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9.5. SUMMARY 179
9.5 Summary
Polygraphic ciphers, or block ciphers, work by enciphering groups of letters at
a time, rather than one letter at a time. In particular, digraphic ciphers treat
pairs of letters, or bigrams, as units when enciphering and deciphering. Since
there are 676 pairs of letters, enciphering and deciphering tables for digraphic
ciphers are large and cumbersome, and hence never used (outside of introductory
examples).
Digraphic ciphers are the “monoalphabetic” ciphers for bigrams. That is,
each plaintext bigram is replaced by the same ciphertext bigram throughout
the entire message. Hence frequency analysis may be used to attack a digraphic
cipher. But whereas e and t, the two most common single letters, make up
nearly 20% of all letters, the two most common bigrams, th and in, make up less
than 5% of bigrams. So using frequency analysis only to break digraphic ciphers
is difficult unless one has a very large amount of ciphertext. In particular, a
good digraphic system is much more secure than a good monoalphabetic system.
The Hill Cipher System enciphers by treating the block of letters as a matrix
with only one column and multiplying the column by another matrix. Using a
2 × 2 involution, a matrix with 2 columns and 2 rows that is its own multiplicative
inverse, is most common, but non-involutions may be used, as may 3 × 3,
4 × 4 or larger matrices. Hill Ciphers can be thought of as a digraphic version
of Decimation Ciphers. In particular, once one knows the plaintext equivalents
of two ciphertext bigrams, then one can use some simple mathematics to determine
the enciphering matrix, and hence break the message. The importance
of the Hill Cipher is that it indicates that by the middle of the 20th century
cryptography was almost entirely a mathematical subject.
Playfair Ciphers were invented by Charles Wheatstone and are the most
common digraphic ciphers, being used in both World Wars. After setting up a
rectangular array of letters, enciphering proceeds by replacing pairs of letters
with the pair to the right if they are in the same column, underneath if they are
in the same row, or the opposite corners of the rectangle produced otherwise.
The strengths of Playfair are its ease of use, its rapidity, and its use of a keyword.
Conversely, each letter can be replaced by only 6 others, and this allows entries
into the system.
9.6 Topics and Techniques
1. What is a digraphic cipher How does it differ from a mono-graphic
cipher
2. What does the use of “di” in digraphic refer to
3. Is a digraphic cipher monoalphabetic or polyalphabetic Explain.
4. What is the general appearance of a digraphic cipher That is, explain