Worst-Case Analysis of Webers Algorithm

Recently, Ken Weber introduced an algorithm for finding the $(a,b)$-pairs satisfying $au+bv\equiv 0\pmod{k}$, with $0<|a|,|b|<\sqrt{k}$, where $(u,k)$ and $(v,k)$ are coprime. It is based on Sorenson’s and Jebelean’s “$k$-ary reduction” algorithms. We provide a formula for $N(k)$, the maximal number of iterations in the loop of Weber’s GCD algorithm.

💡 Research Summary

The paper conducts a rigorous worst‑case analysis of the “Weber algorithm” (also called the Jebelian‑Weber algorithm, JW A), which finds integer pairs ((a,b)) satisfying (au + bv \equiv 0 \pmod{k}) with the size constraints (0<|a|,|b|<\sqrt{k}). The algorithm is a variant of the (k)-ary reduction techniques originally introduced by Sorenson and later refined by Jebelean. The authors focus on the maximal number of loop iterations, denoted (N(k)), that can occur for a given modulus (k).

Algorithmic setting.
The inputs are two coprime integers (u) and (v) together with a modulus (k>1). By computing the modular ratio (c \equiv u/v \pmod{k}) (with (0<c<k) and (\gcd(c,k)=1)), the algorithm can be expressed in terms of a pair of sequences ((n_i,d_i)) generated by the loop:

1. Initialise (f_1=(k,0)) and (f_2=(c,1)).
2. While (n_{i+1}\ge\sqrt{k}) replace (f_1) by (f_1-\lfloor n_i/n_{i+1}\rfloor f_2) and swap the two registers.

The invariant (n_i|d_{i+1}|+n_{i+1}|d_i|=k) holds for every iteration, mirroring the invariant of the extended Euclidean algorithm. The loop terminates when the new (n)-value drops below (\sqrt{k}); the number of performed iterations is denoted (t(k,c)). The worst‑case count for a fixed modulus is \