Bài giảng Mật mã học: Tổng quan về mật mã học - Huỳnh Trọng Thưa

Bài giảng Mật mã học: Tổng quan về mật mã học cung cấp cho người học các kiến thức: Introduction, information security and cryptography, cryptographic goals, one-way functions, basic terminology and concepts,... Mời các bạn cùng tham khảo.

Tổng quan về mật mã học

Huỳnh Trọng Thưa htthua@ptithcm.edu.vn

• Cryptography was used as a

tool to protect national

secrets and strategies.

• 1960s (computers and

communications systems) ->

means to protect information

and to provide security

services.

Information security and

Information security and

Information security and

cryptography (cont.)

• Cryptography is the study of mathematical

techniques related to aspects of information security such as confidentiality, data integrity, entity authentication, and data origin

authentication.

• Cryptography is not the only means of

providing information security, but rather one set of techniques.

A taxonomy of

cryptographic

primitives

1-1 functions

• A function is 1 − 1 (injection - đơn ánh) if each

element in Y is the image of at most one

element in X

• A function is onto (toàn ánh) if each element

in Y is the image of at least one element in X,

i.e Im(f)=Y

• If a function f: X → Y is 1−1 and Im(f)=Y, then f

is called a bijection (song ánh).

Inverse function

• f:XY and g:YX; g(y)=x where f(x)=y

• g obtained from f, called the inverse function of f, g = f −1

• Ex: Let X = {a, b, c, d, e}, and Y = {1, 2, 3, 4, 5}

One-way functions

• A function f from a set X to a set Y is called a

one-way function if f(x) is “easy” to compute

for all x ∈ X but for “essentially all” elements y

∈ Im(f) it is “computationally infeasible” to

find any x ∈ X such that f(x)= y.

– Ex: X = {1, 2, 3, , 16}, f(x)= rx for all x ∈ X where rx

is the remainder when 3x is divided by 17

(Hoán vị)

• Let S be a finite set of elements.

– A permutation p on S is a bijection from S to itself (i.e., p: S→S)

• Ex: S = {1, 2, 3, 4, 5} A permutation p: S→S is defined as follows:

p(1) = 3,p(2) = 5,p(3) = 4,p(4) = 2,p(5) = 1.

Basic terminology and concepts

• M denotes a set called the message space

– An element of M is called a plaintext message

– Ex: M may consist of binary strings, English text, computer code, etc.

• C denotes a set called the ciphertext space

– C consists of strings of symbols from an alphabet of

definition, which may differ from the alphabet of definition for M.

– An element of C is called a ciphertext.

Encrypt and decrypt transformations

• For each d ∈ K, D d denotes a bijection from C to M

(i.e., Dd : C→M) D d is called a decryption function or decryption transformation

Encrypt and decrypt transformations

(cont.)

• Ee : e ∈ K ; Dd : d ∈ K

– for each e ∈ K there is a unique key d ∈ K such that Dd = Ee-1 ;

– that is, Dd(Ee(m)) = m for all m ∈ M

• The keys e and d in the preceding definition are referred to as a

key pair and some times denoted by (e, d).

• To construct an encryption scheme requires one to select

– a message space M,

– a ciphertext space C,

Ex of encryption scheme

• Let M = {m1,m2,m3} and C = {c1,c2,c3}

– There are precisely 3! = 6 bijections from M to C.

– The key space K = {1, 2, 3, 4, 5, 6} has six elements in it,

each specifying one of the transformations.

Communication participants

• A channel is a means of conveying information from

one entity to another

• An unsecured channel is one from which parties

other than those for which the information is

intended can reorder, delete, insert, or read

• A secured channel is one from which an adversary

does not have the ability to reorder, delete, insert, or read

• A fundamental premise in cryptography is that the sets M, C,K, {Ee : e ∈ K}, {Dd : d ∈ K} are

public knowledge.

• When two parties wish to communicate

securely using an encryption scheme, the only thing that they keep secret is the particular

key pair (e, d) which they are using, and which

Security (cont.)

• An encryption scheme is said to be breakable

if a third party, without prior knowledge of the key pair (e, d), can systematically recover

plaintext from corresponding ciphertext

within some appropriate time frame.

• The number of keys (i.e., the size of the key

space) should be large enough to make this

approach computationally infeasible.

• Cryptanalysis is the study of mathematical techniques for attempting to defeat cryptographic techniques,

and, more generally, information security services

• A cryptanalyst is someone who engages in

Symmetric-key encryption

• Block ciphers

• Stream ciphers

Overview of block ciphers and

stream ciphers

• Let {Ee : e ∈K} and {Dd : d ∈K}, K is the key space

– The encryption scheme is said to be symmetric-key if for each associated encryption/decryption key pair (e, d), it is computationally “easy” to determine d knowing only e,

and to determine e from d.

• Since e = d in most practical symmetric-key

encryption schemes, the term symmetric-key

becomes appropriate

Ex of symmetric-key encryption

• Let A = {A,B,C, ,X,Y, Z} be the English alphabet

• Let M and C be the set of all strings of length five

over A

• The key e is chosen to be a permutation on A

Ex (cont.)

Block ciphers

• A block cipher is an encryption scheme which breaks up the plaintext messages to be

transmitted into strings (called blocks) of a

fixed length t over an alphabet A, and

encrypts one block at a time.

• Two important classes of block ciphers are

substitution ciphers and transposition ciphers.

Simple substitution ciphers

• Let A be an alphabet of q symbols and M be the set of all strings of length t over A.

• K be the set of all permutations on the set A.

where m =(m1m2 ···mt) ∈ M

• To decrypt c =(c1c2 ··· ct), compute the inverse permutation d = e−1.

Polyalphabetic substitution ciphers

(đa chữ cái)

i the key space K consists of all ordered sets of t

permutations (p1,p2, ,pt), where each

permutation pi is defined on the set A;

ii encryption of the message m =(m1m2 ···mt) under

the key e =(p1,p2, ,pt) is given by

Ee(m)=(p1(m1)p2(m2) ··· pt(mt)); and

iii the decryption key associated with e =(p1,p2, ,pt)

is d =(p1−1,p2−1, ,pt−1)

Ex of Polyalphabetic (Vigenère cipher)

• Let A = {A,B,C, ,X,Y, Z} and t =3 Choose e =

(p1,p2,p3), where p1 maps each letter to the letter

three positions to its right in the alphabet, p2 to the one seven positions to its right, and p3 ten positions

to its right If

Ex of transposition ciphers

e:

d = e−1 :

Plaintext m:

Stream ciphers

• Let K be the key space,

– A sequence of symbols e1e2 ··· ei ∈ K, is called a keystream.

• Let Ee be a simple substitution cipher with block

length 1 where e ∈ K.

• Let m1m2 ··· be a plaintext string

• A stream cipher takes the plaintext string and

produces a ciphertext string c1c2 ··· where ci = Eei(mi)

– If di denotes the inverse of ei, then Ddi (ci)= mi decrypts the ciphertext string.

The Vernam cipher

• The Vernam Cipher is a stream cipher defined on the alphabet A = {0, 1}

• A binary message m1m2 ···mt is operated on by a

binary key string k1k2 ··· kt of the same length to

produce a ciphertext string c1c2 ··· ct where

• If the key string is randomly chosen and never used again, the Vernam cipher is called a one-time pad

Digital signatures

• M is the set of messages which can be signed.

• S is a set of elements called signatures, possibly binary strings

of a fixed length.

• SA is a transformation from the message set M to the

signature set S, and is called a signing transformation for

Ex of digital signature scheme

• M= {m1,m2,m3} and S = {s1,s2,s3}.

Digital signature mechanism

– Accept the signature as having been created by A if u =

true, and reject the signature if u = false.

Public-key cryptography

Public-key encryption scheme

Hash functions

• A hash function is a computationally efficient

function mapping binary strings of arbitrary length to binary strings of some fixed length, called hash-

values

• It is computationally infeasible to find two distinct

inputs which hash to a common value

• It is computationally infeasible to find an input

(pre-image) x such that h(x)= y.