The Playfair Cipher

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The Playfair Cipher

The Playfair Cipher was invented by Sir Charles Wheatstone and Baron Lyon Playfair in 1854 and was used by the British War Office up to the beginning of the 20th century, including use in the Boer War. It is an example of a 'Bigram' Cipher. This means that letters are encrypted in pairs, as opposed to individually. The key is a 5 by 5 square (with entries comprised of the 25 letters obtained by deleting J from the alphabet) and thus there are 25! or

15,511,210,043,330,985,984,000,000

keys. Before encrypting using a Playfair Cipher the message has to be rearranged slightly. To do this you:

●replace Js with Is;

●write message in pairs of letters;

●do not allow identical pairs - if they occur insert Z between them;

●if the number of letters is odd, add Z to the end.

In order to illustrate how the cipher works we choose a specific key. However, there is nothing special about our choice.

Once the message has been suitably rearranged we give the rule for encryption. In order to clarify our description we extend the key by adding a sixth column and a sixth row to the original key. The sixth row is identical to the first row, while the sixth column is identical to the first column. Thus, for our example, the extended key can be set out as in the diagram.

The rule for encryption is as follows.

●If the two letters lie in the same row of the key then each letter is replaced by the one on its right in the extended key.

●If two letters lie in the same column of the key then each letter is replaced by the one below it in the extended key.

●If the two letters are not in the same row or column then the first letter is replaced by the letter that is in the row of the first letter and the column of the second letter. The second letter is replaced by the fourth corner of the rectangle formed by the three letters used so far.

We now encrypt the message: GOOD BROOMS SWEEP CLEAN

Since there are no Js in the message we have only to write the message in pairs of letters with the appropriate insertion of extra Zs. This

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The Playfair Cipher

gives:

GO OD BR OZ OM SZ SW EZ EP CL EA NZ

Thus, for our chosen key, GO becomes FP; OD becomes UT; OM becomes PO. The complete cryptogram is

FP UT EC UW PO DV TV BV CM BG CS DY

As with Simple Substitution Ciphers, users tended to use a key phrase to determine the key matrix. The method was the same as that used for Simple Substitution Ciphers, that is, write out the key phrase, remove repeated letters and then add the unused letters in alphabetical order. So, if the key phrase is UNIVERSITY OF LONDON then removing repeated letters gives UNIVERSTYOFLD and the square can be set out as in the next diagram.

Decryption is, as always, merely the reverse process to encryption. Any reader wishing to ensure they understand how the Playfair Cipher works should decrypt MBOUBTZE using the square below as key. (The answer is a seven-letter English word that, we hope, does not reflect the reader's state of mind.) We do not intend to discuss the cryptanalysis of this cipher. There are many other examples of ciphers that are easy to describe and fun to play with. Suitable references are provided at the end of this book.

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Homophonic Coding

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Homophonic Coding

Another option for improving on the Simple Substitution Cipher might be to expand the alphabet by introducing some extra characters so that, for instance, the plaintext letter E is represented by more than one ciphertext character.

These extra characters are known as randomizing elements and the process of expanding the alphabet is called homophonic coding. As an illustration we might introduce a cipher where the alphabet for the cryptogram would be the numbers 00, 01, 02,..., 31. Each cryptogram number represents a unique plaintext letter but the letters A, E, N, O, R, T are each represented by two distinct letters.

As an illustration we might assign numbers to letters as in the diagram.

If we do this then the word TEETH, which has two pairs of repeated letters, might be written as 24 27 13 08 31. To anyone who does not know the key, the five characters of the cryptogram are different but the genuine receiver is in no danger of confusion.

The six letters chosen are likely to be the six most popular letters in the plaintext. If, for instance, the decision about which of the two chosen numbers represents a particular incidence of the letter E is random, then we would expect each of the two numbers to 'occupy' about 6 per cent of the cryptogram. In general the effect of homophonic coding is to ensure that the expected frequency histogram of the cryptogram is flatter than that of the plaintext. This makes the statistical attack more difficult.

Note 1: In this cipher we write 00, 01, 02 for 0, 1, 2 etc. Whenever spaces are not used this type of notation is needed to distinguish between, for example, 'twelve' and 'one followed by two'.

Note 2: Breaking ordinary Simple Substitution Ciphers is relatively easy and we hope all readers managed to break the paragraph-long cryptogram above. Breaking the type of cipher we are now discussing requires considerably more patience and luck. Anyone who needs convincing or enjoys this type of puzzle should try to read the following cryptogram. The only information is that English text has been encrypted using a Simple Substitution Cipher with homophonic coding as discussed above. The key is unknown, but is not the one listed above. Furthermore the letters have been written in groups of five. (This means the attacker cannot identify short words, notably those of one letter.) It is not easy and readers should not feel obliged to try.

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