by Simon Singh
(6) On some occasions the commonest letter in the ciphertext might be E, the next commonest could be T, and so on. In other words, the frequency of letters in the ciphertext already matches those in the frequency table. The E in the ciphertext appears to be a genuine e, and the same seems to be true for all the other letters, yet the ciphertext looks like gibberish. In this case you are faced not with a substitution cipher, but with a transposition cipher. All the letters do represent themselves, but they are in the wrong positions.
Cryptanalysis by Helen Fouché Gaines (Dover) is a good introductory text. As well as giving tips, it also contains tables of letter frequencies in different languages, and provides lists of the most common words in English.
Appendix C
The So-called Bible Code
In 1997 The Bible Code by Michael Drosnin caused headlines around the world. Drosnin claimed that the Bible contains hidden messages which could be discovered by searching for equidistant letter sequences (EDLSs). An EDLS is found by taking any text, picking a particular starting letter, then jumping forward a set number of letters at a time. So, for example, with this paragraph we could start with the “M” in Michael and jump, say, five spaces at a time. If we noted every fifth letter, we would generate the EDLS mesahirt.…
Although this particular EDLS does not contain any sensible words, Drosnin described the discovery of an astonishing number of Biblical EDLSs that not only form sensible words, but result in complete sentences. According to Drosnin, these sentences are biblical predictions. For example, he claims to have found references to the assassinations of John F. Kennedy, Robert Kennedy and Anwar Sadat. In one EDLS the name of Newton is mentioned next to gravity, and in another Edison is linked with the lightbulb. Although Drosnin’s book is based on a paper published by Doron Witzum, Eliyahu Rips and Yoav Rosenberg, it is far more ambitious in its claims, and has attracted a great deal of criticism. The main cause of concern is that the text being studied is enormous: in a large enough text, it is hardly surprising that by varying both the starting place and the size of the jump, sensible phrases can be made to appear.
Brendan McKay at the Australian National University tried to demonstrate the weakness of Drosnin’s approach by searching for EDLSs in Moby Dick, and discovered thirteen statements pertaining to assassinations of famous people, including Trotsky, Gandhi and Robert Kennedy. Furthermore, Hebrew texts are bound to be particularly rich in EDLSs, because they are largely devoid of vowels. This means that interpreters can insert vowels as they see fit, which makes it easier to extract predictions.
Appendix D
The Pigpen Cipher
The monoalphabetic substitution cipher persisted through the centuries in various forms. For example, the pigpen cipher was used by Freemasons in the 1700s to keep their records private, and is still used today by schoolchildren. The cipher does not substitute one letter for another, rather it substitutes each letter for a symbol according to the following pattern.
To encrypt a particular letter, find its position in one of the four grids, then sketch that portion of the grid to represent that letter. Hence:
If you know the key, then the pigpen cipher is easy to decipher. If not, then it is easily broken by:
Appendix E
The Playfair Cipher
The Playfair cipher was popularized by Lyon Playfair, first Baron Playfair of St. Andrews, but it was invented by Sir Charles Wheatstone, one of the pioneers of the electric telegraph. The two men lived close to each other, either side of Hammersmith Bridge, and they often met to discuss their ideas on cryptography.
The cipher replaces each pair of letters in the plaintext with another pair of letters. In order to encrypt and transmit a message, the sender and receiver must first agree on a keyword. For example, we can use Wheatstone’s own name, CHARLES, as a keyword. Next, before encryption, the letters of the alphabet are written in a 5 × 5 square, beginning with the keyword, and combining the letters I and J into a single element:
Next, the message is broken up into pairs of letters, or digraphs. The two letters in any digraph should be different, achieved in the following example by inserting an extra x between the double m in hammersmith, and an extra x is added at the end to make a digraph from the single final letter:
Encryption can now begin. All the digraphs fall into one of three categories—both letters are in the same row, or the same column, or neither. If both letters are in the same row, then they are replaced by the letter to the immediate right of each one; thus mi becomes NK. If one of the letters is at the end of the row, it is replaced by the letter at the beginning; thus ni becomes GK. If both letters are in the same column, they are replaced by the letter immediately beneath each one; thus ge becomes OG. If one of the letters is at the bottom of the column, then it is replaced by the letter at the top; thus ve becomes CG.
If the letters of the digraph are neither in the same row nor the same column, the encipherer follows a different rule. To encipher the first letter, look along its row until you reach the column containing the second letter; the letter at this intersection then replaces the first letter. To encipher the second letter, look along its row until you reach the column containing the first letter; the letter at this intersection replaces the second letter. Hence, me becomes GD, and et becomes DO. The complete encryption is:
The recipient, who also knows the keyword, can easily decipher the ciphertext by simply reversing the process: for example, enciphered letters in the same row are deciphered by replacing them by the letters to their left.
As well as being a scientist, Playfair was also a notable public figure (Deputy Speaker of the House of Commons, postmaster general, and a commissioner on public health who helped to develop the modern basis of sanitation) and he was determined to promote Wheatstone’s idea among the most senior politicians. He first mentioned it at a dinner in 1854 in front of Prince Albert and the future Prime Minister, Lord Palmerston, and later he introduced Wheatstone to the Under Secretary of the Foreign Office. Unfortunately, the Under Secretary complained that the system was too complicated for use in battle conditions, whereupon Wheatstone stated that he could teach the method to boys from the nearest elementary school in 15 minutes. “That is very possible,” replied the Under Secretary, “but you could never teach it to attachés.”
Playfair persisted, and eventually the British War Office secretly adopted the technique, probably using it first in the Boer War. Although it proved effective for a while, the Playfair cipher was far from impregnable. It can be attacked by looking for the most frequently occurring digraphs in the ciphertext, and assuming that they represent the commonest digraphs in English: th, he, an, in, er, re, es.
Appendix F
The ADFGVX Cipher
The ADFGVX cipher features both substitution and transposition. Encryption begins by drawing up a 6 × 6 grid, and filling the 36 squares with a random arrangement of the 26 letters and the 10 digits. Each row and column of the grid is identified by one of the six letters A, D, F, G, V or X. The arrangement of the elements in the grid acts as part of the key, so the receiver needs to know the details of the grid in order to decipher messages.
The first stage of encryption is to take each letter of the message, locate its position in the grid and substitute it with the letters that label its row and column. For example, 8 would be substituted by AA, and p would be replaced by AD. Here is a short message encrypted according to this system:
So far this is a simple monoalphabetic substitution cipher, and frequency analysis would be enough to crack it. However, the second stage of the ADFGVX is a transposition, which makes cryptanalysis much harder. The transposition depends on a keyword, which in this case happens to be the word MARK, and which must be shared with the receiver. Transposition is carried out according to the following recipe. First, the letters of the keyword are written in the top row of a fresh grid. Next, the stage 1 ciphertext is written underneath it in a series of rows, as shown below. The columns of the grid a
re then rearranged so that the letters of the keyword are in alphabetical order. The final ciphertext is achieved by going down each column and then writing out the letters in this new order.
The final ciphertext would then be transmitted in Morse code, and the receiver would reverse the encryption process in order to retrieve the original text. The entire ciphertext is made up of just six letters (i.e. A, D, F, G, V, X), because these are the labels of the rows and columns of the initial 6 × 6 grid. People often wonder why these letters were chosen as labels, as opposed to, say, A, B, C, D, E and F. The answer is that A, D, F, G, V and X are highly dissimilar from one another when translated into Morse dots and dashes, so this choice of letters minimizes the risk of errors during transmission.
Appendix G
The Weaknesses of Recycling a Onetime Pad
For the reasons explained in Chapter 3, ciphertexts encrypted according to a onetime pad cipher are unbreakable. However, this relies on each onetime pad being used once and only once. If we were to intercept two distinct ciphertexts which have been encrypted with the same onetime pad, we could decipher them in the following way.
We would probably be correct in assuming that the first ciphertext contains the word the somewhere, and so cryptanalysis begins by assuming that the entire message consists of a series of the’s. Next, we work out the onetime pad that would be required to turn a whole series of the’s into the first ciphertext. This becomes our first guess at the onetime pad. How do we know which parts of this onetime pad are correct?
We can apply our first guess at the onetime pad to the second ciphertext, and see if the resulting plaintext makes any sense. If we are lucky, we will be able to discern a few fragments of words in the second plaintext, indicating that the corresponding parts of the onetime pad are correct. This in turn shows us which parts of the first message should be the.
By expanding the fragments we have found in the second plaintext, we can work out more of the onetime pad, and then deduce new fragments in the first plaintext. By expanding these fragments in the first plaintext, we can work out more about the onetime pad, and then deduce new fragments in the second plaintext. We can continue this process until we have deciphered both plaintexts.
This process is very similar to the decipherment of a message enciphered with a Vigenère cipher using a key that consists of a series of words, such as the example in Chapter 3, in which the key was CANADABRAZILEGYPTCUBA.
Appendix H
The Daily Telegraph Crossword Solution
ACROSS DOWN
1. Troupe 1. Tipstaff
4. Short Cut 2. Olive oil
9. Privet 3. Pseudonym
10. Aromatic 5. Horde
12. Trend 6. Remit
13. Great deal 7. Cutter
15. Owe 8. Tackle
16. Feign 11. Agenda
17. Newark 14. Ada
22. Impale 18. Wreath
24. Guise 19. Right nail
27. Ash 20. Tinkling
28. Centre bit 21. Sennight
31. Token 23. Pie
32. Lame dogs 25. Scales
33. Racing 26. Enamel
34. Silencer 29. Rodin
35. Alight 30. Bogie
Appendix I
Exercises for the Interested Reader
Some of the greatest decipherments in history have been achieved by amateurs. For example, Georg Grotefend, who made the first breakthrough in interpreting cuneiform, was a schoolteacher. For those readers who feel the urge to follow in his footsteps, there are several scripts that remain a mystery. Linear A, a Minoan script, has defied all attempts at decipherment, partly due to a paucity of material. Etruscan does not suffer from this problem, with over 10,000 inscriptions available for study, but it has also baffled the world’s greatest scholars. Iberian, another pre-Roman script, is equally unfathomable.
The most intriguing ancient European script appears on the unique Phaistos Disk, discovered in southern Crete in 1908. It is a circular tablet dating from around 1700 B.C. bearing writing in the form of two spirals, one on each side. The signs are not handmade impressions, but were made using a variety of stamps, making this the world’s oldest example of typewriting. Remarkably, no other similar document has ever been found, so decipherment relies on very limited information-there are 242 characters divided into 61 groups. However, a typewritten document implies mass production, so the hope is that archaeologists will eventually discover a hoard of similar disks, and shed light on this intractable script.
One of the great challenges outside Europe is the decipherment of the Bronze Age script of the Indus civilization, which can be found on thousands of seals dating from the third millennium B.C. Each seal depicts an animal accompanied by a short inscription, but the meaning of these inscriptions has so far evaded all the experts. In one exceptional example the script has been found on a large wooden board with giant letters 37 cm in height. This could be the world’s oldest billboard. It implies that literacy was not restricted to the elite, and raises the question as to what was being advertised. The most likely answer is that it was part of a promotional campaign for the king, and if the identity of the king can be established, then the billboard could provide a way into the rest of the script.
Appendix J
The Mathematics of RSA
What follows is a straightforward mathematical description of the mechanics of RSA encryption and decryption.
(1) Alice picks two giant prime numbers, p and q. The primes should be enormous, but for simplicity we assume that Alice chooses p = 17, q = 11. She must keep these numbers secret.
(2) Alice multiplies them together to get another number, N. In this case N = 187. She now picks another number e, and in this case she chooses e = 7.
(e and (p − 1) × (q − 1) should be relatively prime, but this is a technicality.)
(3) Alice can now publish e and N in something akin to a telephone directory. Since these two numbers are necessary for encryption, they must be available to anybody who might want to encrypt a message to Alice. Together these numbers are called the public key. (As well as being part of Alice’s public key, e could also be part of everybody else’s public key. However, everybody must have a different value of N, which depends on their choice of p and q.)
(4) To encrypt a message, the message must first be converted into a number, M. For example, a word is changed into ASCII binary digits, and the binary digits can be considered as a decimal number. M is then encrypted to give the ciphertext, C, according to the formula
C = Me (mod N)
(5) Imagine that Bob wants to send Alice a simple kiss: just the letter X. In ASCII this is represented by 1011000, which is equivalent to 88 in decimal. So, M = 88.
(6) To encrypt this message, Bob begins by looking up Alice’s public key, and discovers that N = 187 and e = 7. This provides him with the encryption formula required to encrypt messages to Alice. With M = 88, the formula gives
C = 887 (mod 187)
(7) Working this out directly on a calculator is not straightforward, because the display cannot cope with such large numbers. However, there is a neat trick for calculating exponentials in modular arithmetic. We know that, since 7 = 4 + 2 + 1,
887 (mod 187) = [884 (mod 187) × 882 (mod 187) × 881 (mod 187)] (mod 187)
881 = 88 = 88 (mod 187)
882 = 7,744 = 77 (mod 187)
884 = 59,969,536 = 132 (mod 187)
887 = 881 × 882 × 884 = 88 × 77 × 132 = 894,432 = 11 (mod 187)
Bob now sends the ciphertext, C = 11, to Alice.
(8) We know that exponentials in modular arithmetic are one-way functions, so it is very difficult to work backward from C = 11 and recover the original message, M. Hence, Eve cannot decipher the message.
(9) However, Alice can decipher the message because she has some special information: she knows the values of p and q. She calculates a special number, d, the decryption key, otherwise known as her private key. The number d is calculated according to the following formu
la
e × d = 1 (mod (p-1) × (q-1))
7 × d = 1 (mod 16 × 10)
7 × d = 1 (mod 160)
d = 23
(Deducing the value of d is not straightforward, but a technique known as Euclid’s algorithm allows Alice to find d quickly and easily.)
(10) To decrypt the message, Alice simply uses the following formula,
M = Cd (mod 187)
M = 1123 (mod 187)
M = [111 (mod 187) × 112 (mod 187) × 114 (mod 187) × 1116 (mod 187)] (mod 187)
M = 11 × 121 × 55 × 154 (mod 187)
M = 88 = X in ASCII.
Rivest, Shamir and Adleman had created a special one-way function, one that could be reversed only by somebody with access to privileged information, namely the values of p and q. Each function can be personalized by choosing p and q, which multiply together to give N. The function allows everybody to encrypt messages to a particular person by using that person’s choice of N, but only the intended recipient can decrypt the message because the recipient is the only person who knows p and q, and hence the only person who knows the decryption key, d.