Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

and the keyword letter is (), then the cipher text letter corresponds to . However, 28 is larger than 26, so we subtract 26 to get 2, which is . Thus, enciphers to .

Keyword:

E

N

C

O

D

E

E

N

C

O

D

E

E

N

C

Message:

t

h

e

s

k

y

i

s

f

a

l

l

i

n

g

Cipher text:

X

U

G

G

N

C

M

F

H

O

O

P

M

A

I

Notice that each letter of the cipher text is shifted by varying amounts, unlike the Caesar cipher. Interpreting this as many monoalphabetic ciphers acting on individual characters, it is easy to see why the Vigenère cipher is referred to as a polyalphabetic cipher.

      To decipher, simply subtract the value of the respective key letters from each cipher text letter. For example, to decrypt the cipher text “XUGGNCMFHOOPMAI” from above, we use the keyword “ENCODE” as follows:

      Repeat the process until all plain text characters have been determined. This enciphering process is easy if one has knowledge of the key. However, it can be difficult to break the cipher without such information. This will be investigated in Section 2.6.

      The Vigenère cipher is known as a type of block cipher, where the block length is equivalent to the key length. In a block cipher procedure, the plain text or message is encrypted block by block, rather than character by character.

      There are a few important remarks to be made regarding the use of the Vigenère cipher. First, because the same plain text character enciphers to different characters depending on the position, the cryptanalysis of such a cipher is much more complex than for a simple monoalphabetic substitution cipher. We also point out that the Vigenère cipher was invented to hide the frequencies of letters in the English language. The Caesar cipher, for example, does not do this. Also, if the key phrase “VIGENERECIPHERX” had been used instead of “ENCODE,” in our previous example, the encrypted message would have had perfect secrecy. Perfect secrecy is achieved if the cipher text yields no information about the plain text, and this occurs, roughly speaking, when the keyword is as long as the message itself. Such a secure system can be obtained using one‐time pads, which we investigate later in the book.

2.5 Frequency Analysis

The idea behind the use of frequency analysis in cryptanalysis is that all human languages have underlying statistical patterns and redundancies that can be exploited to help break a variety of ciphers. For the English language, it is well documented that the distribution of the most frequent characters is remarkably similar throughout texts of diverse style and length, as indicated in Table 2.1.

Table 2.1 Approximate frequencies of letters in the English language.

Letter	Frequency (%)	Letter	Frequency (%)
a	7.44	b	1.46
c	2.52	d	3.53
e	12.22	f	2.68
g	1.84	h	5.97
i	6.82	j	0.20
k	0.65	l	4.28
m	2.71	n	6.32
o	8.25	p	1.97
q	0.12	r	6.21
s	6.99	t	9.85
u	3.67	v	0.12
w	2.09	x	0.18
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