Faktorisasi polinomial x11-1 atas lapangan berhingg

Abstract

Misalkan F adalah lapangan berhingga dengan q elemen dan dinotasikan dengan GF(q), a suatu elemen primitif dan i GF(q) dan suatu akar dan i x" — 1, maka = 1 dan a disebut akar satuan primitif ke n. Dengan demikian .1(x) = x"— 1 mempunyai akar satuan primitif ke — n. Misalkan /3 suatu akar satuan primitf n .dari- .GF(e);..polinomial -minimal dari... 13- -atas-GF(q) .adalah " • ." • • • --t-i dan dapat dinyatakan dalam bentuk : 1,(x) =fJ(x - dimana t bilangan bulat positif terkecil sedemikian sehingga = fl . Selanjutnya mfi(x) adalah suatu faktor dari x" — 1 atas, GF(q) Dengan meaggunakan tabel log Zech's (1 + = ce(i) ), perluasan in/x) mudah dievaluasi. Metode lain untuk pemfaktoraq j(x) = xn — 1 atas GF(q) adalah dengan inenggunakan operasi — operasi PPT (pthnbagi persekutuan terbesar) antara ./(x) dan g(x) dengan derajat g(x) < ..n 1 ..dan memenuhi [g(x)]' g(x)(mod f(x)). Selanjutnya W=FIPPT (f(x),g(i)- • • • • Let F is finite field with q elements and denoted by GF(q), a is a primitive element of GF(q) and a root of e - 1, then an= 1 and a is called a primitive nth root of unity. Hence f(x) = x" — 1 has primitive nth root of unity. Let f3 is a • primitive nth root of unity of GF(e)•,- mithrrial polynomial of 13 over GF(4)..i: Ax) and can be written in the form : p(x) =fi(x 18(/) where t is the smallest i.0 positive integer such that fle = . Then ny(x) is i factor of x" — I over GF(q). Using the Zech's 14 table .(1 + a = a'ti)), expanding mix) is easy to evaluate. • Another method for factoring f(x) = x" —1 over GF(q) is using appropriate gcd (the great common divisor) operations between .f(x) and g(x) where deg g(x) < n — 1 and satisfying [g(x)]1 g(x)(mod fix)). Then. f(x) = fl gcd (/(x), g(x) - 4 sEF •• • This document is Undip Institutional Repository Collection. The author xii copyright owner(s) changing the content, translate the submission to any medium or form,. the purpose of presi " rP r c) 1-1"t INIDIP-!R may keep more than one copy of this submission for purpose of secur httn'tinrHt'