Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over $\mathbb{Z}_4$. We then specialise to planar monomial functions $f(x)=cx^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=cx^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first versio

Computational investigation of 0-APN monomials

This thesis is dedicated to exploring methods for deciding whether a power function $F(x) = x^d$ is 0-APN. Any APN function is 0-APN, and so 0-APN-ness is a necessary condition for APN-ness. APN functions are cryptographically optimal, and are thus an object of significant interest. Deciding whether a given power function is 0-APN, or APN, is a very difficult computational problem in dimensions greater than e.g. 30. Methods which allow this to be resolved more efficiently are thus instrumental to resolving open problems such as Dobbertin's conjecture. Dobbertin's conjecture states that any APN power function must be equivalent to a representative from one of the six known infinite families. This has been verified for all dimensions up to 34, and up to 42 for even dimensions. There have, however, been no further developments, and so Dobbertin's conjecture remains one of the oldest and most well-known open problems in the area. In this work, we investigate some methods for efficiently testing 0-APN-ness. A 0-APN function can be characterized as one that does not vanish on any 2-dimensional linear subspace. We determine the minimum number of linear subspaces that have to be considered in order to check whether a power function is 0-APN. We characterize the elements of this minimal set of linear subspaces, and formulate and implement efficient procedures for generating it. We computationally test the efficiency of this method for dimension 35, and conclude that it can be used to decide 0-APN-ness much faster than by conventional methods, although a dedicated effort would be needed to exploit this further due to the huge number of exponents that need to be checked in high dimensions such as 35. Based on our computational results, we observe that most of the cubic power functions are 0-APN. We generalize this observation into a ``doubly infinite'' family of 0-APN functions, i.e. a construction giving infinitely many exponents, each of which is 0-APN over infinitely many dimensions. We also present some computational results on the differential uniformity of these exponents, and observe that the Gold and Inverse power functions can be expressed using the doubly infinite family.Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN

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 $x^3 + \beta x^{36}$ over $F_{2^{10}}$ 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 $x^3 + \beta x^{36}$ over $F_{2^{10}}$); 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 $F_{2^n}$ for $n \le 9$; 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

Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents

International audienceWe prove a necessary condition for some polynomials of Gold and Kasami degree to be APN over F q n for large n

A new large class of functions not APN infinitely often

In this paper, we show that there is no vectorial Boolean function of degree 4e, with e satisfaying certain conditions, which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodie