Nowadays almost everyone is a frequent sender of secret messages, albeit often without realising it. Every time we purchase an item online or send an email our details are encrypted; we assume that this is secure and do not question the algorithms used. But have you ever wondered how these algorithms work and where they originated?

In The Mathematics of Secrets, Joshua Holden takes the reader on a chronological journey from Julius Caesar¡¯s substitution cipher to modern day public-key algorithms and beyond, showing how Alice and Bob can communicate without eavesdropping Eve deciphering their messages. With the introduction of each cipher, Holden explains the underlying mathematics required to encrypt the message and subsequently decipher it, usually with a key of some description.

Potential readers should not be overly concerned about their mathematical ability. In the book¡¯s first half, no more than school mathematics is needed, as the ciphers described are underpinned by modular arithmetic (which can be described as the mathematics of the 24-hour clock and is fully explained by Holden). Even when the maths becomes more complex, lay readers can still appreciate the beauty of the cipher while skipping some of the theory.

Written for anyone with an interest in cryptography, The Mathematics of Secrets is substantial enough to be recommended background reading for undergraduate or postgraduate modules on this topic. However, it is not a book that can be read quickly. Readers will want to stop and think, work out the messages and, in some cases, encrypt their own. Indeed, I challenge anyone to read Holden¡¯s delightful description of Alberti¡¯s cipher disk without wanting to stop and make one themselves.

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As the book progresses, the mathematics becomes more demanding, with the introduction of Fermat¡¯s Little Theorem (different from the one that Andrew Wiles famously proved), Euclid¡¯s Algorithm and elliptic curves. For me, it is these later chapters that are the most gripping. They focus on several algorithms within public-key cryptography, which Holden describes as being ¡°systems that allow Alice and Bob to communicate securely without an initial secure meeting¡± to agree a key. The key is often in the public domain, but for various reasons, Eve cannot use it to decipher the messages, usually because it would take too long to compute a missing part of the key. Many of these algorithms involve the multiplication of large prime numbers resulting in a number that is hard to factorise. However, it is often not that simple, and there are situations when public-key encryption can be broken, if sufficient care is not taken over the numbers used in the encryption process.

Despite modern-day encryption algorithms, correctly implemented, being difficult to crack, there is always the danger that someone will discover a way to break them. In the book¡¯s final chapter, Holden focuses on quantum computing and the considerable repercussions that developments in this field might have in rendering public-key cryptography insecure, and thus the enormous impact that this will have on global data security. Alice, Bob and Eve are not going to go away; there will always be a need to transmit messages secretly, and therefore there will always be those who seek to decipher them.

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Noel-Ann Bradshaw is principal lecturer in mathematics, University of Greenwich.

The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption
By Joshua Holden
Princeton University Press, 392pp, ?24.95
ISBN 9780691141756 and 9781400885626 (e-book)
Published 22 February 2017