EA-example

# EA-example - A Euclidean Algorithm example We now calculate...

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A Euclidean Algorithm example We now calculate gcd( z 6 , 9 z 5 + 8 z 4 + 2 z 3 + 7 z 2 + 6) = 1 over F 11 using the Euclidean Algorithm. At Step i we deﬁne q i ( z ), r i ( z ), s i ( z ), and t i ( z ) using r i - 2 ( z ) = q i ( z ) r i - 1 ( z ) + r i ( z ) s i ( z ) = s i - 2 ( z ) - q i ( z ) s i - 1 ( z ) t i ( z ) = t i - 2 ( z ) - q i ( z ) t i - 1 ( z ) . Step i q i ( z ) r i ( z ) s i ( z ) t i ( z ) - 1 - z 6 1 0 0 - 9 z 5 + 8 z 4 + 2 z 3 + 7 z 2 + 6 0 1 1 5 z + 9 6 z 4 + 2 z 3 + 3 z 2 + 3 z + 1 1 6 z + 2 2 7 z + 10 5 z 3 + 7 z + 7 4 z + 1 2 z 2 + 3 z + 3 3 10 z + 7 10z 2 + 5z + 7 4 z 2 + 6 z + 5 2z 3 + 10z + 3 4 6 z + 8 2 z + 6 9 z 3 + 9 z 2 + 10 z 4 + 6 z 3 + 8 z 2 + +3 z + 5 +4 z + 1 5 5 z + 4 5 10 z 4 + 7 z 3 + 8 z 2 + 5 z 5 + 7 z 4 + 4 z 3 + +2 z + 7 +3 z 2 + 10 6 7 z + 10 0 - - Thus 5 = (10 z 4 +7 z 3 +8 z 2 +2 z +7) z 6 +(5 z 5 +7 z 4 +4 z 3 +3 z 2 +10)(9 z 5 +8 z 4 +2 z 3 +7 z 2 +6) and, after dividing by 5 (that is, multiplying by 5 - 1 = 9), we have 1 = gcd( z 6 , 9 z 5 + 8 z 4 + 2 z 3

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EA-example - A Euclidean Algorithm example We now calculate...

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