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Quantum computational number theory
Yan S., Springer International Publishing, New York, NY, 2015. 252 pp. Type: Book (978-3-319258-21-8)
Date Reviewed: Apr 26 2017

The idea of quantum computing including the quantum Turing machine originated in the last century. However, real interest in quantum computing developed after the seminal paper by Peter Shor [1], where he gave polynomial-time algorithms for the integer factorization and discrete log problems. This drew the immediate attention of security experts, since it posed a risk to many computer systems that were using cryptographic algorithms based on the hardness of these problems. This paper also inspired people to look for other problems that can be solved by quantum computers. Over the last two decades, the field of quantum computational number theory (QCNT) has grown, and this book summarizes the major developments in the area.

Indexes and references are given in each of the six chapters in this interesting book. Each section also contains very good problems. What I like about this book is that each chapter contains sufficient material to understand the topic. Chapter 1 gives the idea of important problems in number theory, and chapter 2 covers the basics of classical and quantum computing from a theoretical computer science perspective. Chapter 3 discusses the idea of integer factorization, its basic algorithms, and the cryptosystem based on it, and finally the work of Peter Shor about his polynomial-time algorithm and its variations. Chapter 4 follows a similar pattern for discrete log problems. Chapter 5 looks at the elliptic curve discrete log problem. Finally, chapter 6 considers some other quantum algorithms. Each chapter concludes with notes and further reading, making the book very useful to students and newcomers to the field.

I feel that in the post-quantum cryptography (PQC) era, when the Google Chrome web browser is based on PQC, this book will be very useful for people who want to understand the area. I strongly recommend the book to all young computer science students and to mathematicians who love number theory so they can enjoy this new field.

Reviewer:  Manish Gupta Review #: CR145222 (1707-0420)
1) Shor, P. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. of FOCS. IEEE Computer Society Press, New York, NY, 1994, 124–134.
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