Deciding EA-equivalence via invariants

We define a family of efficiently computable invariants for (n,m)-functions under EA-equivalence, and observe that, unlike the known invariants such as the differential spectrum, algebraic degree, and extended Walsh spectrum, in the case of quadratic APN functions over $\mathbb {F}_{2^n}$ with n even, these invariants take on many different values for functions belonging to distinct equivalence classes. We show how the values of these invariants can be used constructively to implement a test for EA-equivalence of functions from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{m}$; to the best of our knowledge, this is the first algorithm for deciding EA-equivalence without resorting to testing the equivalence of associated linear codes.publishedVersio

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

Generalization of a class of APN binomials to Gold-like functions

In 2008 Budaghyan, Carlet and Leander generalized a known instance of an APN function over the finite field F212 and constructed two new infinite families of APN binomials over the finite field F2n , one for n divisible by 3, and one for n divisible by 4. By relaxing conditions, the family of APN binomials for n divisible by 3 was generalized to a family of differentially 2t -uniform functions in 2012 by Bracken, Tan and Tan; in this sense, the binomials behave in the same way as the Gold functions. In this paper, we show that when relaxing conditions on the APN binomials for n divisible by 4, they also behave in the same way as the Gold function x2s+1 (with s and n not necessarily coprime). As a counterexample, we also show that a family of APN quadrinomials obtained as a generalization of a known APN instance over F210 cannot be generalized to functions with 2t -to-1 derivatives by relaxing conditions in a similar way.acceptedVersio

Triplicate functions

We define the class of triplicate functions as a generalization of 3-to-1 functions over $\mathbb {F}_{2^{n}}$ for even values of n. We investigate the properties and behavior of triplicate functions, and of 3-to-1 among triplicate functions, with particular attention to the conditions under which such functions can be APN. We compute the exact number of distinct differential sets of power APN functions and quadratic 3-to-1 functions; we show that, in this sense, quadratic 3-to-1 functions are a generalization of quadratic power APN functions for even dimensions, in the same way that quadratic APN permutations are generalizations of quadratic power APN functions for odd dimensions. We show that quadratic 3-to-1 APN functions cannot be CCZ-equivalent to permutations in the case of doubly-even dimensions. We compute a lower bound on the Hamming distance between any two quadratic 3-to-1 APN functions, and give an upper bound on the number of such functions over $\mathbb {F}_{2^{n}}$ for any even n. We survey all known infinite families of APN functions with respect to the presence of 3-to-1 functions among them, and conclude that for even n almost all of the known infinite families contain functions that are quadratic 3-to-1 or are EA-equivalent to quadratic 3-to-1 functions. We also give a simpler univariate representation in the case of singly-even dimensions of the family recently introduced by Göloglu than the ones currently available in the literature. We conduct a computational search for quadratic 3-to-1 functions in even dimensions n ≤ 12. We find six new APN instances for n = 10, and the first sporadic APN instance for n = 12 since 2006. We provide a list of all known 3-to-1 APN functions for n ≤ 12.publishedVersio

A New Family of APN Quadrinomials

The binomial B(x) = x 3 +βx 36 (where β is primitive in F 2 2) over F 2 10 is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form x 3 +a(x 2i+1 )2 k +bx 3·2m +c(x2 i+m+2m ) 2k from which B(x) can be obtained by setting a = β, b = c = 0, i = 3, k = 2. We show that for any dimension n = 2m with m odd and 3 + m,setting(a, b, c)=(β, β 2 , 1) and i =m -2 or i = (m - 2) -1 mod n yields an APN function, and verify that for n = 10 the quadrinomials obtained in this way for i = m - 2 and i = (m - 2) -1 mod n are CCZ-inequivalent to each other, to B(x), and to any other known APN function over F 2 10.acceptedVersio

Searching for APN functions by polynomial expansion

In pape

Partially APN functions with APN-like polynomial representations

In this paper we investigate several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field F2. We also investigate the differential uniformity of some binomial partial APN functions. Furthermore, the partial APN-ness for some classes of multinomial functions is investigated. We show also that the size of the pAPN spectrum is preserved under CCZ-equivalence.acceptedVersio

On Two Fundamental Problems on APN Power Functions

The six infinite families of power APN functions are among the oldest known instances of APN functions, and it has been conjectured in 2000 that they exhaust all possible power APN functions. Another long-standing open problem is that of the Walsh spectrum of the Dobbertin power family, which is still unknown. Those of Kasami, Niho and Welch functions are known, but not the precise values of their Walsh transform, with rare exceptions. One promising approach that could lead to the resolution of these problems is to consider alternative representations of the functions in questions. We derive alternative representations for the infinite APN monomial families. We show how the Niho, Welch, and Dobbertin functions can be represented as the composition xi∘x1/j of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can be used to represent the functions in this way. We investigate compositions xi∘L∘x1/j for a linear polynomial L , show how the Kasami functions in odd dimension can be expressed in this way with i=j being a Gold exponent and compute all APN functions of this form for n≤9 and for L with binary coefficients, thereby showing that our theoretical constructions exhaust all possible cases. We present observations and data on power functions with exponent ∑k−1i=122ni−1 which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for n≤35 , and conjecture its exact form. As an application of our results, we determine the exact values of the Walsh transform of the Kasami function at all points of a special form. Computations performed for n≤21 show that these points cover about 2/3 of the field.acceptedVersio