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Multiplicative Group • • • • The multiplicative group of Zn includes every a, 0<a<n, such that (a,n)=1. The number of elements is Euler’s Totient function (n) If n is prime, (n)=n-1 If n=PQ, and P, Q are prime then (n)=(P-1)(Q-1) Order of Elements • Let an denote a,…,a n times • We say that a is of order n if an=1, and for any 0<m<n, am1 • Examples • Euler theorem: in the multiplicative group of Zn any element is of order at most (n) • Generalization: in a finite group every element has finite order and it is at most the size of the group. Sub-groups • Let (G,) be a group. (H,) is a sub-group of (G,) if it is a group, and HG • Claim: If (G,) is a finite group and (H,) is closed, where HG, then (H,) is a sub-group of (G,). • Examples • Lagrange theorem: if G is finite and (H,) is a sub-group of (G,) then |H| divides |G| • Examples Cyclic Groups • Claim: let G be a group and a be an element of order n. The set [a]={1, a,…,an-1} is a sub-group of G, and is called the sub-group generated by a. • a is the generator of [a] • If G is generated by some a, G is called cyclic. • Theorem: for any prime p, the multiplicative group of Zp is cyclic Rings and Fields Rings • A ring <R,,> has properties as follows: – <R,> is a commutative group, identity 0 – <R,> is associative, identity 1 – is distributive: a(bc)=(ab)(ac) and (bc)a=(ba)(ca) • A ring is called commutative if is a commutative operation • Claim: 0 is a multiplicative annihilator in a ring • Examples: Z, Zn Fields • A field is a commutative ring in which all non-zero elements have multiplicative inverses • Example: Rational numbers, Zp • Claim: In a field there are no r,s0, rs=0 • Theorem: the multiplicative group of a finite field is cyclic Polynomials over Rings • A polynomial is an expression: a(x)=amxm…a0 over a commutative ring <R,,> (where xm denotes xx) • Degree of a polynomial is m, for the largest non-zero am Polynomial Ring • The Polynomial Ring R[x] is a commutative ring over a commutative ring R: – Addition: c(x)=a(x)+b(x) if ci=aibi – Multiplication: d(x)=a(x)b(x) if di=a0bia1 bi-1 … ai b0 • Examples of operations over Z2. Is Z2[x] finite? Analogies Polynomials-Integers • Henceforth we consider polynomials over finite fields • If h(x)0, there is a unique representation for g(x) as g(x)=q(x)h(x)+r(x) such that degree r(x) < degree h(x) • Example • If for g(x) there is no h(x) of degree>0 s.t. h(x)|g(x) then g(x) is irreducible • Example x4+x+1 is irreducible over Z2(not over Z3!) F[x]/f(x) • F[x]/f(x): includes all polynomials over field F of degree less than the degree of f(x). Addition and multiplication are computed modulo f(x) • F[x]/f(x) is a commutative ring for any f(x) • GCD theorems carry over to F[x]/f(x) – Euclidean algorithm finds (g(x),h(x)) – Extended Euclid finds a(x), b(x) s.t. a(x)g(x)+b(x)h(x)=(g(x),h(x)) • Theorem: If f(x) is irreducible F[x]/f(x) is a field Finite Fields • The characteristic of a field F is the smallest m such that 11 m times is 0 • Claim: In a finite field the characteristic is prime • Example: The characteristic of Zp is p • Theorem: All finite fields with the same number of elements are identical up to isomorphism • Theorem: the number of elements of a finite field is pn for a prime p and a natural number n. • Example: GF(24) with irreducible poly. x4+x+1