Computer and Network Security

Rabie A. Ramadan

Lecture 4

Table of Contents

2

Mathematics of cryptography• Groups

• Rings

• Polynomials

3

Mathematics of Cryptography

Groups

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A group G is a set of elements with a binary operation

that satisfies four properties:

Closure : if a and b are elements of G , then c = a b

is also an element of G

Associatively : if a, b , and c are elements of G , then

(a b) c = a ( b c )

Groups (Cont.)

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Existence of Identity : for all a in G , there exists an element e , called the identity element , such that

e a = a e = a

Existence of Inverse : for each a in G , there exists an , called the inverse of a , such that

a = a = e

An Commutative or Abelian group is a group that satisfies Commutativity property

Commutativity : for all a and b in G a b = b a

a a

a

Group Activity

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Given the following group , is it abelian group?

G = <{a,b,c,d} , >

a b c d

a a b c d

b b c d a

c c d a b

d d a b c

Answer

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Yes it is an abelian group Closure

• Applying the operation on any element ,

results an element in the group. Associativity

• Check the combination of any 3 elements

• (a b) c = a ( b c) = d

Commutative • a b = b a

Identity element is a

• Using it with any element gives the same element

Inverse • Each element has an inverse e.g. (a , a), (b, d), (c , c)

a b c d

a a b c d

b b c d a

c c d a b

d d a b c

Groups (Cont.)

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Finite Group

• If it has a finite number of elements

Order of a group |G|

• Number of elements in the group

Subgroups • A subset H is a subgroup of G if H is a group with respect to G

• The two groups must be under the same operations

• Both H and G will have the same properties

Cyclic Group

Define exponentiation as repeated application of operator• example: a3 = a.a.a

a is said to be a generator of the group

Ring A set of “numbers” with two operations (addition and

multiplication) which are:

An Abelian group with addition operation

Multiplication:• has closure• is associative• distributive over addition: a(b+c) = ab + ac

If multiplication operation is commutative, it forms a commutative ring

Field

A set of numbers with two operations:

• abelian group for addition

• abelian group for multiplication (ignoring 0)

• ring

Galois Fields(Cont.)

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finite fields play a key role in cryptography can show number of elements in a finite field must be a power of a

prime pn

known as Galois fields denoted GF(pn) in particular often use the fields:

• GF(p)

• GF(2n) Hence arithmetic is “well-behaved” and can do addition,

subtraction, multiplication, and division without leaving the field GF(p)

GF(2n )

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The elements in the set is n-bit words

Example • GF(23) the set is {000, 001,010, 011, 100, 101, 110, 111}

GF(p)

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The field includes the set Z = {0,1,2,3,4,…,p-1} Operations Add / Subtract E.g.

• GF(2)

{0,1} + X+ 0 1

0 0 1

1 1 0

x 0 1

0 0 0

1 0 1

a 0 1 a 0 1

-a 1 0 a-1 - 1

Addition/Subtraction is the same as XOR operations Multiplication/Division is the same as AND operations

Group Activity

15

Show a GF(7) using multiply operation?

X 0 1 2 3 4 5 6

0

1

2

3

4

5

6

Example GF(7)

Modular Arithmetic

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We care about the reminder of a given operation An operation

• a = q x n + r

• a and n (modulus) are inputs, q is quotient , and r is the residue

modmod

Z = {…., -2,-1, 0, 1, 2, …..}

n

r (nonnegative)

a

Example

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25 mod 3• a =25 and n = 3

• Divide a by n and get the reminder r = 1

-7 mod 10• a = -7 and n = 10

• Divide -7 by 10 r = -7

• Add the modulus (n= 10) to remove the negative sign -7 + 10 = 3 r =3

The result of a mod n is always nonnegative number less than n

Set of Residues Zn

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Modulo operation creates a set with numbers always less than n

The output of a mod operation for a number n must fall in the set

E.g.

• Z2 = { 0, 1}

• Z5 = {0,1,2,3,4}

• Z11 = {0,1,2,3,4,5,6,7,8, 9,10}

Congruence

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It is the same as the equality

Using mod , we can get an infinite number of residue when we map Z to Zn

E.g. • 2 mod 10 = 2 12 mod 10 22 mod 2

• 2 , 12, 22 are congruentcongruent of mod 10

Residue Classes

21

Residue class [a] or [a]n

Set of all integers such that x =a (mod n)

E.g. If n = 5 • [0] = {.., -15, -10 , -5, 0, 5, 10,15,…}

• [1] = {…., -14, -9, -4, 1, 6, 11, 16,...}

• ..

• [4] = {.., -11, -6, -1, 4, 9, 14, 19,…}

Operations on Zn

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(a + b) mod n = [ (a mod n) + b mod n)] mod n

(a - b) mod n = [ (a mod n) - b mod n)] mod n

(a x b ) mod n = [ (a mod n) x (b mod n ) ] mod n

Group Activity

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Find :• 10 mod 3

• 102 mod 3

• 103 mod 3

Answer • 10 mod 3 = 1

• 102 mod 3 = [10 mod 3 X 10 mod 3] mod 3

• 103 mod 3 = [ 10 mod 3 X 10 mod 3 X 10 mod 3] mod 3

Then 10n mod 3 = (10 mod 3) n mod 3

an mod x = ( a mod x) n mod x

Group Activity

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Draw a table that shows the addition Modulo 8?

Modulo 8

Inverses

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Additive inverse Multiplicative inverse

In Z (Integer Numbers ) • a + b = 0 , b is the additive inverse of a and a is the inverse of b

• a X b = 1, b is the multiplicative inverse of a and a is the inverse of b

In Zn (Modulo )

• a + b = 0 (mod n) Simply : b = n – a

• a X b = 1 (mod n) Simply : (a X b) mod n = 1

Inverses (Cont. )

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Examples:• Find the additive inverse of 4 in Z10? find b?

• b = n – a = 10 – 4 = 6

• Find all additive inverse of Z10 ?

• (0,0), (1, 9), (2, 8), (3,7) , (4, 6),..

• Find the multiplicative inverse of 8 in Z10?

• (a X 8) mod 10 = 1 no multiplicative inverse can be found

• Find all multiplicative inverse of Z10 ?

• (1,1), (3, 7), and (9, 9)

The greatest common divisor

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The greatest common divisor (gcd) of two non-zero integers, is :

The largest positive integer that divides both numbers without remainder

gcd(42, 56)=14 where

Euclid's GCD Algorithm

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An efficient way to find the GCD(a,b) Uses theorem that:

• GCD(a,b) = GCD(b, a mod b) • GCD(55,22)= GCD(22,55 mod 22)= GCD (22,11) = 11

Euclid's Algorithm to compute GCD(a,b):

• A=a, B=b

• while B>0• R = A mod B

• A = B, B = R

• return A

Example GCD(1970,1066)

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1970 = 1 x 1066 + 904 gcd(1066, 904)1066 = 1 x 904 + 162 gcd(904, 162)904 = 5 x 162 + 94 gcd(162, 94)162 = 1 x 94 + 68 gcd(94, 68)94 = 1 x 68 + 26 gcd(68, 26)68 = 2 x 26 + 16 gcd(26, 16)26 = 1 x 16 + 10 gcd(16, 10)16 = 1 x 10 + 6 gcd(10, 6)10 = 1 x 6 + 4 gcd(6, 4)6 = 1 x 4 + 2 gcd(4, 2)4 = 2 x 2 + 0 gcd(2, 0)

gcd(1970, 1066) = 2

Polynomial Arithmetic Can compute using polynomials

Several alternatives available• ordinary polynomial arithmetic

• poly arithmetic with coords mod p

• poly arithmetic with coords mod p and polynomials mod M(x)

Ordinary Polynomial Arithmetic

add or subtract corresponding coefficients multiply all terms by each other eg

• let f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1

f(x) + g(x) = x3 + 2x2 – x + 3

f(x) – g(x) = x3 + x + 1

f(x) x g(x) = x5 + 3x2 – 2x + 2

Polynomial Arithmetic with Modulo Coefficients

when computing value of each coefficient do calculation modulo some value

could be modulo any prime but we are most interested in mod 2

• ie all coefficients are 0 or 1

• eg. let f(x) = x3 + x2 and g(x) = x2 + x + 1

f(x) + g(x) = x3 + x + 1

f(x) x g(x) = x5 + x2

Modular Polynomial Arithmetic

can write any polynomial in the form:• f(x) = q(x) g(x) + r(x)

• can interpret r(x) as being a remainder

• r(x) = f(x) mod g(x) if have no remainder say g(x) divides f(x) if g(x) has no divisors other than itself & 1

say it is irreducible (or prime) polynomial arithmetic modulo an irreducible polynomial

forms a field

Modular Polynomial Arithmetic

can compute in field GF(2n) • polynomials with coefficients modulo 2

• whose degree is less than n

• hence must reduce modulo an irreducible poly of degree n (for multiplication only)

form a finite field

Example GF(23)

Group Activity

Find the results of (x5+x2+x) * x7+x4+x3+x2+x in GF(28) with irreducible polynomial x8+x4+x3+x+1

Answer Multiply the two polynomials

• (x5+x2+x) * x7+x4+x3+x2+x = x5 * (x7+x4+x3+x2+x ) + x2 * (x7+x4+x3+x2+x ) + x * (x7+x4+x3+x2+x ) = (x12+x7+x2)

Get the results of

• (x12+x7+x2) mod (x8+x4+x3+x+1) = (x5+x3+x2+x+1)

1

1Re1

xxx1xxxx

4

235

348

2458

457812

2712348

x

xxxxmainderxxxx

xxxx

xxxxx

Group Activity

3DES (Tripple Data Encryption Standard) is based on which of the following?• A.      Hashing algorithm

•  B.     Symmetric key-based algorithm

•  C.      Asymmetric key-based algorithm

•  D.      None of these

Viruses, Worms, Zombies, and others

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41

What is Computer Security?

What is Computer Security?

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Computer Security is the protection of computing systems and the data that they store or access

Why is Computer SecurityImportant?

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Computer Security allows the University to carry out its mission by:

• Enabling people to carry out their jobs, education, and research.

• Supporting critical business processes

• Protecting personal and sensitive information

Why do I need to learn aboutComputer Security?

44

Isn’t this just an IT Problem?

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Good Security Standards follow the “90 / 10” Rule:• 10% of security safeguards are technical

• 90% of security safeguards rely on the computer user (“YOU”) to adhere to good computing practices

What Does This Mean for Me?

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This means that everyone who uses a computer or mobile device needs to understand how to keep their computer, device and data secure.• Information Technology Security is everyone’s

responsibility

What’s at Stake?

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Kinds of damage caused by insecurity• Nuisance: spam, …

• Data erased, corrupted, or held hostage

• Valuable information stolen(credit card numbers, trade secrets, etc.)

• Services made unavailable (email and web site outages, lost business)

Breaking into a Computer

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Breaking into a Computer

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What?• Run unauthorized software

How?• Trick the user into running bad software

(“social engineering”)

• Exploit software bugs to run bad software without the user’s help

Example of “social engineering”: Trojan Horse

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CoolScreenSaver.exe

Viruses and Worms

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Automated ways of breaking in;

Use self-replicating programs

(Recall self-replicating programs:

Print the following line twice, the second time in quotes. “Print the following line twice, the second time in quotes.” )

Computer Viruses

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Self-replicating programs that spread by infecting other programs or data files

Must fool users into opening the infected file

Payload

Cool Screen Saver

Notepad Solitaire Paint

PayloadPayloadPayload

Email Viruses

53

Infected program, screen saver, or Word document launches virus when opened

Use social engineering to entice you to open the virus attachment

Self-spreading: after you open it, automatically emails copies to everyone in your address book

Other forms of social engineering: downloadable software/games, P2P software, etc.

The Melissa Virus (1999)

54

Social engineering: Email says attachment contains porn site passwords

Self-spreading: Random 50 people from address book

Traffic forced shutdown of many email servers

$80 million damage 20 months and $5000 fine

David L. SmithAberdeen, NJ

Computer Worms

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Self-replicating programs like viruses, except exploit security holes in OS (e.g., bugs in networking software) to spread on their own without human intervention

PayloadPayloadPayload

PayloadPayloadPayloadPayload

Robert Tappan Morris

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First Internet worm Created by student at Cornell Exploited holes in email

servers, other programs Infected ~10% of the net Spawned multiple copies,

crippling infected servers Sentenced to 3 years

probation, $10,000 fine, 400 hours community service

Robert Tappan Morris

“Can we just develop software to detect a virus/worm?”

57

[Adleman’88] This task is undecidable.(so no software can work with 100% guarantee)

No real guarantee

Current methods: (i) Look for snippets of known virus programs on hard

drive (ii) maintain log of activities such as network requests,

read/writes to hard-drive and look for “suspicious” trends (iii) look for changes to OS code.

Spyware/Adware

58

Hidden but not self-replicating

Tracks web activity for marketing,

shows popup ads, etc.

Usually written by businesses: Legal gray area

Zombies

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Bot

Bot program runs silently in the background, awaiting instructions from the attacker

Attacker’s Program

Can we stop computer crime?

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Probably not! Wild West nature of the Internet Software will always have bugs Rapid exponential spread of attacks

But we can take steps to reduce risks…

Protecting Your Computer

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Six easy things you can do…

Keep your software up-to-date Use safe programs to surf the ‘net Run anti-virus and anti-spyware regularly Add an external firewall Back up your data Learn to be “street smart” online

Keep Software Up-to-Date

Use Safe Software to Go Online

Firefox(web browser)

Thunderbird(email)

Anti-virus / Anti-spyware Scans

Symantec Antivirus(Free from OIT)

Spybot Search & Destroy(Free download)

Add an External Firewall

Provides layered security (think: castle walls, moat)

(Recent operating systems have built-in firewall features)

Back Up Your Data

Tivoli Storage Manager(Free from OIT)

Learn Online “Street Smarts” Be aware of your surroundings

• Is the web site being spoofed? Don’t accept candy from strangers

• How do you know an attachment or download isn’t a virus, Trojan, or spyware?

Don’t believe everything you read• Email may contain viruses or phishing attack – remember,

bad guys can forge email from your friends

Quiz: A hacked computer can be used to… (select all that apply)

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1. Send spam and phishing emails.

2. Harvest and sell email addresses and passwords.

3. Illegally distribute music, movies and software.

4. Distribute child pornography.

5. Infect other systems.

6. Hide programs that launch attacks on other computers.

7. Record keystrokes and steal passwords.

8. Access restricted or personal information on your computer or other systems that you have access to.

9. Generate large volumes of traffic, slowing down the entire system

Next time is an exam

69

Assignment• Survey some of viruses and security incidents

that

• ha been found in the last five years ?