Lecture Data security and encryption - Chapter 12: Message authentication codes

This chapter presents the following content: Number theory, divisibility & GCD, modular arithmetic with integers, Euclid’s algorithm for GCD & inverse, Group, Ring, Field, finite fields GF(p), polynomial arithmetic in general and in GF(2n).

(CSE348)

Lecture # 12

– Number Theory

– divisibility & GCD

– modular arithmetic with integers

– Euclid’s algorithm for GCD & Inverse

• Groups, rings, and fields are the fundamental

elements of a branch of mathematics

• known as abstract algebra, or modern algebra

• In abstract algebra, we are concerned with sets

on whose elements we can operate algebraically

• That is, we can combine two elements of the set, perhaps in several ways, to obtain a third

element of the set

• A group G, denoted by {G, • }, is a set of

elements with a binary operation

• Denoted by •, that associates to each ordered pair (a, b) of elements in G an element (a • b) in G

• Such that the following axioms are obeyed:

– Closure, Associative, Identity element, Inverse

element

• we have used as operator: could be addition +, multiplication x or any other mathematical

operator

• A group can have a finite (fixed) number of

elements, or it may be infinite

• Integers (+ve, -ve and 0) using addition form an infinite abelian group

• a set of elements or “numbers”

– may be finite or infinite

• with some operation whose result is also

in the set (closure)

• obeys:

– associative law: (a.b).c = a.(b.c)

– has identity e: e.a = a.e = a

– has inverses a-1: a.a-1 = e

• if commutative a.b = b.a

Cyclic Group

• Define exponentiation as repeated

application of operator

– example: a-3 = a.a.a

• and let identity be: e=a0

• a group is cyclic if every element is a

power of some fixed element

– ie b = ak for some a and every b in group

• a is said to be a generator of the group

• and multiplication without leaving the set

• and which obeys the associative and distributive laws

• With respect to addition and multiplication

• The set of all n-square matrices over the real

numbers form a ring

• The set of integers with addition & multiplication form an integral domain

• If multiplication operation is commutative, it

forms a commutative ring

• If multiplication operation has an identity and no

zero divisors, it forms an integral domain

• Lastly define a field, We denote a Field as {F,+,.}

• In essence, a field is a set in which we can do addition, subtraction, multiplication

• and division without leaving the set

• Division is defined with the following rule:

a/b = a (b–1)

• Examples of fields are:

– rational numbers, real numbers, complex numbers

• Integers are NOT a field since there are no

multiplicative inverses (except for 1)

 a set of numbers

 with two operations which form:

abelian group for addition

abelian group for multiplication (ignoring 0)

ring

 have hierarchy with more axioms/laws

group -> ring -> field

Group, Ring, Field

• As a memory aid

• Can use the acronym for groups: CAIN

(Closure Associative Identity iNverse)

• Mostly we need to compute with Rings, if not

Fields

• When we do arithmetic modulo a prime, we

have a field

Group, Ring, Field

• These are terms we use for different sorts of

"number systems“

• ones obeying different sets of laws

• From group to ring to field we get more and

more laws being obeyed

• as shown here in Stallings Figure 4.2

Group, Ring, Field

Finite (Galois) Fields

• Infinite fields are not of particular interest in the context of cryptography

• However, finite fields play a crucial role in many cryptographic algorithms

• It can be shown that the order of a finite field

(number of elements in the field) must be a

positive power of a prime

Finite (Galois) Fields

• These are known as Galois fields

• In honor of the mathematician who first studied finite fields, & are denoted GF(p^n)

• We are most interested in the cases where

either n=1 - GF(p), or p=2 - GF(2^n)

Finite (Galois) Fields

• 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)

Galois Fields GF(p)

• GF(p) is the set of integers {0,1, … , p-1} with

arithmetic operations modulo prime p

• These form a finite field

– since have multiplicative inverses

– find inverse with Extended Euclidean algorithm

• Hence arithmetic is “well-behaved” and can do

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

• As can be seen, it satisfies all of the

properties required of a field (Figure 4.2)

GF(7) Multiplication Example

• Compare this table with Table 4.2

• In the latter case, we see that using modular

arithmetic modulo 8, is not a field

Polynomial Arithmetic

• can compute using polynomials

f(x) = anxn + an-1xn-1 + … + a1x + a0 = ∑ aixi

• nb not interested in any specific value of x

• which is known as the indeterminate

• several alternatives available

– ordinary polynomial arithmetic

– poly arithmetic with coords mod p

– poly arithmetic with coords mod p and

Ordinary Polynomial Arithmetic

• Includes:

• add or subtract corresponding coefficients

• multiply all terms by each other

Polynomial Arithmetic with Modulo

Coefficients

• Consider variant where now when computing

value of each coefficient

• Do the calculation modulo some value, usually a prime

• If the coefficients are computed in a field (eg

GF(p)), then division on the polynomials is

possible

Polynomial Arithmetic with Modulo

Coefficients

• Are most interested in using GF(2)

• i.e all coefficients are 0 or 1

• and any addition/subtraction of coefficients is

done mod 2 (ie 2x is the same as 0x!)

• which is just the common XOR function

Polynomial Arithmetic with Modulo

Coefficients

 when computing value of each coefficient do

calculation modulo some value

forms a polynomial ring

 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

Polynomial Division

• and show that the Euclidean algorithm can be extended to find the greatest common divisor of two polynomials

• whose coefficients are elements of a field

Polynomial Division

• Define an irreducible (or prime) polynomial as

one with no divisors other than itself & 1

• If compute polynomial arithmetic modulo an

irreducible polynomial

• This forms a finite field, and the GCD & Inverse algorithms can be adapted for it

• 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

• Arithmetic operations are performed on

polynomials using the ordinary rules of algebra

Polynomial GCD

• Polynomial division is not allowed unless the

coefficients are elements of a field

• Next, we discussed polynomial arithmetic in

which the coefficients are elements of GF(p)

• In this case, polynomial addition, subtraction,

multiplication, and division are allowed

• However, division is not exact; that is, in general

Polynomial GCD

• Finally, we showed that the Euclidean algorithm can be extended

• To find the greatest common divisor of two

polynomials whose coefficients are elements of

Polynomial GCD

• can find greatest common divisor for polys

– c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest

degree which divides both a(x), b(x)

• can adapt Euclid’s Algorithm to find it:

Euclid(a(x), b(x))

if (b(x)=0) then return a(x);

else return

Euclid(b(x), a(x) mod b(x));

• all foundation for polynomial fields as see next

Modular Polynomial Arithmetic

• Consider now the case of polynomial arithmetic with coordinates mod 2 and polynomials mod an irreducible polynomial m(x)

• That is Modular Polynomial Arithmetic uses the set S of all polynomials of degree n-1 or less over the field Zp

• With the appropriate definition of arithmetic

operations, each such set S is a finite field

Modular Polynomial Arithmetic

• The definition consists of the following elements:

1 Arithmetic follows the ordinary rules of polynomial

arithmetic using the basic rules of algebra, with the

following two refinements.

2 Arithmetic on the coefficients is performed modulo p.

3 If multiplication results in a polynomial of degree

greater than n-1, then the polynomial is reduced

modulo some irreducible polynomial m(x) of degree n That is, we divide by m(x) and keep the remainder.

Modular Polynomial Arithmetic

• This forms a finite field

• And just as the Euclidean algorithm can be

adapted to find the greatest common divisor of two polynomials

• The extended Euclidean algorithm can be

adapted to find the multiplicative inverse of a

polynomial

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

• can always find an inverse

– can extend Euclid’s Inverse algorithm to find

Using a Generator

• Equivalent definition of a finite field

• A generator g is an element whose powers

generate all non-zero elements

• using all possible bit values

• and the calculations only use simple common

machine instructions

Computational Considerations

• The shortcut for polynomial reduction comes

from the observation

• That if in GF(2n) then irreducible poly g(x) has

highest term xn

• and if compute xn mod g(x) answer is g(x)- xn

Computational Considerations

• since coefficients are 0 or 1, can represent any such polynomial as a bit string

• addition becomes XOR of these bit strings

• multiplication is shift & XOR

– cf long-hand multiplication

• modulo reduction done by repeatedly

substituting highest power with remainder of

Computational Example

• Show here a few simple examples of addition,

• multiplication & modulo reduction in GF(23)

• The long form modulo reduction finds

p(x)=q(x).m(x)+r(x) with r(x) being the desired remainder

Computational Example

• Polynomial modulo reduction (get q(x) & r(x)) is

–  (x 3 +x 2 +x+1 ) mod (x 3 +x+1) = 1.(x 3 +x+1) + (x 2 ) = x 2

–  1111 mod 1011 = 1111 XOR 1011 = 01002