Hard Mathematical Problems in Cryptography and Coding Theory

In this thesis, we are concerned with certain interesting computationally hard problems and the complexities of their associated algorithms. All of these problems share a common feature in that they all arise from, or have applications to, cryptography, or the theory of error correcting codes. Each chapter in the thesis is based on a stand-alone paper which attacks a particular hard problem. The problems and the techniques employed in attacking them are described in detail. The first problem concerns integer factorization: given a positive integer $N$. the problem is to find the unique prime factors of $N$. This problem, which was historically of only academic interest to number theorists, has in recent decades assumed a central importance in public-key cryptography. We propose a method for factorizing a given integer using a graph-theoretic algorithm employing Binary Decision Diagrams (BDD). The second problem that we consider is related to the classification of certain naturally arising classes of error correcting codes, called self-dual additive codes over the finite field of four elements, $GF(4)$. We address the problem of classifying self-dual additive codes, determining their weight enumerators, and computing their minimum distance. There is a natural relation between self-dual additive codes over $GF(4)$ and graphs via isotropic systems. Utilizing the properties of the corresponding graphs, and again employing Binary Decision Diagrams (BDD) to compute the weight enumerators, we can obtain a theoretical speed up of the previously developed algorithm for the classification of these codes. The third problem that we investigate deals with one of the central issues in cryptography, which has historical origins in the theory of geometry of numbers, namely the shortest vector problem in lattices. One method which is used both in theory and practice to solve the shortest vector problem is by enumeration algorithms. Lattice enumeration is an exhaustive search whose goal is to find the shortest vector given a lattice basis as input. In our work, we focus on speeding up the lattice enumeration algorithm, and we propose two new ideas to this end. The shortest vector in a lattice can be written as ${\bf s} = v_1{\bf b}_1+v_2{\bf b}_2+\ldots+v_n{\bf b}_n$. where $v_i \in \mathbb{Z}$ are integer coefficients and ${\bf b}_i$ are the lattice basis vectors. We propose an enumeration algorithm, called hybrid enumeration, which is a greedy approach for computing a short interval of possible integer values for the coefficients $v_i$ of a shortest lattice vector. Second, we provide an algorithm for estimating the signs $+$ or $-$ of the coefficients $v_1,v_2,\ldots,v_n$ of a shortest vector ${\bf s}=\sum_{i=1}^{n} v_i{\bf b}_i$. Both of these algorithms results in a reduction in the number of nodes in the search tree. Finally, the fourth problem that we deal with arises in the arithmetic of the class groups of imaginary quadratic fields. We follow the results of Soleng and Gillibert pertaining to the class numbers of some sequence of imaginary quadratic fields arising in the arithmetic of elliptic and hyperelliptic curves and compute a bound on the effective estimates for the orders of class groups of a family of imaginary quadratic number fields. That is, suppose $f(n)$ is a sequence of positive numbers tending to infinity. Given any positive real number $L$. an effective estimate is to find the smallest positive integer $N = N(L)$ depending on $L$ such that $f(n) > L$ for all $n > N$. In other words, given a constant $M > 0$. we find a value $N$ such that the order of the ideal class $I_n$ in the ring $R_n$ (provided by the homomorphism in Soleng's paper) is greater than $M$ for any $n>N$. In summary, in this thesis we attack some hard problems in computer science arising from arithmetic, geometry of numbers, and coding theory, which have applications in the mathematical foundations of cryptography and error correcting codes

A suite of quantum algorithms for the shortestvector problem

Crytography has come to be an essential part of the cybersecurity infrastructure that provides a safe environment for communications in an increasingly connected world. The advent of quantum computing poses a threat to the foundations of the current widely-used cryptographic model, due to the breaking of most of the cryptographic algorithms used to provide confidentiality, authenticity, and more. Consequently a new set of cryptographic protocols have been designed to be secure against quantum computers, and are collectively known as post-quantum cryptography (PQC). A forerunner among PQC is lattice-based cryptography, whose security relies upon the hardness of a number of closely related mathematical problems, one of which is known as the shortest vector problem (SVP). In this thesis I describe a suite of quantum algorithms that utilize the energy minimization principle to attack the shortest vector problem. The algorithms outlined span the gate-model and continuous time quantum computing, and explore methods of parameter optimization via variational methods, which are thought to be effective on near-term quantum computers. The performance of the algorithms are analyzed numerically, analytically, and on quantum hardware where possible. I explain how the results obtained in the pursuit of solving SVP apply more broadly to quantum algorithms seeking to solve general real-world problems; minimize the effect of noise on imperfect hardware; and improve efficiency of parameter optimization.Open Acces

Методи розв’язання задачі LPN над скінченними кільцями для оцінювання стійкості симетричних постквантових шифросистем

Дисертація на здобуття наукового ступеня кандидата технічних наук за спеціальністю 05.13.21 – Системи захисту інформації. – Харківський національний університет імені В. Н. Каразіна, Міністерства освіти і науки України. – Харків, 2021. У дисертації розв’язано актуальну наукову задачу розробки більш ефективних (в порівнянні з перебірним) методів розв’язання задачі LPN над скінченними кільцями для оцінювання стійкості симетричних постквантових шифросистем. Вперше отримано аналітичні оцінки обсягу матеріалу, достатнього для розв’язання із заданою достовірністю задачі LPN над довільним скінченним кільцем, які дозволяють визначити часову складність узагальненого алгоритму BKW. Розроблено два методи підвищення ефективності розв’язання задачі LPN за допомогою ММП. Вперше розроблено метод побудови нових алгоритмів розв’язання СР над кільцем за довільною скінченною сукупністю вхідних таких алгоритмів. Наведено аналітичні вирази оцінок достовірності та часової складності алгоритмів розв’язання СР, які будуються за допомогою розробленого методу, через відповідні характеристики вхідних алгоритмів. Головним практичним результатом роботи є можливість оцінювати стійкість симетричних шифросистем, які будуються над скінченними кільцями та базуються на складності розв’язання задачі LPN

Semi-Quantum Conference Key Agreement (SQCKA)

A need in the development of secure quantum communications is the scalable extension of key distribution protocols. The greatest advantage of these protocols is the fact that its security does not rely on mathematical assumptions and can achieve perfect secrecy. In order to make these protocols scalable, has been developed the concept of Conference Key Agreements, among multiple users. In this thesis we propose a key distribution protocol among several users using a semi-quantum approach. We assume that only one of the users is equipped with quantum devices and generates quantum states, while the other users are classical, i.e., they are only equipped with a device capable of measuring or reflecting the information. This approach has the advantage of simplicity and reduced costs. We prove our proposal is secure and we present some numerical results on the lower bounds for the key rate. The security proof applies new techniques derived from some already well established work. From the practical point of view, we developed a toolkit called Qis|krypt⟩ that is able to simulate not only our protocol but also some well-known quantum key distribution protocols. The source-code is available on the following link: - https://github.com/qiskrypt/qiskrypt/.Uma das necessidades no desenvolvimento de comunicações quânticas seguras é a extensão escalável de protocolos de distribuição de chaves. A grande vantagem destes protocolos é o facto da sua segurança não depender de suposições matemáticas e poder atingir segurança perfeita. Para tornar estes protocolos escaláveis, desenvolveu-se o conceito de Acordo de Chaves de Conferência, entre múltiplos utilizadores. Nesta tese propomos um protocolo para distribuição de chaves entre vários utilizadores usando uma abordagem semi-quântica. Assumimos que apenas um dos utilizadores está equipado com dispositivos quânticos e é capaz de gerar estados quânticos, enquanto que os outros utilizadores são clássicos, isto é, estão apenas equipados com dispositivos capazes de efectuar uma medição ou refletir a informação. Esta abordagem tem a vantagem de ser mais simples e de reduzir custos. Provamos que a nossa proposta é segura e apresentamos alguns resultados numéricos sobre limites inferiores para o rácio de geração de chaves. A prova de segurança aplica novas técnicas derivadas de alguns resultados já bem estabelecidos. Do ponto de vista prático, desenvolvemos uma ferramenta chamada Qis|krypt⟩ que é capaz de simular não só o nosso protocolo como também outros protocolos distribuição de chaves bem conhecidos. O código fonte encontra-se disponível no seguinte link: - https://github.com/qiskrypt/qiskrypt/

Space station data system analysis/architecture study. Task 2: Options development, DR-5. Volume 2: Design options

The primary objective of Task 2 is the development of an information base that will support the conduct of trade studies and provide sufficient data to make key design/programmatic decisions. This includes: (1) the establishment of option categories that are most likely to influence Space Station Data System (SSDS) definition; (2) the identification of preferred options in each category; and (3) the characterization of these options with respect to performance attributes, constraints, cost and risk. This volume contains the options development for the design category. This category comprises alternative structures, configurations and techniques that can be used to develop designs that are responsive to the SSDS requirements. The specific areas discussed are software, including data base management and distributed operating systems; system architecture, including fault tolerance and system growth/automation/autonomy and system interfaces; time management; and system security/privacy. Also discussed are space communications and local area networking