Jebelean-Weber's algorithm without spurious factors

Tudor Jebelean and Ken Weber introduced an algorithm for finding (a,b)-pairs satisfying au+bv≡0 (mod k), with . It is based on Sorenson's “k-ary reduction”. This algorithm does not preserve the GCD and its related GCD algorithm has an O(n²) time bit complexity in the worst case. We present a modified version which avoids this problem. We show that a slightly modified GCD algorithm has an O(n²/logn) running time in the worst case, where n is the number of bits of the larger input.