Symmetric-Key Cryptography - Codes: The Guide to Secrecy from Ancient to Modern Times - page 108

inverses, greatest common divisors, classes, prime factoring and related ideas,
should digress, at this juncture, to review the material presented for them in
order to ease their introduction to the following topics.
First we introduce some preliminary terminology. Both plaintext and cipher-
text are written symbols from some finite set , called the alphabetofdefinition ,

which may consist of letters from an alphabet such as English, Greek, Hebrew,
or Russian, and may include symbols such as

A

, or any symbols
we choose to use to send messages. The message space, which we will denote
by

✃ , ➫ , , ✆ , ✇

from now on, is a finite set of strings of symbols from the alphabet of
definition, which we will denote by

M

, then m is called
a plaintext message unit . The ciphertext space, which we will denote by

A henceforth. If m ∈ M

C in

what follows, consists of strings of symbols from A for the ciphertext. If c ∈ C ,

then c is called a ciphertext message unit . The symbol, called the keyspace ,

will be used to denote the set of parameters from which we choose our keys for
a given cryptosystem.
On pages 12 and 13, we introduced the notion of a generic cipher and il-
lustrated the process of encryption and decryption. We also gave an informal
verbal description of the enciphering and deciphering transformations. The
reader needing a refresher of these concepts should review those pages before
proceeding to the following formal definition.

K

Ciphers/Cryptosystems
An enciphering transformation (also called an enciphering function or en-
cryption function ) is a bijective function

E e : M → C

where the key e ∈ K uniquely determines E e acting upon plaintext message

units m .A deciphering

transformation (also called a deciphering function or decryption function )is
a bijective function determined by a given key d

∈ M to get ciphertext message units E e ( m )= c ∈ C

∈ K

, acting upon ciphertext

∈ C

message units c to get plaintext message units D d ( c )= m . The application

of E e to m , namely, the operation E e ( m ), is called enciphering m ∈ M . The

application of D d to c is called deciphering c .

A cryptosystem or cipher consists of a set of enciphering transformations

∈ C

{ E e : e ∈ K }

and the corresponding set of deciphering transformations

E − e : e

{ D d : d ∈ K } = { ∈ K } .

such that D d = E − e , In other words, for each e ∈ K , there exists a unique d ∈ K

with

.
The keys ( e,d ) are called a key pair . The pairs of plaintext symbols and their
ciphertext equivalents:

D d ( E e ( m )) = m for all m ∈ M

{ ( m,E e ( m )) = ( m,c ): m ∈ M } is called a cipher table .