Pariah Moonshine Part II: For Whom the Moon Shines

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenI ended Part I with the observation that the Monster group was connected with the symmetries of a group sitting in 196883-dimensional space, whereas the number 196884 appeared as part of a function used in number theory, the study of the properties of whole numbers.  In particular, a mathematician named John McKay noticed the number as one of the coefficients of a modular form.  Modular forms also exhibit a type of symmetry, namely if F is a modular form then there is some number k for which

Figure 1

for every set of whole numbers a, b, c, and d such that adbc=1.  (There are also some conditions as the real part of z goes to infinity.)

Modular forms, elliptic curves, and Fermat’s Last Theorem

In 1954, Martin Eichler was studying modular forms and observing patterns in their coefficients.  For example, take the modular form

Figure 2

(I don’t know whether Eichler actually looked at this particular form, but he definitely looked at similar ones.)  The coefficients of this modular form seem to be related to the number of whole number solutions of the equation

y2 = x3 – 4 x2 + 16

This equation is an example of what is known as an elliptic curve, which is a curve given by an equation of the form

y2 = x3 + ax2 + bx + c

Note that elliptic curves are not ellipses!  Elliptic curves have one line of symmetry, two open ends, and either one or two pieces, as shown in Figures 1 and 2. They are called elliptic curves because the equations came up in the seventeenth century when mathematicians started studying the arc length of an ellipse.  These curves are considered the next most complicated type of curve after lines and conic sections, both of which have been understood pretty well since at least the ancient Greeks.   They are useful for a lot of things, including cryptography, as I describe in Section 8.3 of The Mathematics of Secrets.

Figure 1

Figure 1. The elliptic curve y2= x3 + x has one line of symmetry, two open ends, and one piece.

Figure 2

Figure 2. The elliptic curve y2 = x3 – x has one line of symmetry, two open ends, and two pieces.

 

In the late 1950’s it was conjectured that every elliptic curve was related to a modular form in the way that the example above is.  Proving this “Modularity Conjecture” took on more urgency in the 1980’s, when it was shown that showing the conjecture was true would also prove Fermat’s famous Last Theorem.  In 1995 Andrew Wiles, with help from Richard Taylor, proved enough of the Modularity Conjecture to show that Fermat’s Last Theorem was true, and the rest of the Modularity Conjecture was filled in over the next six years by Taylor and several of his associates.

Modular forms, the Monster, and Moonshine

Modular forms are also related to other shapes besides elliptic curves, and in the 1970’s John McKay and John Thompson became convinced that the modular form

J(z) = e -2 π i z + 196884 e 2 π i z + 21493760 e 4 π i z  + 864299970 e 6 π i z  + …

was related to the Monster.  Not only was 196884 equal to 196883 + 1, but 21493760 was equal to 21296876 + 196883 + 1, and 21296876 was also a number that came up in the study of the Monster.  Thompson suggested that there should be a natural way of associating the Monster with an infinite-dimensional shape, where the infinite-dimensional shape broke up into finite-dimensional pieces with each piece having a dimension corresponding to one of the coefficients of J(z).   This shape was (later) given the name V♮, using the natural sign from musical notation in a typically mathematical pun.  (Terry Gannon points out that there is also a hint that the conjectures “distill information illegally” from the Monster.) John Conway and Simon Norton formulated some guesses about the exact connection between the Monster and V♮, and gave them the name “Moonshine Conjectures” to reflect their speculative and rather unlikely-seeming nature. A plausible candidate for V♮ was constructed in the 1980’s and Richard Borcherds proved in 1992 that the candidate satisfied the Moonshine Conjectures.  This work was specifically cited when Borcherds was awarded the Fields medal in 1998.

The construction of V♮ turned out also to have a close connection with mathematical physics.  The reconciliation of gravity with quantum mechanics is one of the central problems of modern physics, and most physicists think that string theory is likely to be key to this resolution.  In string theory, the objects we traditionally think of as particles, like electrons and quarks, are really tiny strings curled up in many dimensions, most of which are two small for us to see.  An important question about this theory is exactly what shape these dimensions curl into.  One possibility is a 24-dimensional shape where the possible configurations of strings in the shape are described by V♮.  However, there are many other possible shapes and it is not known how to determine which one really corresponds to our world.

More Moonshine?

Since Borcherds’ proof, many variations of the original “Monstrous Moonshine” have been explored.  The other members of the Happy Family can be shown to have Moonshine relationships similar to those of the Monster.  “Modular Moonshine” says that certain elements of the Monster group should have their own infinite dimensional shapes, related to but not the same as V♮.  (The “modular” in “Modular Moonshine” is related to the one in “modular form” because they are both related to modular arithmetic, although the chain of connections is kind of long. )  “Mathieu Moonshine” shows that one particular group in the Happy Family has its own shape, entirely different from V♮, and “Umbral Moonshine” extends this to 23 other related groups which are not simple groups.  But the Pariah groups remained outsiders, rejected by both the Happy Family and by Moonshine — until September 2017.

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Pariah Moonshine Part I: The Happy Family and the Pariah Groups

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenBeing a mathematician, I often get asked if I’m good at calculating tips. I’m not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries. The recent announcement of “Pariah Moonshine” has been one of the most exciting developments in the relationship between numbers and symmetries in quite some time. In this blog post I hope to explain the “Pariah” part, which deals mostly with symmetries. The “Moonshine”, which connects the symmetries to numbers, will follow in the next post.

What is a symmetry?

First I want to give a little more detail about what I mean by the symmetries of shapes. If you have a square made out of paper, there are basically eight ways you can pick it up, turn it, and put it down in exactly the same place. You can rotate it 90 degrees clockwise or counterclockwise. You can rotate it 180 degrees. You can turn it over, so the front becomes the back and vice versa. You can turn in over and then rotate it 90 degrees either way, or 180 degrees. And you can rotate it 360 degrees, which basically does nothing. We call these the eight symmetries of the square, and they are shown in Figure 1.

Figure1

Figure 1. The square can be rotated into four different positions, without or without being flipped over, for eight symmetries total.

If you have an equilateral triangle, there are six symmetries. If you have a pentagon, there are ten. If you have a pinwheel with four arms, there are only four symmetries, as shown in Figure 2, because now you can rotate it but if you turn it over it looks different. If you have a pinwheel with six arms, there are six ways. If you have a cube, there are 24 if the cube is solid, as shown in Figure 3. If the cube is just a wire frame and you are allowed to turn it inside out, then you get 24 more, for a total of 48.

Figure 2

Figure 2. The pinwheel can be rotated but not flipped, for four symmetries total.

Figure 3

Figure 3. The cube can be rotated along three different axes, resulting in 24 different symmetries.

These symmetries don’t just come with a count, they also come with a structure. If you turn a square over and then rotate it 90 degrees, it’s not the same thing as if you rotate it first and then flip it over. (Try it and see.) In this way, symmetries of shapes are like the permutations I discuss in Chapter 3 of my book, The Mathematics of Secrets: you can take products, which obey some of the same rules as products of numbers but not all of them. These sets of symmetries, which their structures, are called groups.

Groups are sets of symmetries with structure

Some sets of symmetries can be placed inside other sets. For example, the symmetries of the four-armed pinwheel are the same as the four rotations in the symmetries of the square. We say the symmetries of the pinwheel are a subgroup of the symmetries of the square. Likewise, the symmetries of the square are a subgroup of the symmetries of the solid cube, if you allow yourself to turn the cube over but not tip it 90 degrees, as shown in Figure 4.

Figure 4

Figure 4. The symmetries of the square are contained inside the symmetries of the cube if you are allowed to rotate and flip the cube but not tip it 90 degrees.

In some cases, ignoring a subgroup of the symmetries of a shape gets us another group, which we call the quotient group. If you ignore the subgroup of how the square is rotated, you get the quotient group where the square is flipped over or not, and that’s it. Those are the same as the symmetries of the capital letter A, so the quotient group is really a group. In other cases, for technical reasons, you can’t get a quotient group. If you ignore the symmetries of a square inside the symmetries of a cube, what’s left turns out not to be the symmetries of any shape.

You can always ignore all the symmetries of a shape and get just the do nothing (or trivial) symmetry, which is the symmetries of the capital letter P, in the quotient group. And you can always ignore none of the nontrivial symmetries, and get all of the original symmetries still in the quotient group. If these are the only two possible quotient groups, we say that the group is simple. The group of symmetries of a pinwheel with a prime number of arms is simple. So is the group of symmetries of a solid icosahedron, like a twenty-sided die in Dungeons and Dragons. The group of symmetries of a square is not simple, because of the subgroup of rotations. The group of symmetries of a solid cube is not simple, not because of the symmetries of the square, but because of the smaller subgroup of symmetries of a square with a line through it, as shown in Figures 5 and 6. The quotient group there is the same as the symmetries of the equilateral triangle created by cutting diagonally through a cube near a corner.

Figure 5

Figure 5. The symmetries of a square with line through it. We can turn the square 180 degrees and/or flip it, but not rotate it 90 degrees, so there are four.

Figure 6

Figure 6. The symmetries of the square with a line through it inside of the symmetries of the cube.

Categorizing the Pariah groups

As early as 1892, Otto Hölder asked if we could categorize all of the finite simple groups. (There are also shapes, like the circle, which have an infinite number of symmetries. We won’t worry about them now.)  It wasn’t until 1972 that Daniel Gorenstein made a concrete proposal for how to make a complete categorization, and the project wasn’t finished until 2002, producing along the way thousands of pages of proofs. The end result was that almost all of the finite simple groups fell into a few infinitely large categories: the cyclic groups, which are the groups of symmetries of pinwheels with a prime number of arms, the alternating groups, which are the groups of symmetries of solid hypertetrahedra in 5 or more dimensions, and the “groups of Lie type”, which are related to matrix multiplication over finite fields and describe certain symmetries of objects known as finite projective planes and finite projective spaces. (Finite fields are used in the AES cipher and I talk about them in Section 4.5 of The Mathematics of Secrets.)

Even before 1892, a few finite simple groups were discovered that didn’t seem to fit into any of these categories. Eventually it was proved that there were 26 “sporadic” groups, which didn’t fit into any of the categories and didn’t describe the symmetries of anything obvious — basically, you had to construct the shape to fit the group of symmetries that you knew existed, instead of starting with the shape and finding the symmetries. The smallest of the sporadic groups has 7920 symmetries in it, and the largest, known as the Monster, has over 800 sexdecillion symmetries. (That’s an 8 with 53 zeros after it!) Nineteen of the other sporadic groups turn out to be subgroups or quotient groups of subgroups of the Monster. These 20 became known as the Happy Family. The other 6 sporadic groups became known as the ‘Pariahs’.

The shape that was constructed to fit the Monster lives in 196883-dimensional space. In the late 1970’s a mathematician named John McKay noticed the number 196884 turning up in a different area of mathematics. It appeared as part of a function used in number theory, the study of the properties of whole numbers. Was there a connection between the Monster and number theory? Or was the idea of a connection just … moonshine?

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Craig Bauer on unsolved ciphers

In 1953, a man was found dead from cyanide poisoning near the Philadelphia airport with a picture of a Nazi aircraft in his wallet. Taped to his abdomen was an enciphered message. In 1912, a book dealer named Wilfrid Voynich came into possession of an illuminated cipher manuscript once belonging to Emperor Rudolf II, who was obsessed with alchemy and the occult. Wartime codebreakers tried—and failed—to unlock the book’s secrets, and it remains an enigma to this day. In Unsolved, Craig Bauer examines these and other vexing ciphers yet to be cracked. Recently he took the time to answer some questions about his new book.

Why focus on unsolved ciphers?

They’re much more intriguing because they could be concealing anything. Some might reveal the identities of serial killers. Others could unmask spies, rewrite history, expose secret societies, or even give the location of buried treasure worth millions. This sense of mystery is very appealing to me.

Did you try to solve the ciphers yourself first?

There are so many unsolved ciphers that I realized I would never finish writing about them if I kept stopping to try to solve them. There’s one that I’m confident I could solve, but instead of doing so, I simply presented the approach I think will work and am leaving it for a reader to pursue. I expect that several of them will be solved by readers and I look forward to seeing their results!

Does someone who wants to attack these mysteries need to know a lot of mathematics or have computer programming skills?

No. Many of the ciphers were created by people with very little knowledge in either area. Also, past solvers of important ciphers have included amateurs. One of the Zodiac killer’s ciphers was solved by a high school history teacher. Some of the ciphers might be solved in a manner that completely bypasses mathematics. A reader may find a solution through papers the cipher’s creator left behind, perhaps in some library’s archives, in government storage, or in a relative’s possession. I think some may be solved by pursuing a paper trail or some other non-mathematical avenue. Of course, there are mathematical challenges as well, for those who have the skills to take them on. The puzzles span thousands of years, from ancient Egypt to today’s online community. Twentieth century challenges come from people as diverse as Richard Feynman (a world-class physicist) and Ricky McCormick (thought to have been illiterate).

Are all of the unsolved ciphers covered in the book?

No, far from it. There are enough unsolved ciphers to fill many volumes. I limited myself to only the most interesting examples, and still there were too many! I originally set out to write a book about half the size of what was ultimately published. The problem was that there was so much fascinating material that I had to go to 600 pages or experience the agony of omitting something fabulous. Also, unsolved ciphers from various eras are constantly coming to light, and new ones are created every year. I will likely return to the topic with a sequel covering the best of these.

Which cipher is your favorite?

I’m the most excited about the Paul Rubin case. It involves a cipher found taped to the abdomen of a teenage whiz-kid who was found dead in a ditch by the Philadelphia airport, way back in 1953. While I like well-known unsolved ciphers like the Voynich Manuscript and Kryptos, I have higher hopes for this one being solved because it hasn’t attracted any attention since the 1950s. The codebreakers have made a lot of progress since then, so it’s time to take another look and see what can be learned about this young man’s death. I felt it was very important to include cases that will be new even to those who have read a great deal about cryptology already and this is one such case.

Should the potential reader have some prior knowledge of the subject?

If he or she does, there will still be much that is new, but for those with no previous exposure to cryptology, everything is explained from the ground up. As a teenager I loved books at the popular level on a wide range of topics. In particular, the nonfiction of Isaac Asimov instilled in me a love for many subjects. He always started at the beginning, assuming his readers were smart, but new to the topic he was covering. This is the approach that I have taken. I hope that the book finds a wide readership among the young and inspires them in the same way Asimov inspired me.

Is there anything that especially qualifies you to write on this topic?

Early work on this book was supported by the National Security Agency through their Scholar-in-Residence program at the Center for Cryptologic History. They wanted me in this role because, while I have a PhD in mathematics and have carried out mathematical research in cryptology, I also have a passion for history and other disciplines. In fact, both of my books have the word “history” in their titles. The journal Cryptologia, for which I serve as the editor-in-chief, is devoted to all aspects of cryptology, mathematical, historical, pedagogical, etc. My love of diverse fields allows me to write with enthusiasm about ciphers in music, art, criminal cases, ancient history, and other areas. The broad approach to the subject is more entertaining and ensures that there’s something in the book for nearly every reader.

BauerCraig Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

Keith Devlin: Fibonacci introduced modern arithmetic —then disappeared

More than a decade ago, Keith Devlin, a math expositor, set out to research the life and legacy of the medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, whose book Liber abbaci has quite literally affected the lives of everyone alive today. Although he is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent—Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death. Finding Fibonacci is a compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin recently answered some questions about his new book for the PUP blog:

You’ve written 33 math books, including many for general readers. What is different about this one?

KD: This is my third book about the history of mathematics, which already makes it different from most of my books where the focus was on abstract concepts and ideas, not how they were discovered. What makes it truly unique is that it’s the first book I have written that I have been in! It is a first-person account, based on a diary I kept during a research project spread over a decade.

If you had to convey the book’s flavor in a few sentences, what would you say?

KD: Finding Fibonacci is a first-person account of a ten-year quest to uncover and tell the story of one of the most influential figures in human history. It started out as a diary, a simple record of events. It turned into a story when it became clear that it was far more than a record of dates, sources consulted, places visited, and facts checked. Like any good story, it has false starts and disappointments, tragedies and unexpected turns, more than a few hilarious episodes, and several lucky breaks. Along the way, I encountered some amazing individuals who, each for their own reasons, became fascinated by Fibonacci: a Yale professor who traced modern finance back to Fibonacci, an Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci’s astonishing story, an American math professor who fought against cancer to complete the world’s first (and only) modern language translation of Liber abbaci, and the widow who took over and brought his efforts to fruition after he lost that battle. And behind it all, the man who was the focus of my quest. Fibonacci played a major role in creating the modern commercial world. Yet he vanished from the pages of history for five hundred years, made “obsolete,” and in consequence all but forgotten forever, by a new technology.

What made you decide to write this book?

KD: There were really two key decisions that led to this book. One was deciding, back in the year 2000, to keep a diary of my experiences writing The Man of Numbers. My first history book was The Unfinished Game. For that, all I had to do was consult a number of reference works. It was not intended to be original research. Basic Books asked me to write a short, readable account of a single mathematical document that changed the course of human history, to form part of a series they were bringing out. I chose the letter Pierre De Fermat wrote to his colleague Blaise Pascal in 1654, which most experts agree established modern probability theory, in particular how it can be used to predict the future.

In The Man of Numbers, in contrast, I set out to tell a story that no one had told before; indeed, the consensus among the historians was that it could not be told—there simply was not enough information available. So writing that book would require engaging in a lot of original historical research. I had never done that. I would be stepping well outside my comfort zone. That was in part why I decided to keep a diary. The other reason for keeping a record was to ensure I had enough anecdotes to use when the time came to promote the book—assuming I was able to complete it, that is. (I had written enough popular mathematics books to appreciate the need for author promotional activities!)

The second decision, to turn my diary into a book (which only at the end found the title, Finding Fibonacci), came after The Man of Numbers was published in 2011. The ten-year process of researching and writing that book had turned out to be so rich, and so full of unexpected twists and turns, including several strokes of immense luck, that it was clear there was a good story to be told. What was not clear was whether I would be able to write such a book. All my other books are third-person accounts, where I am simply the messenger. In Finding Fibonacci, I would of necessity be a central character. Once again, I would be stepping outside my comfort zone. In particular, I would be laying out on the printed page, part of my inner self. It took five years and a lot of help from my agent Ted Weinstein and then my Princeton University Press editor Vickie Kearn to find the right voice and make it work.

Who do you expect will enjoy reading this book?

KD: I have a solid readership around the world. I am sure they will all read it. In particular, everyone who read The Man of Numbers will likely end up taking a look. Not least because, in addition to providing a window into the process of writing that earlier book, I also put in some details of that story that I did not fully appreciate until after the book had been published. But I hope, and in fact expect, that Finding Fibonacci will appeal to a whole new group of readers. Whereas the star of all my previous books was a discipline, mathematics, this is a book about people, for the most part people alive today. It’s a human story. It has a number of stars, all people, connected by having embarked on a quest to try to tell parts of the story of one of the most influential figures in human history: Leonardo of Pisa, popularly known as Fibonacci.

Now that the book is out, in one sentence if you can, how would you summarize writing it?

KD: Leaving my author’s comfort zone. Without a doubt. I’ve never been less certain how a book would be received.

DevlinKeith Devlin is a mathematician at Stanford University and cofounder and president of BrainQuake, an educational technology company that creates mathematics learning video games. His many books include The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern and The Man of Numbers: Fibonacci’s Arithmetic Revolution. He is the author of Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.

Cipher challenge #2 from Joshua Holden: Subliminal channels

The Mathematics of Secrets by Joshua Holden takes readers on a tour of the mathematics behind cryptography. Most books about cryptography are organized historically, or around how codes and ciphers have been used in government and military intelligence or bank transactions. Holden instead focuses on how mathematical principles underpin the ways that different codes and ciphers operate. Discussing the majority of ancient and modern ciphers currently known, The Mathematics of Secrets sheds light on both code making and code breaking. Over the next few weeks, we’ll be running a series of cipher challenges from Joshua Holden. The first was on Merkle’s puzzles. Today’s focuses on subliminal channels:

As I explain in Section 1.6 of The Mathematics of Secrets, in 1929 Lester Hill invented the first general method for encrypting messages using a set of multiple equations in multiple unknowns.  A less general version, however, had already appeared in 1926, submitted by an 18-year-old to a cryptography column in a detective magazine.  This was Jack Levine, who would later become a prolific researcher in several areas of mathematics, including cryptography.

Levine’s system was billed as a way of encrypting two different messages at the same time.  Maybe one of them was the real message and the other was a dummy message–if the message was intercepted, the interceptor could be thrown off the scent by showing them the dummy message.  This sort of system is now known as a subliminal channel.

The system starts with numbering the letters of the alphabet in two different ways:

   a  b  c  d  e  f  g  h  i  j  k  l  m
  27 28 29 30 31 32 33 34 35 36 37 38 39
   1  2  3  4  5  6  7  8  9 10 11 12 13
  
   n  o  p  q  r  s  t  u  v  w  x  y  z
  40 41 42 43 44 45 46 47 48 49 50 51 52
  14 15 16 17 18 19 20 21 22 23 24 25 26

Suppose the first plaintext, or unencrypted message, is “tuesday” and the second plaintext is “tonight.”  We use the first set of numbers for the first plaintext:

   t  u  e  s  d  a  y
  46 47 31 45 30 27 51

and the second set for the second plaintext:

   t  o  n  i  g  h  t
  20 15 14  9  7  8 20

The encrypted message, or ciphertext, is made up of pairs of numbers.  The first number in each pair is half the sum of the two message numbers, and the second number is half the difference:

    t       u        e       s       d       a        y
   46      47       31      45      30      27       51
  
    t       o        n       i       g       h        t
   20      15       14       9       7       8       20
  
33,13    31,16  22½,8½   27,18 18½,11½  17½,9½  35½,15½

To decrypt the first message, just take the sum of the two numbers in the pair, and to decrypt the second message just take the difference.  This works because if P1 is the first plaintext number and P2 is the second, then the first ciphertext number is

and the second is

Then the plaintext can be recovered from the ciphertext using

and

This system is not as secure as Hill’s because it gives away too much information.  For starters, the existence and nature of the fractions is a clue to the encryption process.  (The editor of the cryptography column suggested doubling the numbers to avoid the fractions, but then the pattern of odd and even numbers would still give information away.)  Also, the first number in each pair is always between 14 and 39 and is always larger than the second number, which is always between ½ and 25 ½.  This suggests that subtraction might be relevant, and the fact that there are twice as many numbers as letters might make a codebreaker suspect the existence of a second message and a second process.  Hill’s system solves some of these issues, but the problem of information leakage continues to be relevant with modern-day ciphers.

With those hints in mind, can you break the cipher used in the following message?

11 3/5, 15 4/5   10 4/5,  9 2/5   17,     11        14 1/5, 16 3/5
 9 4/5,  7 2/5   12 3/5,  7 4/5    9 2/5, 12  1/5   13 1/5, 13 3/5
18,     11       12 2/5, 14 1/5    8 4/5, 10  2/5   12 1/5,  6 3/5
15 4/5, 12 2/5   13 3/5, 13 4/5   12,     16        11 2/5,  8 1/5
 9 1/5, 16 3/5   14,     17       16 3/5, 12  4/5    9 4/5, 14 2/5
12 1/5,  6 3/5   11 3/5, 15 4/5   10,     11        11 4/5,  6 2/5
10 2/5, 14 1/5   17 2/5, 12 1/5   14 3/5,  9  4/5

Once you have the two plaintexts, can you deduce the process used to encrypt them?

 

Answer to Cipher Challenge #1: Merkle’s Puzzles

The hole in the version of Merkle’s puzzles is that the shift we used for encrypting is vulnerable to a known-plaintext attack. That means that if Eve knows the ciphertext and part of the plaintext, she can get the rest of the plaintext. In Cipher Challenge #1, she knew that the word “ten” is part of the plaintext. So she shifts it until she finds a ciphertext that matches one of the puzzles:

ten
UFO
VGP

“Aha!” says Eve. “The first puzzle starts with VGP, so it must decrypt to ten!” Then she decrypts the rest of the puzzle:

VGPVY QUGXG PVYGP VAQPG UKZVG GPUGX GPVGG PBTPU XSNHT JZFEB
whqwz rvhyh qwzhq wbrqh vlawh hqvhy hqwhh qcuqv ytoiu kagfc
xirxa swizi rxair xcsri wmbxi irwiz irxii rdvrw zupjv lbhgd
yjsyb txjaj sybjs ydtsj xncyj jsxja jsyjj sewsx avqkw mcihe
                             ⋮
qbkqt lpbsb kqtbk qvlkb pfuqb bkpbs bkqbb kwokp snico euazw
rclru mqctc lrucl rwmlc qgvrc clqct clrcc lxplq tojdp fvbax
sdmsv nrdud msvdm sxnmd rhwsd dmrdu dmsdd myqmr upkeq gwcby
tentw oseve ntwen tyone sixte ensev entee nzrns vqlfr hxdcz

So the secret key is 2, 7, 21, 16.

The hole can be fixed by using a cipher that is less vulnerable to known-plaintext attacks. Sections 4.4 and 4.5 of The Mathematics of Secrets give examples of ciphers that would be more secure.

Cipher challenge #1 from Joshua Holden: Merkle’s Puzzles

The Mathematics of Secrets by Joshua Holden takes readers on a tour of the mathematics behind cryptography. Most books about cryptography are organized historically, or around how codes and ciphers have been used in government and military intelligence or bank transactions. Holden instead focuses on how mathematical principles underpin the ways that different codes and ciphers operate. Discussing the majority of ancient and modern ciphers currently known, The Mathematics of Secrets sheds light on both code making and code breaking. Over the next few weeks, we’ll be running a series of cipher challenges from Joshua Holden. Presenting the first, on Merkle’s puzzles. 

For over two thousand years, everyone assumed that before Alice and Bob start sending secret messages, they’d need to get together somewhere where an eavesdropper couldn’t overhear them in order to agree on the secret key they would use. In the fall of 1974, Ralph Merkle was an undergraduate at the University of California, Berkeley, and taking a class in computer security. He began wondering if there was a way around the assumption that everyone had always made. Was it possible for Alice to send Bob a message without having them agree on a key beforehand? Systems that do this are now called public-key cryptography, and they are a key ingredient in Internet commerce. Maybe Alice and Bob could agree on a key through some process that the eavesdropper couldn’t understand, even if she could overhear it.

Merkle’s idea, which is now commonly known as Merkle’s puzzles, was slow to be accepted and went through several revisions. Here is the version that was finally published. Alice starts by creating a large number of encrypted messages (the puzzles) and sends them to Bob.

The beginning of Merkle’s puzzles.

Merkle suggested that the encryption should be chosen so that breaking each puzzle by brute force is “tedious, but quite possible.” For our very small example, we will just use a cipher which shifts each letter in the message by a specified number of letters. Here are ten puzzles:

VGPVY QUGXG PVYGP VAQPG UKZVG GPUGX GPVGG PBTPU XSNHT JZFEB
GJBAV ARSVI RFRIR AGRRA GJRYI RFRIR AGRRA VTDHC BMABD QMPUP
AFSPO JOFUF FOUFO TFWFO UXFOU ZGJWF TFWFO UFFOI RCXJQ EHHZF
JIZJI ZNDSO RZIOT ADAOZ ZINZQ ZIOZZ IWOPL KDWJH SEXRJ IKAVV
YBJSY DSNSJ YJJSY BJSYD KNAJX JAJSK TZWXJ AJSYJ JSFNY UZAKM
QCTCL RFPCC RUCLR WDMSP RCCLD GDRCC LQCTC LRCCL JLXUW HAYDT
ADLUA FMVBY ALUVU LVULZ LCLUZ LCLUA LLUGE AMPWB PSEQG IKDSV
JXHUU VYLUJ XHUUJ UDDYD UIULU DJUUD AUTRC SGBOD ALQUS ERDWN
RDUDM SDDMS VDMSX RDUDM SDDMM HMDSD DMRHW SDDMR DUDMS DDMAW
BEMTD MBEMV BGBPZ MMMQO PBMMV AMDMV NQDMA MDMVB MMVUR YCEZC

Alice explains to Bob that each puzzle consists of three sets of numbers. The first number is an ID number to identify the puzzle. The second set of numbers is a secret key from a more secure cipher which Alice and Bob could actually use to communicate. The last number is the same for all puzzles and is a check so that Bob can make sure he has solved the puzzle correctly. Finally, the puzzles are padded with random letters so that they are all the same length, and each puzzle is encrypted by shifting a different number of letters.

Bob picks one of the puzzles at random and solves it by a brute force search. He then sends Alice the ID number encrypted in the puzzle.

Bob solves the puzzle.

For example, if he picked the puzzle on the fifth line above, he might try shifting the letters:

YBJSY DSNSJ YJJSY BJSYD KNAJX JAJSK TZWXJ AJSYJ JSFNY UZAKM
zcktz etotk zkktz cktze lobky kbktl uaxyk bktzk ktgoz vabln
adlua fupul allua dluaf mpclz lclum vbyzl clual luhpa wbcmo
bemvb gvqvm bmmvb emvbg nqdma mdmvn wczam dmvbm mviqb xcdnp

qtbkq vkfkb qbbkq tbkqv cfsbp bsbkc lropb sbkqb bkxfq mrsce
ruclr wlglc rcclr uclrw dgtcq ctcld mspqc tclrc clygr nstdf
svdms xmhmd sddms vdmsx ehudr dudme ntqrd udmsd dmzhs otueg
twent ynine teent wenty fives evenf ourse vente enait puvfh

Now he knows the ID number is “twenty” and the secret key is 19, 25, 7, 4. He sends Alice “twenty”.

Alice has a list of the decrypted puzzles, sorted by ID number:

ID secret key check
zero nineteen ten seven twentyfive seventeen
one one six twenty fifteen seventeen
two nine five seventeen twelve seventeen
three five three ten nine seventeen
seventeen twenty seventeen nineteen sixteen seventeen
twenty nineteen twentyfive seven four seventeen
twentyfour ten one one seven seventeen

So she can also look up the secret key and find that it is 19, 25, 7, 4. Now Alice and Bob both know a secret key to a secure cipher, and they can start sending encrypted messages. (For examples of ciphers they might use, see Sections 1.6, 4.4, and 4.5 of The Mathematics of Secrets.)

Alice and Bob both have the secret key.

Can Eve the eavesdropper figure out the secret key? Let’s see what she has overheard. She has the encryptions of all of the puzzles, and the check number. She doesn’t know which puzzle Bob picked, but she does know that the ID number was “twenty”. And she doesn’t have Alice’s list of decrypted puzzles. It looks like she has to solve all of the puzzles before she can figure out which one Bob picked and get the secret key. This of course is possible, but will take her a lot longer than the procedure took Alice or Bob.

Eve can’t keep up.

Merkle’s puzzles were always a proof of concept — even Merkle knew that they wouldn’t work in practice. Alice and Bob’s advantage over Eve just isn’t large enough. Nevertheless, they had a direct impact on the development of public-key systems that are still very much in use on the Internet, such as the ones in Chapters 7 and 8 of The Mathematics of Secrets.

Actually, the version of Merkle’s puzzles that I’ve given here has a hole in it. The shift cipher has a weakness that lets Eve use Bob’s ID number to figure out which puzzle he solved without solving them herself. Can you use it to find the secret key which goes with ID number “ten”?