Cryptography is the science of analyzing and deciphering codes and ciphers. Today, cryptography is widely used in many day-to-day applications. This book was produced as part of the CREST Crypto-Math Project funded by the Japan Science and Technology Agency. Many older cryptosystems have become vulnerable to quantum computing techniques. Hence, the goal of the CREST project is to construct mathematical models for the post-quantum cryptography era. This book, edited by six editors, is volume 29 of the “Mathematics for Industry” series by Springer.

The book is divided into four parts, and 19 chapters in total. Each chapter is a research paper that was selected for publication by the editors. An introductory chapter on the CREST project serves as the starting point and brings out the goals of the project. The first part of the book is on mathematical cryptography. It has five chapters that focus on varied topics such as multivariate public key cryptosystems, zero-knowledge protocols that are based on codes, the application of Ramanujan graphs for hash functions, hyperelliptic curves, and isogeny sequences and their applications to cryptography.

The second part of the book has the curious title “Mathematics Towards Cryptography.” This part contains five chapters. The topics discussed are the quantum Rabi model, group-subgroup pair graphs, quaternion groups, random number generation and the use of sparse matrices for agreement of secret keys, and the process of recovering a secret key from noise.

The third part of the book is on lattices and their applications to cryptography. This part has five chapters. The first chapter of this part talks about the problem of learning with errors and studies key recovery attacks. The next chapter is on the shortest vector problem. For this problem, the authors discuss a mixed integer quadratic conceptualization. The next two chapters focus on short-generator problems. The last chapter of this part of the book is a survey on Don Coppersmith’s method involving lattices.

The fourth and final part of the book is on cryptographic protocols. There are three chapters that focus on signature schemes and their security, the IND-CCA1 homomorphic encryption method, and identity-based encryption using lattices. The book contains a short index. There are numerous references to the literature at the end of the chapters.

The book is meant for graduate students and researchers. However, in my opinion only mathematically sophisticated graduate students and those pursuing their research programs in the field of cryptography will be able to fully benefit from this book. It is simply not suitable for those who are likely to be put off by advanced mathematical concepts. I agree with the contributors who emphasize the need for exploring areas of mathematics such as representation theory, mathematical physics, and topology in the post-quantum cryptography era. Many papers in the book are surveys, which indicate that there is plenty of scope for further research.

The book lists many open problems that will be of interest to researchers. This engaging book looks at novel ideas that are worth exploring. The length of the book seems to be quite okay for teaching advanced courses on cryptography, although there are no exercises in the book for pedagogy. However, it will be useful for self-study and also as a reference book. The style of writing in the book is satisfactory, although the book is mathematically heavy. I strongly recommend it for the intended audience: graduate students and researchers. Practitioners and libraries will also benefit from this book, which provides new perspectives.