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The ciphertext is read from the columns in alphabetical order (as determined by the keyword), so that, in this example, the ciphertext is ROUPSXCTFYIN. Is it possible to conduct a ciphertext-only attack on a keyword columnar transposition cipher It is certainly not as straightforward as attacking a non-keyword columnar cipher. Suppose we obtain the ciphertext
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which we believe was encrypted using a keyword columnar transposition. Our goal is to recover the key and the plaintext. First, note that there are 45 letters in the ciphertext. Assuming the array is not a single column or row, the array could have any of the following dimensions: 9 x 5. 5 x 9. 15 x 3 or 3 x 15. Suppose that we first try a 9 x 5 array. Then we have the ciphertext array in Table 1.1. We focus our attention on the top row of the array in Table 1.1. If we permute the columns as shown in Table 1.2, we see the word GIVE in the first row and we see words or partial words in the other rows. Therefore, we have almost certainly recovered the key. This method is somewhat ad hoc, but the process could be automated, provided we can automatically recognize likely plaintexts. In this example, we have recovered the encryption key 24013 and the plaintext is
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1.4 SELECTED CLASSIC C R Y P T 0 TOPICS
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Table 1.1: Ciphertext Array 0 1 2
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Table 1.2: Permuted Ciphertext Array
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There are many ways to systematically mix the letters of the plaintext. For example, we can strengthen the columnar transposition cipher by allowing the permutation of columns and rows. Since two transpositions are involved, this is known as a double transposition cipher, which we briefly describe next.
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Double Transposition Cipher
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To encrypt with a double transposition cipher, we first write the plaintext into an array of a given size and then permute the rows and columns according to specified permutations. For example, suppose we write the plaintext ATTACKATDAWN into a 3 x 4 array:
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Now if we transpose the rows according to (0,1,2) + (2,1,0) and then transpose the columns according to (0,1,2,3) ( 3 , 1 , 0 , 2 ) , we obtain
The ciphertext is read directly from the final array:
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NADWTKCAATAT.
For the double t,ransposition, the key consists of the size of the matrix and the row and column permutations. The recipient who knows the key can simply put the ciphertext into the appropriate sized matrix and undo the permutations to recover the plaintext. If Trudy happens to know the size of the matrix used in a double transposition, she can insert the ciphertext into a matrix of the appropriate size. She can then try to unscramble the columns to reveal words (or partial words). Once the column transposition has been undone, she can easily unscramble the rows; see Problem 12 for an example. This attack illustrates the fundamental principle of divide and conquer. That is, Trudy can recover the double transposition key in parts, instead of attacking the entire key all at once. There are many exarnples of divide and conquer attacks throughout the remainder of this book. in spite of the inherent divide and conquer attack, the double transposition cipher is relatively strong---at least in comparison to many other classic cipher. The interested reader is directed to [88] for a thorough cryptanalysis of the double transposition.
Substitution Ciphers
Like transposition, substitution is a crucial concept in the design of modern ciphers. in fact, Shannon s [133] two fundamental principles for the design of symmetric ciphers are confusion and diflusion, which, roughly, correspond to the classic concepts of substitution and transposition, respectively. These are still the guiding principles in the design of symmetric ciphers. In this section we discuss several classic substitution ciphers. We highlight, some of the clever techniques that can be brought to bear to attack such ciphers.