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

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, IEEE Transactions on Information Theory, 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.     

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.     

关于丢番图方程(an)x+(bn)y=(cn)z

设n,a,b,c是正整数,gcd(a,b,c)=1,a,b≥3,且丢番图方程a~x+b~y=c~z只有正整数解(x,y,z)=(1,1,1).证明了若(x,y,z)是丢番图方程(an)~x+(bn)~y=(cn)~z的正整数解且(x,y,z)≠(1,1,1),则yzz或xzy.还证明了当(a,b,c)=(3,5,8),(5,8,13),(8,13,21),(13,21,34)时,丢番图方程(an)~x+(bn)~y=(cn)~z只有正整数解(x,y,z)=(1,1,1).     

Some Notes on the Linear Complexity of Sidel’nikov-Lempel-Cohn-Eastman Sequences

We continue the study of the linear complexity of binary sequences, independently introduced by Sidel’nikov and Lempel, Cohn, and Eastman. These investigations were originated by Helleseth and Yang and extended by Kyureghyan and Pott. We determine the exact linear complexity of several families of these sequences using well-known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences.     

On the limit of maximal density of sequences with a Perfect Linear Complexity Profile

A binary sequence with a Perfect Linear Complexity Profile (PLCP) has all the jumps in its linear complexity profile of height exactly 1. We prove the conjecture made in Ho that the limit of the maximum density (proportion of ones) over all PLCPs of a particular length is 2/3 as the length tends to infinity.     

二阶锥线性互补问题内点法的复杂性

本文提出一个二阶锥线性互补问题的长步原始对偶内点法,搜索方向由一个一般的核函数来定义.如果给出初始的严格内点,可以得到本算法的复杂性为O((1+2k)llog(lμ0/ε)).     

On the k-error linear complexity over \mathbb{F}_p of Legendre and Sidelnikov sequences

We determine exact values for the k-error linear complexity L _k over the finite field of the Legendre sequence of period p and the Sidelnikov sequence of period p ^m  − 1. The results are

环Z/(2~e)上压缩序列的0,1分布

本文研究环 Z/( 2 e)上本原序列最高权位的 0 ,1分布 ,证明了当 e≥ 8,次数 n≥2 0时 ,本原序列 a的最高权位序列 ae- 1 在一个周期中 0 (或 1 )所占的比例λ( ae- 1 )满足 43.6 76 8 <λ( ae- 1 ) <5 6 .32 32     

Estimates of the Entropy of the Set of Means for Some Classes of Stationary and Quasistationary Sequences

We consider the set of arithmetic means for some classes of stationary and quasistationary sequences. Order-sharp estimates of the entropy of this set for the classes under consideration are given.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 52–58.Original Russian Text Copyright © 2005 by V. F. Gaposhkin.     

Erratum to "More Results on Hermite-Hadamard type inequality through (α,m)-preinvexity"

In this paper, we present some corrections to definitions of $m$-preinvex, $\left( \alpha,m\right) $-preinvex functions and statements of the theorems of the results proved in [7].     

On the Generation of Some Embeddable GF(2) Geometries

The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has generating rank 22. We also include a survey of all instances in which either the generating or embedding rank of an embeddable GF(2) geometry is known.     

On the functional equationf(x) g(y) = p(x + y) q(x/y)

Aequationes mathematicae -     

On the Applications of the Linear Recurrence Relationships to Pseudoprimes

We present here some results on the applications of linear recursive sequences of order $2$ to the Fermat pseudoprimes, Fibonacci pseudoprimes, and Dickson pseudoprimes.     

More results on Hermite-Hadamard type inequality through} $( \alpha ,m)$-preinvexity

We establish various inequalities for n-times differentiable mappings that are connected with illustrious Hermite-Hadamard integral inequality for mapping whose absolute values of derivatives are $\left( {\alpha ,m} \right)$-preinvex function. The new integral inequalities are then applied to some special means     

On the Complexity of Parallel Algorithms for Computing Inverses in GF(2^m) with m Prime

Inversion over a finite field is usually an expensive operation which is a crucial issue in many applications, such as cryptography and error-control codes. In this paper, three different strategies for computing the inverse over binary finite fields , called Eulerian, Gaussian, and Euclidean, respectively, are discussed and compared in terms of time and space complexity. In particular, some new upper and lower bounds to the respective complexities are evaluated.     

On the Clausen integral Cl2(Θ) and a related integral

In this article, the Clausen integral $\text{Cl}{}_2\text{(Θ)}\text{=-∫}{}^{\mathrm{\Theta }}{}_0\text{ln(2 sin}\text{t}\text{2}\text{)}\text{d}\text{t}$ whereby Θ is equal to a rational multiple of π belonging to ]0,2π[, is developed into a finite series of derivatives of the di-gamma function ψ. For the same values of Θ, a method is presented to calculate the related integral $\text{∫}{}^1{}_0\text{ln?}\text{?}{}^2\text{?2?cosΘ+1}\text{d?}$.     

On the number of pairwise non-isomorphic minimal blocking sets in PG(2, q)

In this paper we examine whether the number of pairwise non-isomorphic minimal blocking sets in PG(2, q) of a certain size is larger than polynomial. Our main result is that there are more than polynomial pairwise non-isomorphic minimal blocking sets for any size in the intervals [2q−1, 3q−4] for q odd and for q square. We can also prove a similar result for certain values of the intervals and .      

On an integral-type operator from a weighted-type space to F(p,q,s) on the unit ball

We characterize the boundedness and compactness of an integral-type operator introduced by S. Stevi? from a weighted-type space to F(p,q,s) space on the unit ball of ?ⁿ.     

关于(α,β) -度量的S -曲率

给出(α,β) -度量F=α\phi(β/α)的S -曲率的计算公式. 证得对一般的(α,β) -度量,当β为关于α长度恒定的Killing1 -形式时,S=0.研究了Matsumoto -度量F=α²/(α-β)和(α,α) -度量F=α+εβ+kβ²/α)的S -曲率, 证得S=0当且仅当β为关于α长度恒定的Killing1 -形式.同时还得到这两类度量成为弱Berwald度量的充要条件.其中\phi(s)为光滑函数,α(y)=\sqrt{a_ij(x)yⁱy^j}为黎曼度量,β(y)=b_i(x)yⁱ为非零1 -形式且ε,k≠ 0为常数.     

On the Embedding of (k,p)-Arcs is Maximal Arcs

Our main result is that a (k,p)-arc in PG (2,q),q = p ^h , with k qp - q + p - can be extended to a maximal arc. Combining this result with the recent Ball, Blokhuis, Mazzocca theorem about the non-existence of maximal arcs for p > 2, it gives an upper bound for the size of a (k,p)-arc. The method can be regarded as a generalization of B. Segre's method for proving similar embeddability theorems for k-arcs (that is when n= 2). It is based on associating an algebraic envelope containing the short lines to the (k,p)-arc. However, the construction of the envelope is independent of Segre's method using the generalization of Menelaus' theorem.