New Kloosterman sums identities over F2m for all m

Recently, Shin and Sung found new identities for Kloosterman sums over F_2^m with odd m. They posed the question whether similar results could be obtained for even m. In this paper, we will give a positive answer to this question. We will present new results that hold for any m and include as special cases the results of Shin and Sung in the case where m is odd.     

Distributions modulo subgroups of GF(q)

The main purpose of this paper is to determine the number of symbols in each equivalence class modulo subgroups of $\text{GF(q)}{}^1$ for a certain class of linear recursive sequences with irreducible generating polynomial. The method used is to map sequences onto maximal linear recursive sequences.     

The fourth and sixth power mean of the classical Kloosterman sums

The main purpose of this paper is using the elementary and algebraic methods to study the computational problems of the fourth and sixth power mean of the classical Kloosterman sums, and to give an exact computation formula and conversion formula for them.     

Polynomial Basis Multiplication over GF(2 m )

In this paper, we describe, analyze and compare various multipliers. Particularly, we investigate the standard modular multiplication, the Montgomery multiplication, and the matrix–vector multiplication techniques.     

A note on the mean value of the general Kloosterman sums

The main purpose of this paper is to use the analytic method to study the calculating problem of the general Kloosterman sums, and give an exact calculating formula for it.     

利用有限域GF（2m）上的三项式构造二元循环码

循环码是一类特殊的线性码,由于循环码快速的编码和译码算法,它被广泛应用于消费电子,数据存储以及通信系统当中.在本文中,利用特征是偶数的有限域上的三项式构造出了两类二元循环码,我们不仅可以确定出这两类循环码最小距离的下界,而且这两类循环码在参数的选取上非常的灵活.     

Exponential sums on (G m ) n

Partially supported by NSF Grant No. DMS-8803085     

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

On Buekenhout-Metz unitals in PG(2,q 2),q even

Research partially supported by M.P.I. (Research project Strutture geometriche combinatorie e loro applicazioni).     

On primitive elements of trace equal to 1 in GF(2m)

The object of this paper is to present a simple proof for the existence of primitive elements of trace equal 1, in GF(2^m).     

Scalable and Systolic Architecture for Computing Double Exponentiation Over GF(2 m )

Double-exponentiation is a crucial arithmetic operation for many cryptographic protocols. Several efficient double-exponentiation algorithms based on systolic architecture have been proposed. However, systolic architectures require large circuit space, thus increasing the cost of the protocol. This would be a drawback when designing circuits in systems requiring low cost and low power consumption. However, some cost savings can be attained by compromising speed, as in portable devices and many embedded systems. This study proposes a scalable and systolic AB ² and a scalable and systolic A × B, which are the core circuit modules of double-exponentiation. A scalable and systolic double-exponentiation can thus be obtained based on the proposed scalable AB ² and A × B architecture. Embedded system engineers may specify a target double-exponentiation with appropriate scaling systolic circuits. The proposed circuit has lower circuit space/cost and low time/propagation than other circuits.     

Bit-Parallel Arithmetic Implementations over Finite Fields GF(2 m ) with Reconfigurable Hardware

Galois (or finite) fields are used in a wide number of technical applications, playing an important role in several areas such as cryptographic schemes and algebraic codes, used in modern digital communication systems. Finite field arithmetic must be fast, due to the increasing performance needed by communication systems, so it might be necessary for the implementation of the modules performing arithmetic over Galois fields on semiconductor integrated circuits. Galois field multiplication is the most costly arithmetic operation and different approaches can be used. In this paper, the fundamentals of Galois fields are reviewed and multiplication of finite-field elements using three different representation bases are considered. These three multipliers have been implemented using a bit-parallel architecture over reconfigurable hardware and experimental results are presented to compare the performance of these multipliers.     

Constructing some designs invariant under PSL2(q), q even

In this paper, we use Key-Moori methods 1 and 2 to construct some designs from the maximal subgroups and conjugacy classes of the group PSL₂(q), where q is a power of 2.     

A characterization of Buekenhout-Metz unitals in PG(2, q 2), q even

The known examples of embedded unitals (i.e. Hermitian arcs) in PG(2, q ²) are B-unitals, i.e. they can be obtained from ovoids of PG(3, q) by a method due to Buekenhout. B-unitals arising from elliptic quadrics are called BM-unitals. Recently, BM-unitals have been classified and their collineation groups have been investigated. A new characterization is given in this paper. We also compute the linear collineation group fixing the B-unital arising from the Segre-Tits ovoid of PG(3, 2 ^r ), r3 odd. It turns out that this group is an Abelian group of order q ².Research supported by MURST.     

The Weight Enumerator of Linear Codes over GF q m) Having Generator Matrix over GF q

We have the relationships between the Hamming weight enumerator of linear codes over GFq ^m which have generator matrices over GFq, the support weight enumerator and the -ply weight enumerator.