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Linear complexity of Ding generalized cyclotomic sequences 被引量:2
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作者 闫统江 陈智雄 肖国镇 《Journal of Shanghai University(English Edition)》 CAS 2007年第1期22-26,共5页
Minimal polynomials and linear complexity of binary Ding generalized cyclotomic sequences of order 2 with the two-prime residue ring Zpq are obtained by Bai in 2005. In this paper, we obtain linear complexity and mini... Minimal polynomials and linear complexity of binary Ding generalized cyclotomic sequences of order 2 with the two-prime residue ring Zpq are obtained by Bai in 2005. In this paper, we obtain linear complexity and minimal polynomials of all Ding generalized cyclotomic sequences. Our result shows that linear complexity of these sequences takes on the values pq and pq-1 on our necessary and sufficient condition with probability 1/4 and the lower bound (pq - 1)/2 with probability 1/8. This shows that most of these sequences are good. We also obtained that linear complexity and minimal polynomials of these sequences are independent of their orders. This makes it no more difficult in choosing proper p and q. 展开更多
关键词 stream cipher generalized cyclotomic sequence linear complexity minimal polynomial
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Linear complexity and autocorrelation of a new class of binary generalized cyclotomic sequences of order two and length pqr 被引量:1
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作者 JIA Wenjuan GAO Juntao ZHANG Peng 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2019年第4期651-661,共11页
Cyclotomic sequences have good cryptographic properties and are closely related to difference sets.This paper proposes a new class of binary generalized cyclotomic sequences of order two and length pqr.Its linear comp... Cyclotomic sequences have good cryptographic properties and are closely related to difference sets.This paper proposes a new class of binary generalized cyclotomic sequences of order two and length pqr.Its linear complexity,minimal polynomial,and autocorrelation are investigated.The results show that these sequences have a large linear complexity when 2∈D1,which means they can resist the Berlekamp-Massey attack.Furthermore,the autocorrelation values are close to 0 with a probability of approximately 1?1/r.Therefore,when r is a big prime,the new sequence has a good autocorrelation. 展开更多
关键词 GENERALIZED cyclotomic sequence linear complexity MINIMAL POLYNOMIAL AUTOCORRELATION value.
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Factorization of Cyclotomic Polynomials with Quadratic Radicals in the Coefficients 被引量:1
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作者 Alfred Wünsche 《Advances in Pure Mathematics》 2017年第9期472-506,共35页
In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the po... In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the possible factorization of the cyclotomic polynomial in polynomial factors which contain not higher than quadratic radicals in the coefficients whereas usually the factorization of the cyclotomic polynomials is considered in products of irreducible factors with integer coefficients. In considering the regular heptagon we find a modified variant of its construction by rhombic bicompasses and ruler. In detail, supported by figures, we investigate the case of the regular tridecagon (n=13) which in addition to n=7 is the only candidate with low n (the next to this is n=769 ) for which such a construction by rhombic bicompasses and ruler seems to be possible. Besides the coordinate origin we find here two points to fix for the possible application of two bicompasses (or even four with the addition of the complex conjugate points to be fixed). With only one bicompass one has in addition the problem of the trisection of an angle which can be solved by a neusis construction that, however, is not in the spirit of constructions by compass and ruler and is difficult to realize during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correlated action of two rhombic bicompasses must be applied in this case which avoids the disadvantages of the neusis construction. Single rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points on one circle determines the positions of all the other points on the second circle in unique way. The known case n=17 embedded in our method is discussed in detail. 展开更多
关键词 Geometrical Constructions by COMPASS and RULER Bicompasses cyclotomic POLYNOMIALS CHEBYSHEV POLYNOMIALS 7-Gon 13-Gon 17-Gon Fermat Numbers
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Linear Complexity of New Generalized Cyclotomic Sequences of Length 2pq
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作者 王宏伟 葛武 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 2014年第6期710-716,共7页
The linear complexity and minimal polynomial of new generalized cyclotomic sequences of order two are investigated.A new generalized cyclotomic sequence Sof length 2pqis defined with an imbalance p+1.The results show ... The linear complexity and minimal polynomial of new generalized cyclotomic sequences of order two are investigated.A new generalized cyclotomic sequence Sof length 2pqis defined with an imbalance p+1.The results show that this sequence has high linear complexity. 展开更多
关键词 generalized cyclotomic sequence linear complexity minimal polynomial
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A note about the cyclotomic polynomial Φ_(pq)(x) and some related results
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作者 ZHU Feng-xiang QI Wen-feng 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2008年第2期245-250,共6页
. The expression of cyclotomic polynomial Фpq (x) is concerned for a long time. A simple and explicit expression of Фpq (x) in Z[x] has been showed. The form of the factors of Фpq (x) over F2 and the upper, l... . The expression of cyclotomic polynomial Фpq (x) is concerned for a long time. A simple and explicit expression of Фpq (x) in Z[x] has been showed. The form of the factors of Фpq (x) over F2 and the upper, lower bounds of their Hamming weight are provided. 展开更多
关键词 cyclotomic polynomial Фpq(x) COEFFICIENT factor.
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CYCLOTOMIC ELEMENTS IN K_2F,REVISITED
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作者 Jerzy Browkin 《Acta Mathematica Scientia》 SCIE CSCD 2010年第1期19-26,共8页
Basing on results of Xu and Qin [10], and Guo [12] on cyclotomic elements in K2F for local fields F, we prove that every element in K2Q is a finite or infinite product of cyclotomic elements. Next, we extend this resu... Basing on results of Xu and Qin [10], and Guo [12] on cyclotomic elements in K2F for local fields F, we prove that every element in K2Q is a finite or infinite product of cyclotomic elements. Next, we extend this result to finite extensions of Q satisfying some additional conditions. 展开更多
关键词 Milnor group cyclotomic elements
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Formulas for Coefficients of Hundred Cyclotomics and Numbers Battle
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作者 Jawad Squalli 《American Journal of Computational Mathematics》 2019年第2期97-115,共19页
In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others wil... In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others will be deduced, we’ll see how. The formula is , without exception if u(n)=-1?or if 4 doesn’t divide and with its 165 exceptions of which 7 when u(n)=0?and 158 when u(n)=1?that will be shared in 154 and 4 pairs (n, i), which we will specify the conditions and values of the coefficients. According to u(n), according to the class of i modulo p, the first factor of the prime factor decomposition of n when u(n)=1?and according to gcd(n, i), the formula will or will not be valid and replaced otherwise by the good value that will be 0 for 152 pairs (n,i) or 1 in the 13 other exceptions. 展开更多
关键词 cyclotomic Polynomial Coefficient EULER MOBIUS MAPLE GCD PRIME Number
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Cyclotomic problem, Gauss sums and Legendre curve
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作者 XIA LingLi YANG Jing 《Science China Mathematics》 SCIE 2013年第7期1485-1508,共24页
In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the correspond... In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented. 展开更多
关键词 cyclotomic number cyclotomic problem Gauss sums Jacobi sums Legendre curve
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Some Notes on Generalized Cyclotomic Sequences of Length pq 被引量:2
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作者 陈智雄 李胜强 《Journal of Computer Science & Technology》 SCIE EI CSCD 2008年第5期843-850,共8页
We review the constructions of two main kinds of generalized cyclotomic binary sequences with length pq (the product with two distinct primes). One is the White-generalized cyclotomic sequences, the other is the Din... We review the constructions of two main kinds of generalized cyclotomic binary sequences with length pq (the product with two distinct primes). One is the White-generalized cyclotomic sequences, the other is the Ding-Helleseth(DH, for short)-generalized cyclotomic sequences. We present some new pseudo-random properties of DH-generalized cyclotomic sequences using the theory of character sums instead of the theory of cyclotomy, which is a conventional method for investigating generalized cyclotomic sequences. 展开更多
关键词 stream cipher generalized cyclotomic sequence pseudo-random binary sequence character sum correlation
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Restricting Specht modules of cyclotomic Hecke algebras 被引量:2
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作者 Andrew Mathas 《Science China Mathematics》 SCIE CSCD 2018年第2期299-310,共12页
This paper proves that the restriction of a Specht module for a(degenerate or non-degenerate)cyclotomic Hecke algebra, or KLR(Khovanov-Lauda-Rouquier) algebra, of type A has a Specht filtration.
关键词 cyclotomic Hecke algebras KLR algebras Specht modules representation theory
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THE REDUCIBILITY OF THE GENERALIZED CYCLOTOMIC EQUATIONS MODULO AN ODD INTEGER
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作者 王军 《Chinese Science Bulletin》 SCIE EI CAS 1992年第4期281-284,共4页
Let q be a power of a prime, F_q a finite field with q elements, b a fixed primitive root of F_q and e a given divisor of q-1. Then the cyclotomic number (h, k )_e of order e in F_q is defined as the number of ordered... Let q be a power of a prime, F_q a finite field with q elements, b a fixed primitive root of F_q and e a given divisor of q-1. Then the cyclotomic number (h, k )_e of order e in F_q is defined as the number of ordered pairs (s, t ) 展开更多
关键词 cyclotomic equation JACOBI SUM DIRICHLET character
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BERNOULLI-GOSS POLYNOMIAL AND CLASS NUMBER OF CYCLOTOMIC FUNCTION FIELDS
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作者 冯克勤 高文云 《Science China Mathematics》 SCIE 1990年第6期654-662,共9页
Let k = Fq(T),q=pn, and let K=k((?)p)be the cyclotomic function field with conduc-tor P = P(T), and suppose K+ is the maximal real subfield of K, hp(h+p) is the class number of divisor group (of degree zero) of K(K+),... Let k = Fq(T),q=pn, and let K=k((?)p)be the cyclotomic function field with conduc-tor P = P(T), and suppose K+ is the maximal real subfield of K, hp(h+p) is the class number of divisor group (of degree zero) of K(K+), and h-p=hp/h+p(∈ Ⅱ). This paper proves that for any fixed q≥3, there exist infinite many irreducible manic polynomial P∈Fq[T] such that p\h+p and pq-2\h-p. In addition, all regular quadratic irreducible polynomials in Fq[T] for 2≤p≤269 are determined. 展开更多
关键词 cyclotomic FUNCTION field CLASS namber of DIVISOR GROUP Bernoulli-Goss POLYNOMIAL
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ON MAXIMAL INDEPENDENT SYSTEM OF UNITS IN CYCLOTOMIC FUNCTION FIELDS
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作者 冯克勤 印林生 《Chinese Science Bulletin》 SCIE EI CAS 1992年第5期353-356,共4页
This note is concerned in constructing a series of maximal independent systems of cyclotomic units in cyclotomic function fields and their subfields. Let us introduce basic facts on cyclotomic function fields briefly ... This note is concerned in constructing a series of maximal independent systems of cyclotomic units in cyclotomic function fields and their subfields. Let us introduce basic facts on cyclotomic function fields briefly (see Refs. [1—3] for detail). 展开更多
关键词 cyclotomic FUNCTION FIELD cyclotomic UNITS
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A specific family of cyclotomic polynomials of order three
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作者 JI ChunGang School of Mathematical Sciences,Nanjing Normal University,Nanjing 210097,China 《Science China Mathematics》 SCIE 2010年第9期2269-2274,共6页
Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p 【 q 【 r and 2r ≡±1(mod pq),we prove that Φpqr(x)... Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p 【 q 【 r and 2r ≡±1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3). 展开更多
关键词 cyclotomic POLYNOMIAL FLAT cyclotomic POLYNOMIAL TERNARY cyclotomic POLYNOMIAL
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MAXIMAL INDEPENDENT SYSTEM OF UNITS IN CYCLOTOMIC FUNCTION FIELDS
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作者 冯克勤 印林生 《Science China Mathematics》 SCIE 1991年第8期908-919,共12页
A series of maximal independent systems of cyclotomic units for cyclotomic functionfields and their subfields are constructed as an analogue of Ramachandra’s and Levesque’scyclotomic unit systems in cyclotomic numbe... A series of maximal independent systems of cyclotomic units for cyclotomic functionfields and their subfields are constructed as an analogue of Ramachandra’s and Levesque’scyclotomic unit systems in cyclotomic number fields. The indexes of subgroups generated bythese unit systems in the whole unit groups are calculated and compared. 展开更多
关键词 cyclotomic UNIT cyc1otomic FUNCTION field.
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Notes on the Borwein-Choi Conjecture of Littlewood Cyclotomic Polynomials
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作者 Shao Fang HONG Wei CAOMathematical College, Sichuan University, Chengdu 610064, P. R. China 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2009年第1期65-76,共12页
Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N - 1 is cyclotomic iffP(x)=±Φp1(±x)ΦP2(±x^p1)…Φpr(±x^p1p2…pr-1),where N = P1P2 … pτ and the... Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N - 1 is cyclotomic iffP(x)=±Φp1(±x)ΦP2(±x^p1)…Φpr(±x^p1p2…pr-1),where N = P1P2 … pτ and the pi are primes, not necessarily distinct. Here Φ(x) := (x^p - 1)/(x - 1) is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the coniecture. 展开更多
关键词 cyclotomic polynomial Littlewood polynomial E-transformation Ramanujan sum least element
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Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form
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作者 Alexander Bors Qiang Wang 《Communications in Mathematical Research》 CSCD 2022年第2期246-318,共73页
This paper is concerned with so-called index d generalized cyclotomic mappings of a finite field F_(q), which are functions F_(q)→F_(q) that agree with a suitable monomial function x↦axr on each coset of the index d ... This paper is concerned with so-called index d generalized cyclotomic mappings of a finite field F_(q), which are functions F_(q)→F_(q) that agree with a suitable monomial function x↦axr on each coset of the index d subgroup of F_(q)^(*). We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index d generalized cyclotomic permutations of F_(q) and pertain to cycle structures, the classification of (q−1)-cycles and involutions, as well as inversion. 展开更多
关键词 Finite fields CYCLOTOMY cyclotomic mappings permutation polynomials wreath product cycle structure INVOLUTION
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Schubert Class and Cyclotomic NilHecke Algebras
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作者 Kai Zhou Jun Hu 《Algebra Colloquium》 SCIE CSCD 2021年第3期379-398,共20页
Let l and n be positive integers such that l≥n,and let Gn,l be the Grassmannian which consists of the set of n-dimensionsil subspaces of C^(l),There is a Z-graded algebra isomorphism between the cohomology H*(Gn,l,Z)... Let l and n be positive integers such that l≥n,and let Gn,l be the Grassmannian which consists of the set of n-dimensionsil subspaces of C^(l),There is a Z-graded algebra isomorphism between the cohomology H*(Gn,l,Z)of Gn,l and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebraH^(0)l,n=<ψ1,…ψn-1,y1,…,yn>.We show that the isomorphism can be chosen such that the image of each(geometrically defined)Schubert class(a1,...,an)coincides with the basis element bλconstructed by Hu and Liang by purely algebraic method,where 0≤q1≤q2≤…≤an≤l-n with ai∈Z for each i,andλis the l-€-multipartition of n associated to(l+1-(an+n),l+1-(an-1+n-),...,l+1-(a1+1)).A similar correspondence between the Schubert class basis of the cohomology of the Grassmanni-an Gl-n,l and the bλ's basis(λis anl-multipartition of n with each component being either(1)or empty)of the natural Z-form B of the Z-graded basic algebra of H^(0)_(l,n)is also obtained.As an application,we obtain a second version of the Giambelli formula for Schubert classes. 展开更多
关键词 cyclotomic nilHecke algebras GRASSMANNIANS Schur polynomials
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On the GF(p) Linear Complexity of Hall's Sextic Sequences and Some Cyclotomic-Set-Based Sequences
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作者 Xianmang HE Liqin HU Dong LI 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2016年第4期515-522,共8页
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 con... 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. 展开更多
关键词 Linear complexity Hall's sextic residues sequence cyclotomic set
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THE CONSTRUCTIONS OF ALMOST BINARY SEQUENCE PAIRS WITH THREE-LEVEL CORRELATION BASED ON CYCLOTOMY 被引量:1
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作者 Peng Xiuping Xu Chengqian 《Journal of Electronics(China)》 2012年第1期9-16,共8页
In this paper,a new class of almost binary sequence pair with a single zero element is presented.The almost binary sequence pairs with three-level correlation are constructed based on cyclotomic numbers of order 2,4,a... In this paper,a new class of almost binary sequence pair with a single zero element is presented.The almost binary sequence pairs with three-level correlation are constructed based on cyclotomic numbers of order 2,4,and 6.Most of them have good correlation and balance property,whose maximum nontrivial correlation magnitudes are 2 and the difference between the numbers of occurrence of +1's and-1's are 0 or 1.In addition,the corresponding binary sequence pairs are investigated as well and we can also get some kinds of binary sequence pairs with optimum balance and good correlation. 展开更多
关键词 CORRELATION cyclotomic numbers Almost binary sequence pairs Binary sequence pair
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