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

A new family of APN quadrinomials

The binomial $B(x) = x^3 + \beta x^{36}$ (where $\beta$ is primitive in $\mathbb{F}_{2^4}$) over $\mathbb{F}_{2^{10}}$ 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 $x^3 + a (x^{2^i+1})^{2^k} + b x^{3 \cdot 2^m} + c (x^{2^{i+m}+2^m})^{2^k}$ from which $B(x)$ can be obtained by setting $a = \beta$, $b = c = 0$, $i = 3$, $k = 2$. We show that for any dimension $n = 2m$ with $m$ odd and $3 \nmid m$, setting $(a,b,c) = (\beta, \beta^2, 1)$ 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 $\mathbb{F}_{2^{10}}$

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

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

Classification and computational search for planar functions in characteristic 3

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Two New Families of Quadratic APN Functions

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Constructing APN functions through isotopic shifts

Almost perfect nonlinear (APN) functions over fields of characteristic 2 play an important role in cryptography, coding theory and, more generally, information theory as well as mathematics. Building new APN families is a challenge which has not been successfully addressed for more than seven years now. The most general known equivalence relation preserving APN property in characteristic 2 is CCZ-equivalence. Extended to general characteristic, it also preserves planarity. In the case of quadratic planar functions, it is a particular case of isotopic equivalence. We apply the idea of isotopic equivalence to transform APN functions in characteristic 2 into other functions, some of which can be APN. We deduce new quadratic APN functions and a new quadratic APN family

On known constructions of APN and AB functions and their relation to each other

This work is dedicated to APN and AB functions which are optimal against differential and linear cryptanlysis when used as Sboxes in block ciphers. They also have numerous applications in other branches of mathematics and information theory such as coding theory, sequence design, combinatorics, algebra and projective geometry. In this paper we give an overview of known constructions of APN and AB functions, in particular, those leading to infinite classes of these functions. Among them, the bivariate construction method, the idea first introduced in 2011 by the third author of the present paper, turned out to be one of the most fruitful. It has been known since 2011 that one of the families derived from the bivariate construction contains the infinite families derived by Dillon’s hexanomial method. Whether the former family is larger than the ones it contains has stayed an open problem which we solve in this paper. Further we consider the general bivariate construction from 2013 by the third author and study its relation to the recently found infinite families of bivariate APN functions

Classification of quadratic APN functions with coefficients in GF(2) for dimensions up to 9

Almost perfect nonlinear (APN) and almost bent (AB) functions are integral components of modern block ciphers and play a fundamental role in symmetric cryptography. In this paper, we describe a procedure for searching for quadratic APN functions with coefficients in GF(2) over the finite fields GF(2^n) and apply this procedure to classify all such functions over GF(2^n) with n up to 9. We discover two new APN functions (which are also AB) over GF(2^9) that are CCZ-inequivalent to any known APN function over this field. We also verify that there are no quadratic APN functions with coefficients in GF(2) over GF(2^n) with n between 6 and 8 other than the currently known ones