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The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption
Explaining the mathematics of cryptography
The Mathematics of Secrets takes readers on a fascinating tour of the mathematics behind cryptography―the science of sending secret messages. Most books about cryptography are organized historically, or around how codes and ciphers have been used, such as in government and military intelligence or bank transactions. Joshua Holden instead shows how mathematical principles underpin the ways that different codes and ciphers operate. Holden focuses on both code making and code breaking and he discusses the majority of ancient and modern ciphers currently known.
Holden begins by looking at substitution ciphers, built by substituting one letter or block of letters for another. Explaining one of the simplest and historically well-known ciphers, the Caesar cipher, Holden establishes the key mathematical idea behind the cipher and discusses how to introduce flexibility and additional notation. Holden goes on to explore polyalphabetic substitution ciphers, transposition ciphers, including one developed by the Spartans, connections between ciphers and computer encryption, stream ciphers, and ciphers involving exponentiation. He also examines public-key ciphers, where the methods used to encrypt messages are public knowledge, and yet, intended recipients are still the only ones who are able to read the message. He concludes with a look at the future of ciphers and where cryptography might be headed. Only basic mathematics up to high school algebra is needed to understand and enjoy the book.
With a plethora of historical anecdotes and real-world examples, The Mathematics of Secrets reveals the mathematics working stealthily in the science of coded messages.
The Mathematics of Secrets takes readers on a fascinating tour of the mathematics behind cryptography―the science of sending secret messages. Most books about cryptography are organized historically, or around how codes and ciphers have been used, such as in government and military intelligence or bank transactions. Joshua Holden instead shows how mathematical principles underpin the ways that different codes and ciphers operate. Holden focuses on both code making and code breaking and he discusses the majority of ancient and modern ciphers currently known.
Holden begins by looking at substitution ciphers, built by substituting one letter or block of letters for another. Explaining one of the simplest and historically well-known ciphers, the Caesar cipher, Holden establishes the key mathematical idea behind the cipher and discusses how to introduce flexibility and additional notation. Holden goes on to explore polyalphabetic substitution ciphers, transposition ciphers, including one developed by the Spartans, connections between ciphers and computer encryption, stream ciphers, and ciphers involving exponentiation. He also examines public-key ciphers, where the methods used to encrypt messages are public knowledge, and yet, intended recipients are still the only ones who are able to read the message. He concludes with a look at the future of ciphers and where cryptography might be headed. Only basic mathematics up to high school algebra is needed to understand and enjoy the book.
With a plethora of historical anecdotes and real-world examples, The Mathematics of Secrets reveals the mathematics working stealthily in the science of coded messages.
392 pages, Hardcover
Published February 14, 2017
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348 people want to read
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Displaying 1 - 7 of 7 reviews
April 15, 2017
There are so many books on cryptography I've read or perused that I'm not going to read one more of them from cover to cover. With this one, I checked what I knew I wanted to read about, and then I tried to read in detail a number of specific topics.
To be frank, given the technological progress, I'm not interested anymore in reading about DES, RSA and the likes (Chapters 4-7). There are many more sources that detail the challenges of contemporary cryptography with more practical details. Then, the quantum computing Chapter 9 is very unsatisfactory by my standards (but my standards say that quantum computing is a hoax, as everything that's quantum is probabilistic hence only statistically relevant, but with very limited practical use where accuracy is needed). Chapter 8 feels more like an improvised one, but it has the extenuating circumstances that other public-key systems are less popular, after all.
Each and every such book is more or less successful in describing the (pre)history of cryptography, starting from Caesar and going through Claude Shannon. This one is no exception. "Classical" naive and historically (and literarily!) famous ciphers are nicely described, sometimes with the typical flaws of an academic. Some obvious steps are described in too much detail, sometimes so much that the elementary appears more complex than it is and hence difficult to understand, whereas the graphical representations sometimes fail to be obvious enough (I especially hate some figures from Chapters 4-5).
The challenge of any such book is to make the transition between the pre-Shannon and the post-Shannon era, from the naive to the mathematical approach. The result is always clumsy because it's a huge paradigm shift. Here, Chapter 4 brings this shift, and the following two chapters are a bit makeshift; for instance, why are Girolamo Cardano's autokey ciphers (keystreams) only mentioned in Chapter 5, when they're known since the 16th century? (The "Looking Forward" end-chapter sections are not to my liking.) Simply put, the book is technically good, but not systematic enough; given it's made more like a university textbook, I simply don't like it (blame it on how much I hated some of my University professors and the memories this book brought to me).
What's always funny, entertaining and instructive in such a book is to read the said (pre)history of cryptography. Being a modern book though, this one fails to provide the necessary historical and literary references, despite the efforts of doing so.
E.A. Poe is only quoted at page 301 for having written that “human ingenuity cannot concoct a cipher which human ingenuity cannot resolve,” but no reference to "The Gold Bug" is given. Unbelievable.
Then, despite the reasonable attention given to the extremely naive ciphers, the author fails to mention the Fleissner (or Fleißner) grille cipher[1]. Girolamo Cardano's grille[2] was used by Cardinal Richelieu, but in his novel "Mathias Sandorf" Jules Verne used the version designed by his contemporary Oberst Eduard Baron Fleissner von Wostrowitz (1825-1885)[3].
As a side note, and typically for an academic, the author isn't able to clearly state the differences between the following: Trithemius’ tabula recta, the Vigenère square, Bellaso’s cipher, and the Vigenère cipher. You'll probably need to read pp. 41-43 more than three times—I've read them three times and I still can't remember which is which! (Blame it on my age.) Commonly, they're all "the Vigenère cipher" anyway.
Going forward, a variation of the Vigenère cipher is encountered in Jules Verne's "Eight Hundred Leagues on the Amazon" aka "The Giant Raft" ("Huit cents lieues sur l’Amazone" aka "La Jangada") in a very spectacular matter, as it opens the book! Verne has actually used the Gronsfeld cipher[4], namely a Vigenère using a key of numbers instead of letters. Gronsfeld is only incidentally mentioned in this book in a note to Chapter 5, at page 168: "Gromark cipher: Gromark stands for GROnsfeld with Mixed Alphabet and Running Key. ... The Gronsfeld cipher is just a name for variants of a tabula recta cipher using a key of numbers instead of letters, such as we use here and in the key autokey cipher of the previous section."
The "good old elementary ciphers" were still used in the two World Wars. Page 116: "the cipher invented by the German officer Lieutenant (later Colonel) Fritz Nebel and used by the German army during World War I. The Germans called this cipher GedeFu 18, short for Geheimschrift der Funker 1918, or Radio Operators’ Cipher 1918. The French, seeing a bunch of cryptograms containing only the letters A, D, F, G, V, and X, called it the ADFGVX cipher. This system starts out with a 6×6 version of the Polybius square, containing both letters and digits in a scrambled order and labeled on the top and side with the letters ADFGVX."
Pp. 163-165: "the key autokey cipher ... is a slightly simplified version of a cipher used by Soviet troops during World War II to encipher numerical code groups." The German Enigma is mentioned in Chapter 2, but details as to how long were the "extremely long periods" of the repeating-key ciphers of such machines are not given; several endnotes send to external sources. Such a pity...
Finally, a bit of paranoia at page 118 just prepares for the transition to modern cryptography: "In a paper written in 1945 and declassified and published in 1949, Shannon defined diffusion..." So, it was written as a classified material!
Let's recap: the "easy parts for the general public" are incomplete, the parts on modern cryptography are underwhelming, and the general impression is that of a mediocre University textbook. Flop.
[1] https://en.wikipedia.org/wiki/Grille_...
[2] https://en.wikipedia.org/wiki/Cardan_...
[3] https://www.kryptographiespielplatz.d...
[4] https://www.apprendre-en-ligne.net/cr...
To be frank, given the technological progress, I'm not interested anymore in reading about DES, RSA and the likes (Chapters 4-7). There are many more sources that detail the challenges of contemporary cryptography with more practical details. Then, the quantum computing Chapter 9 is very unsatisfactory by my standards (but my standards say that quantum computing is a hoax, as everything that's quantum is probabilistic hence only statistically relevant, but with very limited practical use where accuracy is needed). Chapter 8 feels more like an improvised one, but it has the extenuating circumstances that other public-key systems are less popular, after all.
Each and every such book is more or less successful in describing the (pre)history of cryptography, starting from Caesar and going through Claude Shannon. This one is no exception. "Classical" naive and historically (and literarily!) famous ciphers are nicely described, sometimes with the typical flaws of an academic. Some obvious steps are described in too much detail, sometimes so much that the elementary appears more complex than it is and hence difficult to understand, whereas the graphical representations sometimes fail to be obvious enough (I especially hate some figures from Chapters 4-5).
The challenge of any such book is to make the transition between the pre-Shannon and the post-Shannon era, from the naive to the mathematical approach. The result is always clumsy because it's a huge paradigm shift. Here, Chapter 4 brings this shift, and the following two chapters are a bit makeshift; for instance, why are Girolamo Cardano's autokey ciphers (keystreams) only mentioned in Chapter 5, when they're known since the 16th century? (The "Looking Forward" end-chapter sections are not to my liking.) Simply put, the book is technically good, but not systematic enough; given it's made more like a university textbook, I simply don't like it (blame it on how much I hated some of my University professors and the memories this book brought to me).
What's always funny, entertaining and instructive in such a book is to read the said (pre)history of cryptography. Being a modern book though, this one fails to provide the necessary historical and literary references, despite the efforts of doing so.
E.A. Poe is only quoted at page 301 for having written that “human ingenuity cannot concoct a cipher which human ingenuity cannot resolve,” but no reference to "The Gold Bug" is given. Unbelievable.
Then, despite the reasonable attention given to the extremely naive ciphers, the author fails to mention the Fleissner (or Fleißner) grille cipher[1]. Girolamo Cardano's grille[2] was used by Cardinal Richelieu, but in his novel "Mathias Sandorf" Jules Verne used the version designed by his contemporary Oberst Eduard Baron Fleissner von Wostrowitz (1825-1885)[3].
As a side note, and typically for an academic, the author isn't able to clearly state the differences between the following: Trithemius’ tabula recta, the Vigenère square, Bellaso’s cipher, and the Vigenère cipher. You'll probably need to read pp. 41-43 more than three times—I've read them three times and I still can't remember which is which! (Blame it on my age.) Commonly, they're all "the Vigenère cipher" anyway.
Going forward, a variation of the Vigenère cipher is encountered in Jules Verne's "Eight Hundred Leagues on the Amazon" aka "The Giant Raft" ("Huit cents lieues sur l’Amazone" aka "La Jangada") in a very spectacular matter, as it opens the book! Verne has actually used the Gronsfeld cipher[4], namely a Vigenère using a key of numbers instead of letters. Gronsfeld is only incidentally mentioned in this book in a note to Chapter 5, at page 168: "Gromark cipher: Gromark stands for GROnsfeld with Mixed Alphabet and Running Key. ... The Gronsfeld cipher is just a name for variants of a tabula recta cipher using a key of numbers instead of letters, such as we use here and in the key autokey cipher of the previous section."
The "good old elementary ciphers" were still used in the two World Wars. Page 116: "the cipher invented by the German officer Lieutenant (later Colonel) Fritz Nebel and used by the German army during World War I. The Germans called this cipher GedeFu 18, short for Geheimschrift der Funker 1918, or Radio Operators’ Cipher 1918. The French, seeing a bunch of cryptograms containing only the letters A, D, F, G, V, and X, called it the ADFGVX cipher. This system starts out with a 6×6 version of the Polybius square, containing both letters and digits in a scrambled order and labeled on the top and side with the letters ADFGVX."
Pp. 163-165: "the key autokey cipher ... is a slightly simplified version of a cipher used by Soviet troops during World War II to encipher numerical code groups." The German Enigma is mentioned in Chapter 2, but details as to how long were the "extremely long periods" of the repeating-key ciphers of such machines are not given; several endnotes send to external sources. Such a pity...
Finally, a bit of paranoia at page 118 just prepares for the transition to modern cryptography: "In a paper written in 1945 and declassified and published in 1949, Shannon defined diffusion..." So, it was written as a classified material!
Let's recap: the "easy parts for the general public" are incomplete, the parts on modern cryptography are underwhelming, and the general impression is that of a mediocre University textbook. Flop.
[1] https://en.wikipedia.org/wiki/Grille_...
[2] https://en.wikipedia.org/wiki/Cardan_...
[3] https://www.kryptographiespielplatz.d...
[4] https://www.apprendre-en-ligne.net/cr...
July 9, 2019
A great book about the history of cryptography. Book starts with Caesar cipher that it is the simplest and most widely known encryption techniques and ends with quantum cryptography that it is the science of exploiting quantum mechanical properties to perform cryptographic tasks. Author examines different encryption algorithms and explains how they are made and works. He also talks about how you can break the encrypted messages. Cryptography is a science and this book is about the mathematics behind this science.
There are good suggestions for further reading at the end of the book.
If you don’t like mathematics it’s better to avoid reading this book because there is a lot of mathematics in it.
There are good suggestions for further reading at the end of the book.
If you don’t like mathematics it’s better to avoid reading this book because there is a lot of mathematics in it.
April 28, 2020
After going over a few different sources in search for a clearer presentation of modern cryptography I stumbled across this book which had a recent publication date and came from a highly respected academic publisher. I placed an online purchase order with great anticipation, and to my delight it has turned out to be as good as I had hoped for. The book progresses methodically, building idea on top of an idea in a logical manner. Examples are beautifully manufactured, chosen and interspersed with descriptions and explanations, so is a saucy amount of history. Most of the mathematics is very straightforward, and where it does get a little bit tricky (such as in modulo multiplicative and exponential inverses, and modulo elliptic curve operations) Joshua’s exposition makes it simple again.
The book does not spend any effort on the practical implementations of cryptography, so for that the reader would have to look elsewhere.
The book does not spend any effort on the practical implementations of cryptography, so for that the reader would have to look elsewhere.
September 20, 2017
Could be clearer in many places, but a good introduction to codes for those mathematically inclined.
February 23, 2025
Really great read for the beginner
October 28, 2017
An excellent introductory book on cryptography. Clear, simple explanations of the concepts. A historical depth that builds an interesting storyline and some remarkable facts. I was amazed that non-carrying binary addition (XOR) was used in 1917. Many concepts are introduced in a historical context where technology was much simpler. Some people will still find the math a bit challenging. But this is a great introduction.
Displaying 1 - 7 of 7 reviews