On the differential equivalence of APN functions

C.~Carlet, P.~Charpin, V.~Zinoviev in 1998 defined the associated Boolean function $\gamma_F(a,b)$ in $2n$ variables for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself. It takes value~$1$ if $a\neq {\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function $G$ in $n$ variables, $n\leq 6$, that is differentially equivalent to a given quadratic APN function $F$, can be represented as $G = F + A$, where $A$ is affine. For the APN Gold function $F$, we completely described all affine functions $A$ such that $F$ and $F+A$ are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial

On a remarkable property of APN Gold functions

In [13] for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself it was defined an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables that takes value~$1$ iff $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. It is an interesting open problem to describe differential equivalence class of a given APN function. We consider the APN Gold function $F(x)=x^{2^k+1}$, where gcd$(k,n)=1$, and prove that there exist exactly $2^{2n+n/2}$ distinct affine functions $A$ such that $F$ and $F+A$ are differentially equivalent if $n=4t$ for some $t$ and $k = n/2 \pm 1$; otherwise the number of such affine functions is equal to $2^{2n}$. This theoretical result and computer calculations obtained show that APN Gold functions for $k=n/2\pm1$ and $n=4t$ are the only functions (except one function in 6 variables) among all known quadratic APN functions in $2,\ldots,8$ variables that have more than $2^{2n}$ trivial affine functions $A^F_{c,d}(x)=F(x)+F(x+c)+d$, where $c,d\in\mathbb{F}_2^n$, preserving the associated Boolean function when adding to $F$

Problems, solutions and experience of the first international student\u27s Olympiad in cryptography

A detailed overview of the problems, solutions and experience of the first international student\u27s Olympiad in cryptography, NSUCRYPTO\u272014, is given. We start with rules of participation and description of rounds. All 15 problems of the Olympiad and their solutions are considered in detail. There are discussed solutions of the mathematical problems related to cipher constructing such as studying of differential characteristics of S-boxes, S-box masking, determining of relations between cyclic rotation and additions modulo $2$ and $2^n$, constructing of special linear subspaces in $\mathbb{F}_2^n$; problems about the number of solutions of the equation $F(x)+F(x+a)=b$ over the finite field $\mathbb{F}_{2^n}$ and APN functions. Some unsolved problems in symmetric cryptography are also considered

Psychrotolerant Strains of <em>Phoma herbarum</em> with Herbicidal Activity

The search for stress-tolerant producer strains is a key factor in the development of biological mycoherbicides. The aim of the study was to assess the herbicidal potential of phoma-like fungi. Morphological and physiological features of two Antarctic psychrotolerant strains 20-A7-1.M19 and 20-A7-1.M29 were studied. Multilocus sequence analysis was used to identify these strains. They happened to belong to Phoma herbarum Westend. The psychrotolerant properties of these strains were suggested not only by ecology, but also by their capability to grow in a wide temperature range from 5 °C to 35 °C, being resistant to high insolation, UV radiation, aridity, and other extreme conditions. It was shown that treatment with their cell-free cultural fugate, crude mycelium extract, and culture liquid significantly reduced the seed germination of troublesome weeds such as dandelion and goldenrod. Cell-free cultural fugate and culture liquid also led to the formation of chlorosis and necrotic spots on leaves. Thus, psychrotolerant strains P. herbarum 20-A7-1.M19 and 20-A7-1.M29 demonstrate high biotechnological potential. Our next step is to determine the structures of biologically active substances and to increase their biosynthesis, as well as the development of biological and biorational mycoherbicides. New mycoherbicides can reduce the chemical load on agroecosystems and increase the effectiveness of applied chemicals

On the Sixth International Olympiad in Cryptography NSUCRYPTO

NSUCRYPTO is the unique cryptographic Olympiad containing scientific mathematical problems for professionals, school and university students from any country. Its aim is to involve young researchers in solving curious and tough scientific problems of modern cryptography. From the very beginning, the concept of the Olympiad was not to focus on solving olympic tasks but on including unsolved research problems at the intersection of mathematics and cryptography. The Olympiad history starts in 2014. In 2019, it was held for the sixth time. In this paper, problems and their solutions of the Sixth International Olympiad in cryptography NSUCRYPTO'2019 are presented. We consider problems related to attacks on ciphers and hash functions, protocols, Boolean functions, Dickson polynomials, prime numbers, rotor machines, etc. We discuss several open problems on mathematical countermeasures to side-channel attacks, APN involutions, S-boxes, etc. The problem of finding a collision for the hash function Curl27 was partially solved during the Olympiad

An overview of the Eight International Olympiad in Cryptography "Non-Stop University CRYPTO"

Non-Stop University CRYPTO is the International Olympiad in Cryptography that was held for the eight time in 2021. Hundreds of university and school students, professionals from 33 countries worked on mathematical problems in cryptography during a week. The aim of the Olympiad is to attract attention to curious and even open scientific problems of modern cryptography. In this paper, problems and their solutions of the Olympiad’2021 are presented. We consider 19 problems of varying difficulty and topics: ciphers, online machines, passwords, binary strings, permutations, quantum circuits, historical ciphers, elliptic curves, masking, implementation on a chip, etc. We discuss several open problems on quantum error correction, finding special permutations and s-Boolean sharing of a function, obtaining new bounds on the distance to affine vectorial functions