A Note on Cyclic Codes from APN Functions

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty}$ defined by $s_t=Tr((1+\alpha^t)^e)$, where $\alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in \cite{D}

The Weight Distributions of Cyclic Codes and Elliptic Curves

Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curves, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases

New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $f(x)=\sum_{i=1}^{\frac{n}{2}-1}Tr^n_1(c_ix^{1+2^i})+ Tr_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$, where $c_i\in GF(2^n)$ for $1\leq i \leq \frac{n}{2}-1$ and $c_{n/2}\in GF(2^{n/2})$. The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1}Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$ , where $c_i\in GF(2^e)$, $n=em$ and $m$ is even. The bentness of these functions are characterized and some methods for constructing new quadratic bent functions are given. Finally, for a special case: $m=2^{v_0}p^r$ and $gcd(e,p-1)=1$, we present the enumeration of quadratic bent functions

An identity-based mutual authentication with key agreement scheme for mobile client-server environment

Electronic commerce transactions on mobile devices develops rapidly and are implemented on Internet or wireless networks. To achieve secure communication, mutual authentication with key agreement schemes are necessary and inspiring. Schemes can be constructed from passwords, traditional public key cryptography, and identity-based cryptography. And identity-based mutual authentication with key agreement schemes based on elliptic curve cryptography are preferable. This paper presents a secure and efficiency identity-based mutual authentication with key agreement scheme for mobile client-server environment.EI

Constructing Vectorial Boolean Functions with High Algebraic Immunity Based on Group Decomposition

Abstract. In this paper, we construct a class of vectorial Boolean functions over F2n with high algebraic immunity based on the decomposition of the multiplicative group of F2n. By viewing F2n as G1G2 {0} (where G1 and G2 are subgroups of F ∗ 2n, (#G1, #G2) = 1 and #G1 × #G2 = 2 2k − 1), we give a generalized description for constructing vectorial Boolean functions with high algebraic immunity. Moreover, when n is even, we provide two special classes of vectorial Boolean functions with high(sometimes optimal) algebraic immunity, one is hyper-bent, and the other is of balancedness and optimal algebraic degree

A New Class of Hyper-bent Boolean Functions with Multiple Trace Terms

Introduced by Rothaus in 1976 as interesting combinatorial objects, bent functions are maximally nonlinear Boolean functions with even numbers of variables whose Hamming distance to the set of all affine functions equals 2 n−1 ± 2 n 2 −1. Not only bent functions are applied in cryptography, such as applications in components of S-box, block cipher and stream cipher, but also they have relations to coding theory. Hence a lot of research have been paid on them. Youssef and Gong introduced a new class of bent functions the so-called hyper-bent functions which have stronger properties and rarer elements. It seems that hyper-bent functions are more difficult to generate. Moreover, (hyper)-bent functions are not classified. Charpin and Gong studied a class of hyper-bent functions f defined on F2n by f = ∑ Tr n 1 (arx r(2m−1)), n = 2m and ar ∈ F2n, where R is a r∈R subset of a set of representatives of the cyclotomic cosets modulo 2 m +1 for which each coset has the full size n. Further, Mesnager contributed to the knowledge of a class of hyper-bent functions fb defined over F2n by fb = ∑ Tr n 1 (arx r(2m−1)

A note on weight distributions of irreducible cyclic codes

Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case that an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.EI