Eighth edition of the N&O column / Spooks newsletter

(Date: Wed, 2 Dec 1998 18:06:07 +0100 MET)

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Transposition ciphers

by Torbjorn Andersson

Table of contents

Introduction

Transposition ciphers are rarely encountered nowadays. They differ from both code systems and substitution ciphers; in a transposition cipher the letters of the cleartext are shifted about to form the cryptogram. This can be done in a number of ways and some systems exist where even whole words are transposed, rather than individual letters. To encrypt Chinese, for instance, one can use a transposition cipher operating on the individual signs of written Chinese (using a substitution cipher for a language like Chinese would be awkward if not impossible).

Single columnar transposition

One of the easiest ways to achieve transposition is the single columnar transposition cipher. To use it one needs a keyword or phrase, whose letters are numbered according to their presence in the alphabet. The keyword 'Osymandias' is numbered in the following way:

O S Y M A N D I A S
7 8 10 5 1 6 3 4 2 9

That is, the first occurrence of the letter A is numbered 1, the second 2. There are no B's or C's so the next letter to be numbered are the D followed by I, and so on.

Next the cleartext are written in rows under the numbered keyword, one letter under each letter of the keyword. Let's say that the cleartext to be encrypted is 'Company have reached primary goal'. It will look like this:

O S Y M A N D I A S
7 8 10 5 1 6 3 4 2 9
c o m p a n y h a v
e r e a c h e d p r
i m a r y g o a l

Now the letters of the cleartext are copied down by reading them off columnwise in the order stated by the enumeration of the keyword and the result is the finished cryptogram, which - of course - are put into groups of five letters, like this:
ACYAP LYEOH DAPAR NHGCE IORMV RMEA

To decrypt a received message enciphered by this method one first must calculate the number of letters present in the cryptogram. This is done to see how many letters there originally were in the last row. As can be seen above, the last column - the one numbered 9 - only contains two letters and this is important. Now the cryptogram above contains 29 letters and as a legitimate user of the crypto system one knows that the keyword is ten letters wide, therefore the last row must consist of nine letters only, the last position being empty. Keeping that in mind - or better still, marking the last position of row three in some way to indicate that it shouldn't be used - one numbers the keyword letters just as when encrypting and then start by writing the first three letters of the cryptogram under keyword letter number one, thus:

O S Y M A N D I A S
7 8 10 5 1 6 3 4 2 9
. . . . a . . . . .
. . . . c . . . . .
. . . . y . . . . *

Continue in the same way by writing the next three letters under keyword letter number two, and so on up to keyword letter eight, it will look like this:

O S Y M A N D I A S
7 8 10 5 1 6 3 4 2 9
c o . p a n y h a .
e r . a c h e d p .
i m . r y g o a l *

 

Now column nine follows and there only two letters should be written as stated above (the position marked by a star being left empty). This leaves three letters of the cryptogram, and these - of course - are written in column ten and then the cleartext can be read in the normal way, row by row.

Usually when employing a transposition cipher like the above, one adds dummy letters to make the final group five letters long if it isn't already full. It is important to do this before transposing the letters, otherwise the receiver can't calculate the columns that haven't a full number of letters if the last row isn't complete. In some cases the last row is always made complete by adding dummy letters, but that reduces the security of the cipher and isn't recommended (now, this cipher is quite easy to break anyhow...).

Double columnar transposition

Double columnar transposition is similar to single columnar transposition, but the process is repeated twice. One either uses the same keyword both times or, preferably, a different one on the second occasion. Let's encrypt the text 'Send armoured car to headquarters' using the keywords 'Agamemnon' and 'Mycenae':

A G A M E M N O N
1 4 2 5 3 6 7 9 8
s e n d a r m o u
r e d c a r t o h
e a d q u a r t e
r s j

(Note dummy letter j added at the end to make the total number of letters a multiple of five)

This first encryption gives:
srer-nddj-aau-eeas-dcq-rra-mtr-uhe-oot.

These letters are written under the second keyword, thus:

M Y C E N A E
5 7 2 3 6 1 4
s r e r n d d
j a a u e e a
s d c q r r a
m t r u h e o
o t

 

And, finally this gives the cryptogram:
DEREE ACRRU QUDAA OSJSM ONERH RADTT

Double columnar transposition is substantially safer against cryptanalysis than single columnar transposition (not impossible, though).

See also the quiz and the exercises in Newsletter 9, or more about polysubstitution ciphers in Newsletter 10 .

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