Cryptography Reference
In-Depth Information
Example 1.12.
Let the key be
k
=(
a
,
b
)=(9
,
13), and the plaintext be
ATTACK
=
x
1
,
x
2
,...,
x
6
= 0
,
19
,
19
,
0
,
2
,
10
.
The inverse
a
−
1
of
a
exists and is given by
a
−
1
= 3. The ciphertext is computed as
y
1
,
y
2
,...,
y
6
= 13
,
2
,
2
,
13
,
5
,
25 =
nccnfz
Is the affine cipher secure? No! The key space is only a bit larger than in the case
of the shift cipher:
key space =(#values for
a
)
×
(#values for
b
)
= 12
×
26 = 312
A key space with 312 elements can, of course, still be searched exhaustively, i.e.,
brute-force attacked, in a fraction of a second with current desktop PCs. In addition,
the affine cipher has the same weakness as the shift and substitution cipher: The
mapping between plaintext letters and ciphertext letters is fixed. Hence, it can easily
be broken with letter frequency analysis.
The remainder of this topic deals with strong cryptographic algorithms which are
of practical relevance.
1.5 Discussion and Further Reading
This topic addresses practical aspects of cryptography and data security and is in-
tended to be used as an introduction; it is suited for classroom use, distance learning
and self-study. At the end of each chapter, we provide a discussion section in which
we briefly describe topics for readers interested in further study of the material.
About This Chapter: Historical Ciphers and Modular Arithmetic
This chapter
introduced a few historical ciphers. However, there are many, many more, ranging
from ciphers in ancient times to WW II encryption methods. To readers who wish to
learn more about historical ciphers and the role they played over the centuries, the
topics by Bauer [13], Kahn [97] and Singh [157] are highly recommended. Besides
making fascinating bedtime reading, these topics help one to understand the role
that military and diplomatic intelligence played in shaping world history. They also
help to show modern cryptography in a larger context.
The mathematics introduced in this chapter, modular arithmetic, belongs to the
field of number theory. This is a fascinating subject area which is, unfortunately,
historically viewed as a “branch of mathematics without applications”. Thus, it is
rarely taught outside mathematics curricula. There is a wealth of topics on number
theory. Among the classic introductory topics are references [129, 148]. A particu-
larly accessible topic written for non-mathematications is [156].
