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Unformatted text preview: 202: Lecture Notes GARSIA April 6, 2010 1 . Computing GCDs 1. The Berlekamp Algorithm The Euclidean algorithm is the process which yields the greatest common divisor d of two given integers A and B or more generally the GCD of any two elements of a Euclidean ring. It is shown by Berlekamp ( * ) that d may be computed by the following modified version of the Euclidean algorithm. We start by setting r- 2 = A , r- 1 = B , p- 2 = 0 , p- 1 = 1 , q- 2 = 1 , q- 1 = 0 1 . 1 then compute a k ,r k ,p k ,q k according to the recursions (**) ( i ) r k- 2 = a k r k- 1 + r k ( integer division ) ( ii ) p k = a k p k- 1 + p k- 2 ( iii ) q k = a k q k- 1 + q k- 2 . ( k = 0 , 1 , 2 ,... ) 1 . 2 since r k decreases at least by one at each step after n < degree ( B ) steps we shall have r n = 0. we will show that these recursions force the following basic identities ( a ) q n p n- 1- p n q n- 1 = (- 1) n ( b ) A = r n- 1 p n ( c ) B = r n- 1 q n . 1 . 3 Multiplying the first of these equations by r n- 1 and using 1.3 (b) and 1.3 (c) we derive that B p n- 1- A q n- 1 = (- 1) n r n- 1 1...
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