Cunsheng Ding

Highly nonlinear mappings

Functions with high nonlinearity have important applications in cryptography, sequences and coding theory. The purpose of this paper is to give a well-rounded treatment of non-Boolean functions with optimal nonlinearity. We summarize and generalize known results, and prove a number of new results. We also present open problems about functions with high nonlinearity.

Linear codes from perfect nonlinear mappings and their secret sharing schemes

In this paper, error-correcting codes from perfect nonlinear mappings are constructed, and then employed to construct secret sharing schemes. The error-correcting codes obtained in this paper are very good in general, and many of them are optimal or almost optimal. The secret sharing schemes obtained in this paper have two types of access structures. The first type is democratic in the sense that every participant is involved in the same number of minimal-access sets. In the second type of access structures, there are a few dictators who are in every minimal access set, while each of the remaining participants is in the same number of minimal-access sets.

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.

Almost difference sets and their sequences with optimal autocorrelation

Almost difference sets have interesting applications in cryptography and coding theory. We give a well-rounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period n/spl equiv/0 (mod 4) with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel (1982) can be used to construct almost difference sets and sequences of period 4l with optimal autocorrelation.

On the linear complexity of Legendre sequences

We determine the linear complexity of all Legendre sequences and the (monic) feedback polynomial of the shortest linear feedback shift register that generates such a Legendre sequence. The result shows that Legendre sequences are quite good from the linear complexity viewpoint.

Hamming weights in irreducible cyclic codes

Abstract The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes.

Secret sharing schemes from three classes of linear codes

Secret sharing has been a subject of study for over 20 years, and has had a number of real-world applications. There are several approaches to the construction of secret sharing schemes. One of them is based on coding theory. In principle, every linear code can be used to construct secret sharing schemes. But determining the access structure is very hard as this requires the complete characterization of the minimal codewords of the underlying linear code, which is a difficult problem in general. In this paper, a sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums. Some linear codes whose covering structure can be determined are constructed, and then used to construct secret sharing schemes with nice access structures.

On almost perfect nonlinear permutations

In this paper basic properties of APN permutations, which can be used in an iterated secret-key block cipher as a round function to protect it from a differential cryptanalysis, are investigated. Several classes of almost perfect nonlinear permutations and other permutations in GF(2)n with good nonlinearity and high nonlinear order are presented. Included here are also three methods for constructing permutations with good nonlinearity.

The weight distribution of a class of linear codes from perfect nonlinear functions

In this correspondence, the weight distribution of a class of linear codes based on perfect nonlinear functions (also called planar functions) is determined. The class of linear codes under study are either optimal or among the best codes known, and have nice applications in cryptography.

Autocorrelation values of generalized cyclotomic sequences of order two

The generalized cyclotomic sequence of order two has several good randomness properties and behaves like the Legendre sequence in several aspects. We calculate the autocorrelation values of the generalized cyclotomic sequence of order two. Our result shows that this sequence could have very good autocorrelation property and pattern distributions of length two if the two primes are chosen properly.