Multi-sequences with Almost Perfect Linear Complexity Profile and Function Fields over Finite Fields

Sequences with almost perfect linear complexity profile defined by Niederreiter (1997, Lecture Notes in Computer Science, Vol. 304, pp. 37–51, Springer-Verlag, Berlin/New York) are quite important for stream ciphers. In this paper, we investigate multi-sequences with almost perfect linear complexity profile and obtain a construction of such multi-sequences by using function fields over finite fields. Some interesting examples from this construction are presented to illustrate our construction.     

具有2n线性复杂度的2n周期二元序列的3错线性复杂度

线性复杂度和k错线性复杂度是度量密钥流序列的密码强度的重要指标.通过研究周期为2n的二元序列线性复杂度,提出将k错线性复杂度的计算转化为求Hamming重量最小的错误序列.基于Games-Chan算法,讨论了线性复杂度为2n的2n周期二元序列的3错线性复杂度分布情况;给出了对应k错线性复杂度序列的完整计数公式, k=3,4.对于一般的线性复杂度为2n-m的2n周期二元序列,也可以使用该方法给出对应k错线性复杂度序列的计数公式.     

On the Linear Complexity Profile of Nonlinear Congruential Pseudorandom Number Generators with Dickson Polynomials

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and Monte-Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. Recently, a weak lower bound on the linear complexity profile of a general nonlinear congruential pseudorandom number generator was proven by Gutierrez, Shparlinski and the first author. For most nonlinear generators a much stronger lower bound is expected. Here, we obtain a much stronger lower bound on the linear complexity profile of nonlinear congruential pseudorandom number generators with Dickson polynomials.     

p元扩展序列的线性复杂度

给出了由周期为p~m-1的p元序列导出的周期为p~(em)-1的p元扩展序列的线性复杂度.作为一个实例,计算了扩展Legendre序列的线性复杂度.     

Linear Complexity of the Discrete Logarithm

We obtain new lower bounds on the linear complexity of several consecutive values of the discrete logarithm modulo a prime p. These bounds generalize and improve several previous results.     

FCSR序列的线性复杂度

§ 1　 IntroductionFeedback with carry shift register(FCSR) was first introduced by Klapper andGoresky in1 994[1 ] .The main idea of FCSR is to add a memory to linearfeedback shiftreg-ister(LFSR) .The structure is depicted in Fig.1 ,Fig.1where mn- 1 ∈Z,ai,qi∈ { 0 ,1 } and qr=1 .We refer to mn- 1 as memory,(mn- 1 ,an- 1 ,...,an- r)as state,r=log(q+ 1 ) as length,and q=-1 + q1 · 2 + ...+ qr· 2 ras connection integerof FCSR.The operation of the shiftregister is defined as follows:(1 …     

On the Linear Complexity of the Power Generator

We obtain a lower bound on the linear complexity of the powergenerator of pseudo-random numbers, which in some special cases is alsoknown as the RSA generator and as the Blum–Blum–Shubgenerator. In some very important cases this bound is essentially thebest possible. In particular, this implies that lattice reductionattacks on such generators are not feasible.     

On statistical limit points of double sequences

We investigate the structure of the set of all statistical limit points of a double sequence and prove certain results, mainly showing that this set can be characterized as a F_σ-set.     

一类新的广义割圆序列的线性复杂度及其自相关值

最近,丁存生基于新的割圆类(V_0,V_1)构造了循环码并研究了其性质.本文利用割圆类(V_0, V_1)构造了周期为pq的2阶二元序列,并计算了其自相关值、线性复杂度和极小多项式.     

一类吸引子的语法复杂性*

本文研究了Feigenbaum吸引子和周期窗口中Feisenbaum吸引子决定的形式语言,讨论了它们的语法复杂性.证明了这类吸引子都是ETOL语言,从而是上下文有关语言(CSL);而不是上下文无关语言(CFL).     

用迹表示研究序列的线性复杂度

本文用迹表示式证明了序列的线性复杂度等于其秩矩阵的秩，并由此导出了正规基的计数公式．     

On the distinctness of modular reductions of maximal length sequences modulo odd prime powers

We discuss the distinctness problem of the reductions modulo of maximal length sequences modulo powers of an odd prime , where the integer has a prime factor different from . For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo are distinct. In other words, the reduction modulo of a maximal length sequence is proved to contain all the information of the original sequence.

     

An Approximate Distribution for the Maximum Order Complexity

In this paper we give an approximate probability distribution for the maximum order complexity of a random binary sequence. This enables the development of statistical tests based on maximum order complexity for the testing of a binary sequence generator. These tests are analogous to those based on linear complexity.     

On the GF(p) Linear Complexity of Hall's Sextic Sequences and Some Cyclotomic-Set-Based

Klapper (1994) showed that there exists a class of geometric sequences with the maximal possible linear complexity when considered as sequences over $GF(2)$, but these sequences have very low linear complexities when considered as sequences over $GF(p)(p$ is an odd prime). This linear complexity of a binary sequence when considered as a sequence over $GF(p)$ is called $GF(p)$ complexity. This indicates that the binary sequences with high $GF(2)$ linear complexities are inadequate for security in the practical application, while, their $GF(p)$ linear complexities are also equally important, even when the only concern is with attacks using the Berlekamp-Massey algorithm [Massey, J. L., Shift-register synthesis and bch decoding, {\it IEEE Transactions on Information Theory}, {\bf 15}(1), 1969, 122--127]. From this perspective, in this paper the authors study the $GF(p)$ linear complexity of Hall''s sextic residue sequences and some known cyclotomic-set-based sequences.     

How Many Bits have to be Changed to Decrease the Linear Complexity?

The k-error linear complexity of periodic binary sequences is defined to be the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period. For the period length p ⁿ, where p is an odd prime and 2 is a primitive root modulo p ², we show a relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity. Moreover, we describe an algorithm to determine the k-error linear complexity of a given p ⁿ-periodic binary sequence.     

A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems

We consider the standard linear complementarity problem (LCP): Find (x, y) R ²ⁿ such that y = M x + q, (x, y) 0 and x _i y _i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P ₀ LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in Newton iterations where is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods.     

单调线性互补问题的Mehrotra型预估-校正算法的迭代复杂性

Mehrotra型预估-校正算法是很多内点算法软件包的算法基础,但它的多项式迭代复杂性直到2007年才被Salahi等人证明.通过选择一个固定的预估步长及与Salahi文中不同的校正方向,本文把Salahi等人的算法拓展到单调线性互补问题,使得新算法的迭代复杂性为O(n log((x0)T s0/ε)),同时,初步的数值实验证明了新算法是有效的.     

Root Counting, the DFT and the Linear Complexity of Nonlinear Filtering

The method of root counting is a well established technique in the study of the linear complexity of sequences. Recently, Massey and Serconek [11] have introduced a Discrete Fourier Transform approach to the study of linear complexity. In this paper, we establish the equivalence of these two approaches. The power of the DFT methods are then harnessed to re-derive Rueppel's Root Presence Test, a key result in the theory of filtering of m-sequences, in an elegant and concise way. The application of Rueppel's Test is then extended to give lower bounds on linear complexity for new classes of filtering functions.     

On a class of Riesz-Fischer sequences

In this note, we give necessary and sufficient conditions for a system of complex exponentials to form a Riesz-Fischer sequence in for every positive number . The result provides a significant strengthening of the sufficient conditions recently stated by R. M. Reid (1995).