Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

The Seventh International Olympiad in Cryptography: problems and solutions

The International Olympiad in Cryptography NSUCRYPTO is the unique 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. In 2020, it was held for the seventh time. Prizes and diplomas were awarded to 84 participants in the first round and 49 teams in the second round from 32 countries. In this paper, problems and their solutions of NSUCRYPTO'2020 are presented. We consider problems related to attacks on ciphers and hash functions, protocols, permutations, primality tests, etc. We discuss several open problems on JPEG encoding, Miller -- Rabin primality test, special bases in the vector space, AES-GCM. The problem of a modified Miller -- Rabin primality test was solved during the Olympiad. The problem for finding special bases was partially solved

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

Simplicity conditions for binary orthogonal arrays

It is known that correlation-immune (CI) Boolean functions used in the framework of side-channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths $4$ and $5$. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5

Simplicity conditions for binary orthogonal arrays

It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths 4 and 5. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5

О классификации n-разделимых функций многих переменных над конечными полями и одной задаче олимпиады NSUCrypto’2019

A problem of great importance that arises in designing and implementation of a cryptosystem is countering side channel attacks. Often an appropriate mathematical algorithm, implemented on a specific physical device to work in the physical environment,   becomes vulnerable to such attacks.The “function sharing” technique is a prospective and efficient way to avoid this problem. In the paper we investigate “non-complete sharing” of Boolean functions and mappings, and functions and mappings over finite fields and provide a complete description of the set of functions with n variables, which have sharing.The main findings are the following: introducing and investigating a new concept of “weak” non-complete n-sharing, establishing its connection with “weak” and “classical” n-sharing, and substantiating its advantages from the algebraic point-of-view as well as establishing and proving a criterion for the existence of weak non-complete n-sharing for an arbitrary function. The results also include an explicit description of a set of functions which have weak sharing in terms of algebraic normal form, obtaining the precise and simple descriptions for the boundary (“border”) cases: n = 2, n=m and binary fields.  Applying these results to the AES S-box allows complete solving the problem,  i.e. a complete answer to the question of a representability of the S-box of the AES cipher as a sharing is available. We believe that the same way can be successful for other cryptographic algorithms.Одной из важных проблем, возникающих при проектировании и практической реализации криптосистем, является противодействие атакам по побочным каналам. Нередко алгоритмы, стойкость которых с чисто математической точки зрения не вызывает больших сомнений, оказываются уязвимыми к таким атакам при их реализации на конкретном физическом устройстве.Техника разделений функций и отображений является эффективным инструментом для создания реализаций криптографических алгоритмов, защищенных от атак по побочным каналам. В настоящей статье исследуются неполные разделения булевых отображений и отображений над конечными полями. В рамках данной работы дано полное описание множества функций от n переменных, допускающих разделения на функции от n-1 переменной. Полученные результаты обобщаются на случай отображений над произвольными конечными полями.Основными результатами данной работы являются следующие: введено и исследовано понятие "слабого неполного n-разделения"; установлена его связь с классическим понятием n-разделения, обоснованы его преимущества с точки зрения алгебраической структуры; установлен и доказан критерий существования слабого неполного n-разделения для произвольной функции над произвольным полем; дано явное описание функций от m переменных, допускающих  n-разделения для случая конечных полей произвольной характеристики в терминах алгебраической нормальной формы; получены точные и простые описания для граничных случаев n = 2, n=m и двоичных полей; получен полный ответ на вопрос о представимости S-блока шифра AES в виде разделения.Указанные результаты могут быть использованы для построения реализаций криптографических алгоритмов, защищенных от атак по побочным каналам. Также данный подход позволяет в ряде случаев обнаружить и количественно оценить принципиальную уязвимость конкретных криптографических конструкций к атакам по побочным каналам.Также данные результаты могут быть интересны с теоретической точки зрения. В частности, представляется интересным их применение для решения  задачи эффективной декомпозиции функций. Данная функциональность может быть применена, например, в современных системах компьютерной алгебры, символьных вычислений и машинного обучения