UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION

The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.

APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.

UGD PROPERTY OF MUSIELAK-ORLICZ SEQUENCE SPACES

By the properties of the Musielak_Orlicz funciton's sequence, the necessary and sufficient condition for uniform Gateaux differential (UGD) property of Musielak_Orlicz sequence spaces equipped with the Luxemburg norm and a criterion for weakly uniform rotundity of Musielak_Orlicz sequence space with Orlicz norm are given.

Trace Representations of S-boxes and Their Application

In the paper, we use trace representations of Boolean functions to obtain that a class mappings including functionsF(x)=x d over field GF(2 n ), withW(d)=n?1, have desirable cryptographic properties. Therefore we generalize an important result of Nyberg. As application, we use these conclusions to analyze cryptographic property of the S-box of AES (the Advanced Encryption Standard) and give its several equivalent representations, proving that the composition of inversion function of AES and any invertible affine transformations is impossible to satisfy strict avalanche criterion, any order propagation criteria and any order correlation immunity. Key words trace function - nonlinearity - differentially uniform - strict avalanche criterion CLC number TP 309 Foundation item: Supported by the National Natural Science Foundation of China (60373089, 60373041), Natural Science Foundation of Hubei Province (2002AB0037) and Chen-guang Plan of Wuhan City (20025001007).Biography: Zeng Xiang-yong (1973-), male, A postdoctoral fellow, research direction: cryptology and the representation theory of algebra.

APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G＆teaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.

Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces

In this paper, the iteration xn＋l =αny ＋ （1 -αn）Ti（n）k（n）xn for a family of asymptotically nonexpansive mappings T1, T2, ..., TN is originally introduced in an uniformly convex Banach space. Motivated by recent papers, we prove that under suitable conditions the iteration scheme converges strongly to the nearest common fixed point of the family of asymptotically nonexpansive mappings. The results presented in this paper expand and improve correponding ones from Hilbert spaces to uniformly convex Banach spaces, or from nonexpansive mappings to asymptotically nonexpansive mappings.

On the differential uniformities of functions over finite fields

In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that, the differential uniformity of a function over Fq can be any even integer between 2 and q when q is even; and it can be any integer between 1 and q except q-1 when q is odd. Moreover, for any possible differential uniformity t, an explicit construction of a differentially t-uniform function is given.

A new family of differentially 4-uniform permutations over F_(2^(2k))for odd k

We study the differential uniformity of a class of permutations over F2 n with n even. These permutations are different from the inverse function as the values x^(-1) are modified to be(γx)^(-1) on some cosets of a fixed subgroup γ of F_(2n)~*. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent(CCZ-equivalent) classes as checked by Magma for small numbers n. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).

Non-Monomial Permutations with Differential Uniformity Six

In this paper, a family of non-monomial permutations over the finite field F2n with differential uniformity at most 6 is proposed, where n is a positive integer. The algebraic degree of these functions is also determined.

Some Functions with Low Differential Uniformity

We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear（PN） functions,including the properties of the derivative of the components of those functions.Some sufficient and necessary conditions have been explored to judge when a function is a PN function.These conditions may be useful in constructing new PN functions.We also construct some functions with differential 4-uniformity that have rarely been studied in the literature.Some of the constructed functions with differential 4-uniformity have high nonlinearity as well.Finally,a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.

On the Differential Uniformity and Nonlinearity of a Class of Permutation Quadrinomials Over F_(2 2 m)

Permutation polynomials with low differential uniformity and high nonlinearity are preferred in cryptographic systems. In 2018, Tu, Zeng and Helleseth constructed a new class of permutation quadrinomials over the finite field F_(2 2m) for an odd integer m. In this paper, we aim to investigate the differential uniformity and nonlinearity of this class of permutation polynomials so as to find 4-uniform permutation polynomials with high nonlinearity.

A Note on Approximating Fixed Points of Pseudocontractive Mappings

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T ： K →K be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Let {λn} C （0,1/2] be a sequence satisfying the conditions： （i） limn→∞λn=0; （ii） ∑n=0^∞ λn=∞. Let the sequence {xn} be generated from arbitrary x1∈K by xn＋1 = （1 -λn）xn ＋ λnTxn -λn（xn - x1）, n ≥ 1. Suppose limn→∞‖xn - Txn‖ = 0. Then {xn} converges strongly to a fixed point of T.