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The book is divided into three parts. The first one provides background information and basic notions of group theory, algorith- mic problems in groups, and classical cryptography. More precisely, in Chapter 1, we survey the basic theory of groups, subgroups, and special classes of groups such as cyclic and nilpotent groups. In this part, we also describe free groups and combinatorial group theory. Chapter 2 lays the groundwork for describing algo- rithmic problems in group theory: we describe group-theoretic decision problems including the word, conjugacy, and isomorphism decision problems. We also de- scribe the power decision problem, the endomorphism decision problem, the n-th root decision problem, and the geodesic length decision problem. We also present many search problems. In Chapter 3 we describe a number of cryptosystems that have been built around several classes of groups in the last decades. In particular, we present digital signature schemes, secret sharing schemes, and multilinear maps. Part 2 is devoted to group-theoretic post-quantum cryptography. In Chapter 4 we present several key exchange protocols based on an underlying noncommutative structure such as Anshel-Anshel-Goldfeld (AAG) key exchange, Ko, Lee, et al. key exchange, and the noncommutative ElGamal key exchange. We also introduce two key exchange protocols based on the semidirect product of groups or semigroups. A discussion of noncommutative digital signatures, such as the Kahrobaei-Koupparis digital signature scheme and the Mohan and Anjaneyulu digital signature scheme,and of noncommutative secret sharing schemes is also included. We discuss the Panagopoulos threshold scheme as it has motivated the construction of some non- commutative threshold schemes. These include the Habeeb-Kahrobaei-Shpilrain (HKS) secret sharing scheme, which is a hybrid of Shamir’s secret sharing scheme that provides some advantages to Shamir’s scheme, and threshold schemes based on Nielsen transformations. Chapter 5 introduces some recent attacks against these schemes: for example the length-based attack, the linear centralizer attack, the linear decomposition attack, and the quotient attack. Chapter 6 deals with quan- tum cryptanalysis of some examples of group-based cryptography, in particular as it relates to complexity of the hidden subgroup problem (HSP). We cover the fa- mous construct of Shor’s algorithm and survey the state of the art for quantum algorithms addressing the hidden subgroup problem in groups not accounted for by Shor’s algorithm. Finally, we provide a “case study”-type exposition of two recent results in group-based quantum cryptanalysis. The last part of the book is devoted to suitable platforms for the cryptographic protocols presented in Part 2. In each chapter we give the definition of the class of groups proposed together with important properties and the status of their al- gorithmic problems. Then, we survey the current status of the respective group in cryptography, and we also include a list of open problems, which we hope will guide researchers who wish to work in this field. The groups that we propose as cryp- tographic platforms are the following: braid groups, hyperbolic groups, small can- cellation groups, polycyclic groups, graph groups, arithmetic groups, Engel groups, and self-similar groups.

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.

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.

This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.

xiv, 542 pages : 24 cm

In this tutorial, selected topics of cryptology and of computational complexity theory are presented. We give a brief overview of the history and the foundations of classical cryptography, and then move on to modern public-key cryptography. Particular attention is paid to cryptographic protocols and the problem of constructing the key components of such protocols such as one-way functions. A function is one-way if it is easy to compute, but hard to invert. We discuss the notion of one-way functions both in a cryptographic and in a complexity-theoretic setting. We also consider interactive proof systems and present some interesting zero-knowledge protocols. In a zero-knowledge protocol one party can convince the other party of knowing some secret information without disclosing any bit of this information. Motivated by these protocols, we survey some complexity-theoretic results on interactive proof systems and related complexity classes.

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.

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.

The objective of this paper is to build the neutrosophic version of the RSA crypto-algorithm, where we use the foundations of fusion neutrosophic number theory such as neutrosophic phi-Euler's function, neutrosophic congruencies, and neutrosophic inverses to build novel algorithms for cryptography depending of famous RSA algorithm.