Definition 1.1: A cryptosystem is a five-tuple (MATH) where the following conditions are satisfied:

1. $\QTR{bs}{P}$ is a finite set of possible plaintexts:

2. $\QTR{bs}{C}$ is a finite set of possible ciphertexts:

3. $\QTR{bs}{K}$ the keyspace, is the a finite set of possible keys:

4. For each KMATH there is and encryption rule e MATH and a corresponding decryption rule dMATH. Where

e MATH and d MATH are functions such that for all MATH ,d$_{\QTR{bs}{K}}$(e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) $\fallingdotseq $ $\QTR{bf}{x}$.

For the purposes of these examples, we use a-z and 0-25 interchangeably.

Examples:

1. The Shift Cipher : MATH and MATH ,

e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) $\fallingdotseq $ MATH

and

d$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) $\fallingdotseq $ MATH

Observations (not as trivial as they seem) :

1. If I knew it was that a message was coded with a shift cipher and I knew one example (e.g. e$_{\QTR{bs}{K}}$($\QTR{bf}{b}$)MATH) then I

could decode any message. (e.g. $\QTR{bf}{k=3}$)

2. If I knew the rule

e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) $\fallingdotseq $ MATH

I could deduce the rule

d$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) $\fallingdotseq $ MATH

2. The Substitution Cipher : MATH and $\QTR{bs}{K}$ is the set of all possible permutations, $\QTR{bf}{\pi }$,of the alphabet ,

For example (a typewriter cipher)

e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$)) is computed from the following table

a b c d e f g h i j k l m n o p q r s t u v w x y z
m l p n k o b j i v h u c g y x f t z d r s e a w q

Observations :

1. If I knew it was that a message was coded with a substitution cipher and I knew 25 of the substitutions then I

could deduce the 26th and decode any message. Can I do better? (To be continued in our discussion

of Information Theory)

2. If I knew the rule

e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$))

I could deduce the rule

d$_{\QTR{bs}{K}}$($\QTR{bf}{x}$))

with a bit of work.

3. A Block Substitution Cipher : MATH and $\QTR{bs}{K}$ is the set of all possible permutations, $\QTR{bf}{\pi }$,of pairs of letters of the alphabet ,

e$_{\QTR{bs}{K}}$($\QTR{bf}{(d,y)}$)) $=\QTR{bf}{(w,a)}$ - Break the message up into pairs of letters and map each pair. (maybe you have to pad the message string with an

addition letter.

Observations :

1. I would have to know a lot more then in 2. above.

2. If I knew the rule

e$_{\QTR{bs}{K}}$($\QTR{bf}{x}$))

I could deduce the rule

d$_{\QTR{bs}{K}}$($\QTR{bf}{x}$))

with a lot of work. A computer would help.

4. A Binary Cipher of Fixed Length : MATH ,MATH for some $\QTR{bf}{n}$ , and $\QTR{bs}{K}$ is the set of injective maps (1 to 1) of the alphabet to

the set of binary sequences of length $\QTR{bf}{n}$.

Observations :

1. ASCII is a Binary Cipher of length 8.

5. A Binary Cipher of Variable Length : MATH ,MATH for some $\QTR{bf}{n}$ , and $\QTR{bs}{K}$ is the set of injective maps (1 to 1) of the alphabet to

the set of binary sequences of length MATH.

Observations :

1. Morse code is a binary cipher of length less than or equal to 6.

Morse Code Alphabet

The International morse code characters are:

A   .-
B   -...
C   -.-.
D   -..
E   .
F   ..-.
G   --.
H   ....
I   ..
J   .---
K   -.-
L   .-..
M   --
N   -.
O   ---
P   .--.
Q   --.-
R   .-.
S   ...
T   -
U   ..-
V   ...-
W   .--
X   -..-
Y   -.--
Z   --..
0   -----
1   .----
2   ..---
3   ...--
4   ....-
5   .....
6   -....
7   --...
8   ---..
9   ----.
Fullstop   .-.-.-
Comma   --..--
Query   ..--..
see http://www.babbage.demon.co.uk/morseabc.html

2. Latter we will consider compression algorithms that create variable length cyphers. When we discuss Information Theory we will we will show the existance of "best possible" codes from the point of view of transmission times.

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