奇异奇特征正交几何中的几个计数定理

设F_q是奇特征的q元有限域,F_q~(2v+δ+l)是F_q上的2v+δ+l维行向量空间,O_(2v+δ+l,△)(F_q)是奇特征有限域F_q上的正交群.F_q~(2v+δ+l)在O_(2v+δ+l,△)(F_q)作用下,导出了它在F_q~(2v+δ+l)的子空间集合上的作用,因而F_q~(2v+δ+l)在O_(2v+δ+l,△)(F_q)的作用下划分成一些轨道M(m,2s+γ,8,Γ,k;2v+δ,△).采用正交群O_(2v+δ+l,△)(F_q)作用在F_q~(2v+δ)上子空间轨道长度的公式,并且利用矩阵初等行变换的方法,给出M(m,2s+γ,s,Γ,k;2u+δ,△)的长度公式,由此给出(m,2s+γ,8,Γ)型子空间和(m,2s+T,k)子空间的计数.     

On the Computation of Square Roots in Finite Fields

In this paper, two improvements for computing square roots in finite fields are presented. Firstly, we give a simple extension of a method by O. Atkin, which requires two exponentiations in FM _q , when q9 mod 16. Our second method gives a major improvement to the Cipolla–Lehmer algorithm, which is both easier to implement and also much faster. While our method is independent of the power of 2 in q–1, its expected running time is equivalent to 1.33 as many multiplications as exponentiation via square and multiply. Several numerical examples are given that show the speed-up of the proposed methods, compared to the routines employed by Mathematica, Maple, respectively Magma.     

利用奇、偶特征的正交几何构作的最优Cartesian认证码

利用奇、偶特征有限域上的正交几何构作出一类C artesian认证码,并且计算了它们的参数.并在假定按照等概率分布来选择编码规则下,求出了认证码的成功的模仿攻击概率和成功的替换攻击概率.作为一个推论,还得到了一些最优的C artesian认证码.     

On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups O _2n+1(q) and O _2n ^± (q) (where q is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.     

On the Admissibility of Some Linear and Projective Groups in Odd Characteristic

A bijective mapping defined on a finite group G is complete if the mapping defined by , , is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL and PGL(2,q),q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n,q GL , admit a complete mapping.     

On Characteristic Polynomials of Formal Groups over Finite Fields

Generalizing the results of Serre, Hill and Koch, we give some classification theorems of higher dimensional simple formal groups over finite fields. A relation between endomorphism rings of formal groups over ?_p and characteristic polynomials of their reductions mod p is studied. A condition of existence of formal groups over ?_p with complex multiplication is given. Some formal groups over ?_p are also constructed.     

On Parameters of Some Graphs from Finite Fields

We generalize the constructon of F. R. Chung of graphs from finite fields and estimate their parameters with the help of a new bound of exponential sums due to G. I. Perel'muter and the author.     

Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic

Let M denote the set of the simple 3-dimensional unitary groups U₃ and the simple linear groups L₂ over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U₃(Q) or L₂(Q) for a suitable locally finite field Q of odd characteristic.     

Fast Discrete Fourier Transform on Local Fields of Zero Characteristic

For the discrete Fourier transform with respect to the system of characters of a local field with zero characteristic, we propose a fast algorithm. We find the complexity of the algorithm.     

On the Pronormality of Subgroups of Odd Index in Some Extensions of Finite Groups

We study finite groups with the following property (*): All subgroups of odd index are pronormal. Suppose that G has a normal subgroup A with property (*), and the Sylow 2-subgroups of G/A are self-normalizing. We prove that G has property (*) if and only if so does N_G(T)/T, where T is a Sylow 2-subgroup of A. This leads to a few results that can be used for the classification of finite simple groups with property (*).     

Smoothness Test for Polynomials Defined Over Small Characteristic Finite Fields

The problem of determining whether a polynomial defined over a finite field ring is smooth or not with respect to a given degree, is the most intensive arithmetic operation of the so-called descent phase of index-calculus algorithms. In this paper, we present an analysis and efficient implementation of Coppersmith’s smoothness test for polynomials defined over finite fields with characteristic three. As a case study, we review the best strategies for obtaining a fast field and polynomial arithmetic for polynomials defined over the ring \(F_q[X],\) with \(q=3^6,\) and report the timings achieved by our library when computing the smoothness test applied to polynomials of several degrees defined in that ring. This software library was recently used in Adj et al. (Cryptology 2016. http://eprint.iacr.org/2016/914), as a building block for achieving a record computation of discrete logarithms over the 4841-bit field \({{\mathbb {F}}}_{3^{6\cdot 509}}\).     

On the Existence for Some Special Primitive Elements in Finite Fields

Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q~n) such that α + α~(-1)is also primitive or α + α~(-1)is primitive and α is a normal element of F_(q~n) over F_q.     

关于有限域上一类方程零点个数的估计

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.     

Polynomial Invariants of Certain Pseudo-Symplectic Groups Over Finite Fields of Characteristic Two

Let F _q be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over F _q and Ps _n (F _q , S) be the associated pseudo-symplectic group. Let Ps _n (F _q , S) act linearly on the polynomial ring F _q [x ₁,…, x _n ]. In this note, we find an explicit set of generators of the ring of invariants of Ps _n (F _q , S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps ₄(F _q , S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps _2ν+1(F _q , S) is a complete intersection.     

特征为2的有限域上二次曲面的结合方案的注记

By using nondegenerate and degenerate quadrics in projective space over finite fields of characteristic 2, some association schemes were constructed and their parameters were computed by the authors (see Adv. in Math., 3(2000), 120-128 and Acta Math. Appl. Sinica, 1(1999), 96-103). In this note, their polynomial properties, eigenmatrices, imprimitivities, association subschemes and related quotient association schemes are studied.     

Finite Sample Behaviour of an Ergodically Fast Line-Search Algorithm

In order to represent a line-search algorithm as a non-convergent dynamic system, we perform a renormalisation of the uncertainty interval at each iteration. Its ergodic behaviour can then be studied, and it happens that, for locally symmetric functions, the asymptotic performances of the algorithm suggested are far better than those of the well-known GoldenSection algorithm. A proper tuning of internal parameters is then performed to obtain good performances for a finite number of iterations. The case of a function symmetric with respect to its optimum is considered first. An algorithm is presented, that only uses function comparisons,with a significant reduction of the number of comparisons required to reach a given precision when compared to the Golden Section algorithm. The robustness of these performances with respectto non-symmetry of the function is then checked by numerical simulations.     

On Polynomials of Special Form over a Finite Field of Odd Characteristic Attaining the Weil Bound

An explicit construction of polynomials over a finite field of odd characteristic, for which the absolute value of trigonometric sums attains the Weil bound, is based on the construction of cyclic matrices of given rank. Dickson polynomials of the second kind play an essential role in the study of such matrices.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 16–25.Original Russian Text Copyright © 2005 by L. A. Bassalygo, V. A. Zinov’ev.     

构造奇次同心幻方的一种方法

幻方是古老的数学游戏,经过几个世纪的发展形成了很多有趣的构造方法.利用行列式的性质和变换得到了构造奇数阶同心幻方原基的一种方法,利用这种方法和排列组合都能得到任意奇数阶幻方的多种形式.     

Cross-Correlations of Quadratic Form Sequences in Odd Characteristic

Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in odd characteristic. These sequences are shown to have low cross-correlation values in certain cases. They also have significantly higher linear spans than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.