cryptography

cryptography

Cryptography

is the science of

encoding

and decoding messages so that they are only understood by intended senders and recipients.

Informally, the basic ingredients in cryptography consist of

1.

An

alphabet

Σ

. A word over

Σ

is called a

plaintext

or

cleartext

. The set

M

:=

Σ

*

of all plaintexts is called the

message space

.

2.

An alphabet

Δ

. A word over

Δ

is called a

ciphertext

or

cryptotext

. The set

C

:=

Δ

*

of all ciphertexts is called the

ciphertext space

.

3.

A set

K

called the key space, such that each

k

∈

K

, called a

key

, determines

functions

e

k

:

M

→

C

and

d

k

:

C

→

M

called

encryption function

and

decryption function

respectively. It is generally required that

d

k

∘

e

k

=

1

M

(and usually

e

k

∘

d

k

=

1

C

so that

e

k

and

d

k

are

inverses

of one another). The pair

(

e

k

,

d

k

)

is usually called a

key pair

.

The triple

(

M

,

C

,

K

)

is called a

cryptosystem

.

Given a cryptosystem, one generally wants both

e

k

and

d

k

to be easily

computable

, so that encryption by the sender and decryption by the intended receiver can be done effortlessly. For example, if

A

were to send

B

an encrypted message,

A

needs to be able to

easily

encrypt a plaintext

u

into a ciphertext

v

=

e

k

⁢

(

u

)

. Upon receiving

v

, the intended receiver

B

needs to be able to

easily

decrypt

v

into plaintext

u

=

d

k

⁢

(

v

)

. On the other hand, one also wants the task of recovering the plaintext from the ciphertext very difficult, or intractible, without knowing the decryption functions

d

k

. For example, if an eavesdropper

C

ever gets hold of

v

, he/she will have a very hard time recovering

u

without knowing

d

k

, sometimes even in the presence of knowing

e

k

.

Here, the adjectives “easy”, “difficult”, “intractible” are measured in terms of the

complexity

involved in the

computations

. For example, “easy” could mean that the time involved in the computations depends linearly on the length of the input

u

(hopefully with a small coefficient), whereas “difficult” could mean the dependence be exponential instead. A function

f

such that it is easy to compute

f

⁢

(

u

)

given

u

but very hard to find

u

given

f

⁢

(

u

)

is usually called a

one-way function

. For example,

f

⁢

(

m

,

n

)

=

m

⁢

n

where

m

,

n

are primes, is one-way, or nearly so. It is easy to multiply, but very hard to factor, particularly when both

m

and

n

are large.

The study of cryptogrpahy thus can be loosely broken up into two main branches: the construction of a good cryptosystem (meaning that the encryption functions should be one-way, and the decryption functions should be easy to compute), and the breaking of a ciphertext in some given setting (for example, where the encryption functions are known). The latter of the two branches is also known as

cryptanalysis

.

The mathematics behind cryptography involves a

variety

of disciplines, of which the main ones are

information theory

,

formal languages

, computational complexity, probability theory, and

number theory

.

Cryptographic methods range from simple

additive

ciphers to sophisticated public key encryption systems, first introduced by Diffie and Hellman in the mid 1970’s.

Title

cryptography

Canonical

name

Cryptography

Date of creation

2013-03-22 15:49:07

Last modified on

2013-03-22 15:49:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 94A60

Synonym

cleartext

Synonym

cyphertext

Synonym

cryptotext

Defines

plaintext

Defines

ciphertext

Defines

message space

Defines

cyphertext space

Defines

key space

Defines

key

Defines

key pair

Defines

encryption function

Defines

decryption function

Defines

cryptosystem

Defines

cryptanalysis

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