This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography.

In recent decades, computing has undergone rapid evolutions and groundbreaking developments that affect almost every sector across the world. The developments of quantum computing and quantum cryptography are similarly revolutionizing computing and security with lasting impacts and implications. Quantum computing and quantum cryptography will pave the path for new opportunities for the future of computing. Quantum Computing and Cryptography in Future Computers discusses quantum computing and quantum cryptography principles and their impact on future computers. It includes coverage of the role of quantum computing to overcome the issues of current security methods. It also discusses the application of quantum computing in various areas like security, blockchain, and more. Covering topics such as attack detection, machine learning, and quantum key distribution, this premier reference source is an ideal resource for developers, engineers, practitioners, security experts, students and educators of higher education, librarians, researchers, and academicians.

This book presents the mathematical background underlying security modeling in the context of next-generation cryptography. By introducing new mathematical results in order to strengthen information security, while simultaneously presenting fresh insights and developing the respective areas of mathematics, it is the first-ever book to focus on areas that have not yet been fully exploited for cryptographic applications such as representation theory and mathematical physics, among others. Recent advances in cryptanalysis, brought about in particular by quantum computation and physical attacks on cryptographic devices, such as side-channel analysis or power analysis, have revealed the growing security risks for state-of-the-art cryptographic schemes. To address these risks, high-performance, next-generation cryptosystems must be studied, which requires the further development of the mathematical background of modern cryptography. More specifically, in order to avoid the security risks posed by adversaries with advanced attack capabilities, cryptosystems must be upgraded, which in turn relies on a wide range of mathematical theories. This book is suitable for use in an advanced graduate course in mathematical cryptography, while also offering a valuable reference guide for experts.

The book provides theoretical methods of connecting discrete-variable quantum information processing to continuous-variable one. It covers the two major fields of quantum information processing, quantum communication and quantum computation, leading to achievement of a long-sought full security of continuous-variable quantum key distribution (QKD) and proposal of a resource-efficient method for optical quantum computing. Firstly, the book provides a security of continuous-variable QKD against arbitrary attacks under a realistic condition such as finite communication rounds and the use of digitized information processing. The book also provides the unified view for conventionally used approximate Gottesman-Kitaev-Preskill (GKP) codes, which encodes qudits on a continuous-variable system, enabling direct comparison between researches based on different approximations. The book finally proposes a resource-efficient method to realize the universal optical quantum computation using the GKP code via the direct preparation of the GKP magic state instead of GKP Pauli states. Feasibility of the proposed protocol is discussed based on the existing experimental proposals for the GKP state preparation.

This book constitutes the refereed proceedings of the 9th International Workshop on Post-Quantum Cryptography, PQCrypto 2018, held in Fort Lauderdale, FL, USA, in April 2018. The 24 revised full papers presented were carefully reviewed and selected from 97 submissions. The papers are organized in topical sections on Code-based Cryptography; Cryptanalysis; Hash-based Cryptography; Isogenies in Cryptography; Lattice-based Cryptography; Multivariate Cryptography; Protocols; Quantum Algorithms.

This book constitutes the refereed proceedings of the 16th International Workshop on Security, IWSEC 2021, held in Tokyo, Japan in September 2021. The conference was held virtually due to COVID-19 pandemic. The 14 regular papers and 3 short paper presented in this volume were carefully reviewed and selected from 37 submissions. They were organized in topical sections named: Lattice-Based Cryptography; System Security; Multiparty Computation; Machine Learning and Security; Post-quantum Cryptography; Symmetric-key Cryptography; Game Theory and Security.

This book offers an introduction to post-quantum cryptography for students, engineers and researchers in the field of information security. Above all, it describes the mathematical concepts underlying the security of post-quantum cryptographic schemes. The first part of the book provides essential background information by briefly introducing the core elements of quantum computation and presenting Shor’s algorithm, which solves the factoring problem and the discrete logarithm problem in polynomial time. In turn, the second part presents a number of candidates for post-quantum public-key encryption and digital signature schemes. The security of these schemes is based on mathematical problems in coding theory, multivariate quadratic equations, and lattices, respectively. The book provides an essential guide for students, researchers and engineers, helping them to quickly grasp this highly promising area of cryptography.

This book explains the group representation theory for quantum theory in the language of quantum theory. As is well known, group representation theory is very strong tool for quantum theory, in particular, angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, quark model, quantum optics, and quantum information processing including quantum error correction. To describe a big picture of application of representation theory to quantum theory, the book needs to contain the following six topics, permutation group, SU(2) and SU(d), Heisenberg representation, squeezing operation, Discrete Heisenberg representation, and the relation with Fourier transform from a unified viewpoint by including projective representation. Unfortunately, although there are so many good mathematical books for a part of six topics, no book contains all of these topics because they are too segmentalized. Further, some of them are written in an abstract way in mathematical style and, often, the materials are too segmented. At least, the notation is not familiar to people working with quantum theory. Others are good elementary books, but do not deal with topics related to quantum theory. In particular, such elementary books do not cover projective representation, which is more important in quantum theory. On the other hand, there are several books for physicists. However, these books are too simple and lack the detailed discussion. Hence, they are not useful for advanced study even in physics. To resolve this issue, this book starts with the basic mathematics for quantum theory. Then, it introduces the basics of group representation and discusses the case of the finite groups, the symmetric group, e.g. Next, this book discusses Lie group and Lie algebra. This part starts with the basics knowledge, and proceeds to the special groups, e.g., SU(2), SU(1,1), and SU(d). After the special groups, it explains concrete applications to physical systems, e.g., angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, and quark model. Then, it proceeds to the general theory for Lie group and Lie algebra. Using this knowledge, this book explains the Bosonic system, which has the symmetries of Heisenberg group and the squeezing symmetry by SL(2,R) and Sp(2n,R). Finally, as the discrete version, this book treats the discrete Heisenberg representation which is related to quantum error correction. To enhance readers' undersnding, this book contains 54 figures, 23 tables, and 111 exercises with solutions.