17 research outputs found
On the weight distributions of several classes of cyclic codes from APN monomials
Let be an odd integer and be an odd prime. % with ,
where is an odd integer.
In this paper, many classes of three-weight cyclic codes over
are presented via an examination of the condition for the
cyclic codes and , which have
parity-check polynomials and respectively, to
have the same weight distribution, where is the minimal polynomial of
over for a primitive element of
. %For , the duals of five classes of the proposed
cyclic codes are optimal in the sense that they meet certain bounds on linear
codes. Furthermore, for and positive integers such
that there exist integers with and satisfying , the value
distributions of the two exponential sums T(a,b)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where , are
settled. As an application, the value distribution of is utilized to
investigate the weight distribution of the cyclic codes
with parity-check polynomial . In the case of and
even satisfying the above condition, the duals of the cyclic codes
have the optimal minimum distance
Towards a deeper understanding of APN functions and related longstanding problems
This dissertation is dedicated to the properties, construction and analysis of APN and AB functions. Being cryptographically optimal, these functions lack any general structure or patterns, which makes their study very challenging. Despite intense work since at least the early 90's, many important questions and conjectures in the area remain open. We present several new results, many of which are directly related to important longstanding open problems; we resolve some of these problems, and make significant progress towards the resolution of others.
More concretely, our research concerns the following open problems: i) the maximum algebraic degree of an APN function, and the Hamming distance between APN functions (open since 1998); ii) the classification of APN and AB functions up to CCZ-equivalence (an ongoing problem since the introduction of APN functions, and one of the main directions of research in the area); iii) the extension of the APN binomial over into an infinite family (open since 2006); iv) the Walsh spectrum of the Dobbertin function (open since 2001); v) the existence of monomial APN functions CCZ-inequivalent to ones from the known families (open since 2001); vi) the problem of efficiently and reliably testing EA- and CCZ-equivalence (ongoing, and open since the introduction of APN functions).
In the course of investigating these problems, we obtain i.a. the following results: 1) a new infinite family of APN quadrinomials (which includes the binomial over ); 2) two new invariants, one under EA-equivalence, and one under CCZ-equivalence; 3) an efficient and easily parallelizable algorithm for computationally testing EA-equivalence; 4) an efficiently computable lower bound on the Hamming distance between a given APN function and any other APN function; 5) a classification of all quadratic APN polynomials with binary coefficients over for ; 6) a construction allowing the CCZ-equivalence class of one monomial APN function to be obtained from that of another; 7) a conjecture giving the exact form of the Walsh spectrum of the Dobbertin power functions; 8) a generalization of an infinite family of APN functions to a family of functions with a two-valued differential spectrum, and an example showing that this Gold-like behavior does not occur for infinite families of quadratic APN functions in general; 9) a new class of functions (the so-called partially APN functions) defined by relaxing the definition of the APN property, and several constructions and non-existence results related to them.Doktorgradsavhandlin