# 4 Finite Fields - 4 FINITE FIELDS Contents Groups Rings and Fields Greatest Common Divisors Modular Arithmetic Finite Fields of the Form GF(p

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4. FINITE FIELDS

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Contents Groups, Rings, and Fields Greatest Common Divisors Modular Arithmetic Finite Fields of the Form GF( p ) Polynomial Arithmetic Finite Fields of the Form GF(2 n )
Notations N : The set of natural numbers Z : The set of integers Q : The set of rational numbers R : The set of real numbers C : The set of complex numbers

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Notations Binary operations A binary operation • on a set S is a function that maps each ordered pair ( a , b ) to some element c . a b = c . 1 + 3 = 5. Closure A set S is closed under a binary operation • if a b is in S for every pair of a and b in S . N is closed under ‘+’ . Z is closed under ‘ × ’.
Notations Associativity A binary operation • on a set S is associative if a • ( b c ) = ( a b ) • c for all a , b , c S. Commutativity A binary operation • on a set S is commutative if a b = b a for all a , b S.

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Notations Identity element An identity element for an operation • on a set S is an element e satisfying a e = e a = a for all a S . For ‘+’ on Z, 0 is an identity element. Inverse element An inverse element of an element a for an operation • on a set S is an element a satisfying a a = a a = e . For ‘+’ on Z, -3 is an inverse element of 3 because 3+(-3) = 0.
Groups A group { G , •}, sometimes called just G , is a set G under a binary operation • satisfying the following conditions. Closure: G is closed under ‘•’. Associative: The binary operation • is associative . Identity : There is an identity element e for • in G. Inverse : For each a in G, there is an inverse element a for • in G.

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Groups Permutation group (text box in p.105)
Finite groups and infinite groups A finite group A group that has a finite number of elements. An infinite group A group that has an infinite number of elements. The order of a finite group The number of elements in the group.

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Abelian groups An abelian group A group whose binary operation is commutative , i.e., a b = b a , for all a , b G. { Z , +} Closure `+’ is associative. identity: 0 inverse of a : - a `+’ is commutative . { R , × }
Abelian groups The permutation group n = 2 {{1,2}, {2,1}} {1,2} • {2,1} = {2,1} {2,1} • {1,2} = {2,1} n > 2 (?) {1,3,2} • {2,3,1} = {2,1,3} {2,3,1} • {1,3,2} = {3,2,1}

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Exponentiation Exponentiation within a group { G, • }. Repeated application of the binary operation •. a n = a a a • ··· • a a 0 = e a - n = ( a ) n. Exponentiation is defined on an associative binary operation.
Cyclic groups A cyclic group A group G is cyclic if every element of G is a power a k ( k is an integer) of a fixed element a G. The element a is said to generate the group G or to be a generator of G . { Z , +} { Z , +} is a group with generators 1 or -1.

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## This note was uploaded on 05/25/2010 for the course MATERIALS [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ taught by Professor Mrpaak during the Spring '10 term at 카이스트, 한국과학기술원.

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4 Finite Fields - 4 FINITE FIELDS Contents Groups Rings and Fields Greatest Common Divisors Modular Arithmetic Finite Fields of the Form GF(p

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