Cryptography Reference
In-Depth Information
Chapter 1
Classical Ciphers and Their Cryptanalysis
The object of this chapter is to examine some symmetric encryption schemes that
are of historical interest. From this analysis it will turn out that all of them are
easy to cryptanalyze. Thus their interest is not because they can be used in practice
but, rather, because their cryptanalysis gives us valuable information about some
kinds of attacks that have been tried and that should be prevented when designing an
encryption scheme. This chapter has an introductory nature and in it we use only very
basic mathematical notions, mainly frommodular arithmetic, elementary probability
and statistics, and linear algebra. Modular arithmetic and probability are reviewed in
Chap. 2 in more detail than is required to understand the ciphers presented here, and
we refer to that chapter for further background. At the same time, we use Maple to
implement and to cryptanalyze these schemes and this also serves as an introduction
to the Maple programming environment that we will be using throughout. We start
with one of the most ancient encryption schemes known: the Caesar cipher, also
known as the shift cipher.
1.1 The Caesar Cipher
The Caesar cipher gets its name from Julius Caesar who, according to Suetonius,
used it to communicate with his generals. The Caesar cipher over the 26-letter Eng-
lish alphabet can be described as follows. First, to make the description easier, we
identify the alphabet with
in the standard way (i.e., the letter
'a' corresponds to 0, 'b' corresponds to 1 and so on, until 'z' which corresponds to
25). Then we define:
Z 26 ={
0
,
1
,...,
25
}
M = C = K = Z 26 (hence both the plaintext and the ciphertext space, as well as
the key space, are identified with the alphabet).
Enc
(
k
,
i
) := (
i
+
k
)
mod 26, for k
,
i
∈ Z 26 .
Dec
(
k
,
i
) := (
i
k
)
mod 26, for k
,
i
∈ Z 26 .
 
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