consequence, we obtain the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences as the second descent point for $k=3,4$. This is the first time for the second and the third descent points to be completely characterized. In fact, the proposed constructive approach has the potential to be used for constructing $2^n$-periodic binary sequences with the given linear complexity and $k$-error linear complexity (or CELCS), which is a challenging problem to be deserved for further investigation in future.

cs.CR↗
arXiv2013

The 4-error linear complexity distribution for $2^n$-periodic binary sequences

By using the sieve method of combinatorics, we study $k$-error linear complexity distribution of $2^n$-periodic binary sequences based on Games-Chan algorithm. For $k=4,5$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic balanced binary sequences (with linear complexity less than $2^n$) are presented. As a consequence of the result, the complete counting functions on the 4-error linear complexity of $2^n$-periodic binary sequences (with linear complexity $2^n$ or less than $2^n$) are obvious. Generally, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences can be obtained with a similar approach.

cs.CR↗
arXiv2013

On the $k$-error linear complexity for $2^n$-periodic binary sequences via Cube Theory

The linear complexity and k-error linear complexity of a sequence have been used as important measures of keystream strength, hence designing a sequence with high linear complexity and $k$-error linear complexity is a popular research topic in cryptography. In this paper, the concept of stable $k$-error linear complexity is proposed to study sequences with stable and large $k$-error linear complexity. In order to study k-error linear complexity of binary sequences with period $2^n$, a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable $k$-error linear complexity. For such purpose, we first prove that a binary sequence with period $2^n$ can be decomposed into some disjoint cubes and further give a general decomposition approach. Second, it is proved that the maximum $k$-error linear complexity is $2^n-(2^l-1)$ over all $2^n$-periodic binary sequences, where $2^{l-1}\le k<2^{l}$. Thirdly, a characterization is presented about the $t$th ($t>1$) decrease in the $k$-error linear complexity for a $2^n$-periodic binary sequence $s$ and this is a continuation of Kurosawa et al. recent work for the first decrease of k-error linear complexity. Finally, A counting formula for $m$-cubes with the same linear complexity is derived, which is equivalent to the counting formula for $k$-error vectors. The counting formula of $2^n$-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al..

cs.CR↗
arXiv2011

Characterization of $2^n$-periodic binary sequences with fixed 3-error or 4-error linear complexity

The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated based on Games-Chan algorithm. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity less than $2^n$ are characterized. Second, for $k=3,4$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$ are presented. Third, for $k=4,5$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity less than $2^n$ are derived. As a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the 3-error linear complexity are obtained, and the complete counting functions on the 4-error linear complexity of $2^n$-periodic binary sequences are obvious.

cs.CR↗
arXiv2011

Periodic sequences with stable $k$-error linear complexity

The linear complexity of a sequence has been used as an important measure of keystream strength, hence designing a sequence which possesses high linear complexity and $k$-error linear complexity is a hot topic in cryptography and communication. Niederreiter first noticed many periodic sequences with high $k$-error linear complexity over GF(q). In this paper, the concept of stable $k$-error linear complexity is presented to study sequences with high $k$-error linear complexity. By studying linear complexity of binary sequences with period $2^n$, the method using cube theory to construct sequences with maximum stable $k$-error linear complexity is presented. It is proved that a binary sequence with period $2^n$ can be decomposed into some disjoint cubes. The cube theory is a new tool to study $k$-error linear complexity. Finally, it is proved that the maximum $k$-error linear complexity is $2^n-(2^l-1)$ over all $2^n$-periodic binary sequences, where $2^{l-1}\le k<2^{l}$.

cs.CR↗
arXiv2011

An Algorithm for Computing $m$-Tight Error Linear Complexity of Sequences over $GF(p^{m})$ with Period $p^{m}$

The linear complexity (LC) of a sequence has been used as a convenient measure of the randomness of a sequence. Based on the theories of linear complexity, $k$-error linear complexity, the minimum error and the $k$-error linear complexity profile, the notion of $m$-tight error linear complexity is presented. An efficient algorithm for computing $m$-tight error linear complexity is derived from the algorithm for computing $k$-error linear complexity of sequences over GF($p^{m}$) with period $p^n$, where $p$ is a prime. The validity of the algorithm is shown. The algorithm is also realized with C language, and an example is presented to illustrate the algorithm.

cs.CR↗
arXiv2011

Embedding Constructions of Tail-Biting Trellises for Linear Block Codes

In this paper, embedding construction of tail-biting trellises for linear block codes is presented. With the new approach of constructing tail-biting trellises, most of the study of tail-biting trellises can be converted into the study of conventional trellises. It is proved that any minimal tail-biting trellis can be constructed by the recursive process of embedding constructions from the well-known Bahl-Cocke-Jelinek-Raviv (BCJR) constructed conventional trellises. Furthermore, several properties of embedding constructions of tail-biting trellises are discussed. Finally, we give four sufficient conditions to reduce the maximum state-complexity of a trellis with one peak.

cs.IT↗
arXiv2011

On discrete cosine transform

The discrete cosine transform (DCT), introduced by Ahmed, Natarajan and Rao, has been used in many applications of digital signal processing, data compression and information hiding. There are four types of the discrete cosine transform. In simulating the discrete cosine transform, we propose a generalized discrete cosine transform with three parameters, and prove its orthogonality for some new cases. A new type of discrete cosine transform is proposed and its orthogonality is proved. Finally, we propose a generalized discrete W transform with three parameters, and prove its orthogonality for some new cases.

cs.IT↗
arXiv2011

The $k$-error linear complexity distribution for $2^n$-periodic binary sequences

The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By studying the linear complexity of binary sequences with period $2^n$, one could convert the computation of $k$-error linear complexity into finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated in this paper. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic balanced binary sequences (with linear complexity less than $2^n$) are characterized. Second, for $k=3,4$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$ are presented. Third, as a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the $k$-error linear complexity for $k = 2$ and 3 are obtained. Further more, an important result in a recent paper is proved to be not completely correct.

cs.CR↗
arXiv2005

A fast algorithm for determining the linear complexity of periodic sequences

A fast algorithm is presented for determining the linear complexity and the minimal polynomial of periodic sequences over GF(q) with period q n p m, where p is a prime, q is a prime and a primitive root modulo p2. The algorithm presented here generalizes both the algorithm in [4] where the period of a sequence over GF(q) is p m and the algorithm in [5] where the period of a binary sequence is 2 n p m . When m=0, the algorithm simplifies the generalized Games-Chan algorithm.

cs.CR↗
arXiv2005

Generalized partially bent functions

Based on the definition of generalized partially bent functions, using the theory of linear transformation, the relationship among generalized partially bent functions over ring Z N, generalized bent functions over ring Z N and affine functions is discussed. When N is a prime number, it is proved that a generalized partially bent function can be decomposed as the addition of a generalized bent function and an affine function. The result obtained here generalizes the main works concerning partially bent functions by Claud Carlet in [1].

cs.CR↗