共查询到20条相似文献,搜索用时 109 毫秒
1.
Robert S. Coulter Marie Henderson Pamela Kosick 《Designs, Codes and Cryptography》2007,44(1-3):275-286
We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32. 相似文献
2.
In this paper we prove that the P(q,?) (q odd prime power and ?>1 odd) commutative semifields constructed by Bierbrauer (Des. Codes Cryptogr. 61:187?C196, 2011) are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth (SETA, pp.?403?C414, 2008). Also, we show that they are strongly isotopic if and only if q??1(mod?4). Consequently, for each q???1(mod?4) there exist isotopic commutative presemifields of order q 2? (?>1 odd) defining CCZ-inequivalent planar DO polynomials. 相似文献
3.
It is proven that any Dembowski–Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used
to explain some known planar Dembowski–Ostrom polynomials. A direct connection between a planar Dembowski–Ostrom polynomial
and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition,
all commutative semifields of order 35 are classified. 相似文献
4.
Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF(2 n ), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p n , we derive bundles of functions over GF(p n ) of the form λ(x)*ρ(x), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each. We compute all bundles of presemifields of orders p n ≤ 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived λ(x)*ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2). 相似文献
5.
Guobiao Weng Weisheng Qiu Zeying Wang Qing Xiang 《Designs, Codes and Cryptography》2007,44(1-3):49-62
Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter–Matthews presemifield and the Ding–Yuan variation of it,
we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result
to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 34, 36, 38, 310, 54, and 74. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of René Peeters [17,
p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of
their relevant p-ranks.
Dedicated to Dan Hughes on the occasion of his 80th birthday. 相似文献
6.
Shabnam Akhtari 《Journal of Number Theory》2008,128(4):884-894
In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp1(±x)Φp2(±xp1)?Φpr(±xp1p2?pr−1), where N=p1p2?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α2pβ−1 with odd prime p or separable polynomials of any odd degree. 相似文献
7.
We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group. 相似文献
8.
Mashhour Al-Ali Bani-Ata Christoph Hering Anni Neumann Aymen Rawashdeh 《Journal of Geometry》2007,86(1-2):1-5
The purpose of this paper is to prove the existence of semifields of order q
4 for any odd prime power q = pr, q > 3, admitting a free automorphism group isomorphic to Z
2 × Z
2. 相似文献
9.
This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q
2 is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p
e
is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033. 相似文献
10.
Dmitri Shakhmatov 《Topology and its Applications》2010,157(8):1518-324
Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed. 相似文献
11.
Jürgen Bierbrauer 《Designs, Codes and Cryptography》2010,54(3):189-200
We describe a method of proving that certain functions ${f:F\longrightarrow F}$ defined on a finite field F are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic. It is shown that a subfamily of order p 4s for odd s > 1 is not isotopic to previously known examples. 相似文献
12.
Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (p
a
, p
a
, p
a
, 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (p
a
, p
a
, p
a
, 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (p
a
, p
a
, p
a
, 1)-RDS, and give a second new construction using presemifields.We use this cohomological approach to identify equivalence classes of central (p
a
, p
a
, p
a
, 1)-RDS with elementary abelian C and E/C. We show that for p = 2, a 3 and p = 3, a 2, every central (p
a
, p
a
, p
a
, 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p = 2, a = 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (p
a
, p
a
, p
a
, 1)-RDS, of which one is abelian and a – 1 are non-abelian. 相似文献
13.
In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (aijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)2+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations. 相似文献
14.
《代数通讯》2013,41(7):3159-3170
Abstract Let R[X] be a polynomial ring in one variable over a commutative ring R. If (R,?) is a local ring then any Weierstrass polynomial in R[X] is contained only in the maximal ideal (?,X) of R[X]. We generalise this property of Weierstrass polynomials and investigate properties of polynomials contained in a finite number of maximal ideals in R[X]. 相似文献
15.
We investigate the structure of a collineation groupG preserving an oval in a finite projective plane of odd order and we obtain several results aboutG, whenG is a2-group. These results are then used to investigate the structure ofG/O(G) in the general case and, in particular, that ofF*(G/O(G)).This research was supported in part by a grant from the M.U.R.S.T. 相似文献
16.
Donald Mills 《Journal of Number Theory》2002,92(1):87-98
Let Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all c∈Fq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑x∈Fq χ1(D(x)) where D(x)=∑ni=1 aixpαi+pβi, αi?βi for each i, is known as a Dembowski-Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1 xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λpe−γXpe−γ+λpγX is a permutation polynomial over Fq. 相似文献
17.
A classification is given of all spreads in PG(3, q), q = pr, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q2 that admit groups of order q(q+1), when q = 23 or 47. 相似文献
18.
19.
《Journal of Computational and Applied Mathematics》2003,155(1):153-162
The classical Eneström–Kakeya Theorem states that if p(z)=∑v=0navzv is a polynomial satisfying 0⩽a0⩽a1⩽⋯⩽an, then all of the zeros of p(z) lie in the region |z|⩽1 in the complex plane. Many generalizations of the Eneström–Kakeya theorem exist which put various conditions on the coefficients of the polynomial (such as monotonicity of the moduli of the coefficients). We will introduce several results which put conditions on the coefficients of even powers of z and on the coefficients of odd powers of z. As a consequence, our results will be applicable to some polynomials to which these related results are not applicable. 相似文献
20.
Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz. 相似文献