The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph ($V$, $E$) where $V$ is the set of all bent functions in $2k$ variables and $(f, g) \in E$ if the Hamming distance between $f$ and $g$ is equal to $2^k$ (it is the minimal possible distance between two different bent functions). The maximum degree of the graph is obtained and it is shown that all its vertices of maximum degree are quadratic. It is proven that a subgraph of the graph induced by all functions affinely equivalent to Maiorana---McFarland bent functions is connected

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

Maximums of the Additive Differential Probability of Exclusive-Or

At FSE 2004, Lipmaa et al. studied the additive differential probability adp⊕(α,β → γ) of exclusive-or where differences α,β,γ ∈ Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that maxα,βadp⊕(α,β → γ) = adp⊕(0,γ → γ) for all γ. Furthermore, we prove that there always exist either two or eight distinct pairs α,β such that adp⊕( α,β → γ) = adp⊕(0,γ → γ), and we obtain recurrence formulas for calculating adp⊕. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp⊕(0,γ → γ), and we find all γ that satisfy this minimum value

Maximums of the Additive Differential Probability of Exclusive-Or

At FSE 2004, Lipmaa et al. studied the additive differential probability adp⊕(α,β → γ) of exclusive-or where differences α,β,γ ∈ Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that maxα,βadp⊕(α,β → γ) = adp⊕(0,γ → γ) for all γ. Furthermore, we prove that there always exist either two or eight distinct pairs α,β such that adp⊕( α,β → γ) = adp⊕(0,γ → γ), and we obtain recurrence formulas for calculating adp⊕. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp⊕(0,γ → γ), and we find all γ that satisfy this minimum value

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