Finite Fields: Concepts, Properties, and Applications, Slides of Cryptography and System Security

An introduction to finite fields, their properties, and applications in cryptography. It covers the concepts of groups, rings, fields, modular arithmetic, divisors, and polynomial arithmetic. The document also includes examples and algorithms for finding greatest common divisors and inverses in finite fields.

Typology: Slides

2011/2012

Uploaded on 07/17/2012

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Chapter 4 – Finite Fields
The next morning at daybreak, Star flew indoors,
seemingly keen for a lesson. I said, "Tap eight." She did
a brilliant exhibition, first tapping it in 4, 4, then giving me
a hasty glance and doing it in 2, 2, 2, 2, before coming
for her nut. It is astonishing that Star learned to count up
to 8 with no difficulty, and of her own accord discovered
that each number could be given with various different
divisions, this leaving no doubt that she was consciously
thinking each number. In fact, she did mental arithmetic,
although unable, like humans, to name the numbers. But
she learned to recognize their spoken names almost
immediately and was able to remember the sounds of
the names. Star is unique as a wild bird, who of her own
free will pursued the science of numbers with keen
interest and astonishing intelligence.
Living with Birds, Len Howard
docsity.com
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Chapter 4 – Finite Fields The next morning at daybreak, Star flew indoors,seemingly keen for a lesson. I said, "Tap eight." She dida brilliant exhibition, first tapping it in 4, 4, then giving mea hasty glance and doing it in 2, 2, 2, 2, before comingfor her nut. It is astonishing that Star learned to count upto 8 with no difficulty, and of her own accord discoveredthat each number could be given with various differentdivisions, this leaving no doubt that she was consciouslythinking each number. In fact, she did mental arithmetic,although unable, like humans, to name the numbers. Butshe learned to recognize their spoken names almostimmediately and was able to remember the sounds ofthe names. Star is unique as a wild bird, who of her ownfree will pursued the science of numbers with keeninterest and astonishing intelligence.^ —^ Living with Birds

, Len Howard

Introduction

^ will now introduce finite fields ^ of increasing importance in cryptography^ ^ AES, Elliptic Curve, IDEA, Public Key ^ concern operations on “numbers”^ ^ where what constitutes a “number” and thetype of operations varies considerably ^ start with concepts of groups, rings, fieldsfrom abstract algebra

Cyclic Group

^ define^ exponentiation

as repeated

application of operator^ ^ example:

-3^ a= a.a.a

^ and let identity be:

0 e=a

^ a group is cyclic if every element is apower of some fixed element^ ^ ie^ b = a

k^ for some

a^ and every

b^ in group

^ a^ is said to be a generator of the group

Ring

^ a set of “numbers” ^ with two operations (addition and multiplication)which form: ^ an abelian group with addition operation ^ and multiplication:^ ^ has closure^ ^ is associative^ ^ distributive over addition:

a(b+c) = ab + ac ^ if multiplication operation is commutative, itforms a^ commutative ring ^ if multiplication operation has an identity and nozero divisors, it forms an

integral domain

Modular Arithmetic  define modulo operator

“a mod n”

to be

remainder when a is divided by n  use the term

congruence

for:^ a = b mod n ^ when divided by

n,^ a & b have same remainder ^ eg. 100 = 34 mod 11  b is called a

residue^ of a mod n ^ since with integers can always write:

a = qn + b

^ usually chose smallest positive remainder as residue• ie.^0 <= b^ <=^ n- ^ process is known as

modulo reduction

  • eg.^ -12^ mod

7 =^ -5^ mod

7 =^2 mod^

7 =^9 mod^7

Divisors

^ say a non-zero number

b^ divides

a^ if for

some^ m^ have

a=mb^ (a,b,m

all integers)

^ that is^ b

divides into

a^ with no remainder

^ denote this

b|a

^ and say that

b^ is a^ divisor

of^ a

^ eg. all of 1,2,3,4,6,8,12,24 divide 24

Modular Arithmetic  can do modular arithmetic with any group ofintegers:^ Z= {0, 1, … , n-1}n^  form a commutative ring for addition  with a multiplicative identity  note some peculiarities  if (a+b)=(a+c) mod n then b=c mod n  but if (a.b)=(a.c) mod n then b=c mod n

only if^ a^ is relatively prime to

n docsity.com

Modulo 8 Addition Example^ + 0

Euclidean Algorithm  an efficient way to find the GCD(a,b)  uses theorem that:  GCD(a,b) = GCD(b, a mod b)  Euclidean Algorithm to compute GCD(a,b) is: EUCLID(a,b)1. A =^ a;^ B^ =^ b 2. if B^ =^0 return

A^ =^ gcd(a,

b)

3.^ R^ =^ A^ mod

B

4.^ A^ =^ B 5.^ B^ =^ R 6.^ goto^2

Example GCD(1970,1066) 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)

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  hence arithmetic is “well-behaved” andcan do addition, subtraction, multiplication,and division without leaving the field GF(p)

GF(7) Multiplication Example

^0 1

Inverse of 550 in GF(1759)^ Q^ A^

A2^ A^

B1^ B^

B

—^1

1 –^

–3^109

–5^16

21 –^

–339^4

–111^355

Polynomial Arithmetic  can compute using polynomialsn^ f ( x ) = a^ x + a^ nn-

n-1^ x + … + a

x +^ a^ =^ ∑ 10

i a x i

  • 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 andpolynomials mod m(x)