Dynamical Analysis of the Parametrized Lehmer-Euclid Algorithm

The Lehmer–Euclid Algorithm is an improvement of the Euclid Algorithm when applied to large integers. The original Lehmer–Euclid Algorithm replaces divisions on multi-precision integers by divisions on single-precision integers. Here we study a slightly different algorithm that replaces computations on n-bit integers by computations on μn-bit integers. This algorithm depends on the truncation degree μ ∈]0, 1[ and is denoted as the LEμ algorithm. The original Lehmer–Euclid Algorithm can be viewed as the limit of the LEμ algorithms for μ → 0. We provide here a precise analysis of the LEμ algorithm. For this purpose, we are led to study what we call the Interrupted Euclid Algorithm . This algorithm depends on some parameter α ∈ [0, 1] and is denoted by Eα. When running with an input (a, b), it performs the same steps as the usual Euclid Algorithm, but it stops as soon as the current integer is smaller than a, so that E0 is the classical Euclid Algorithm. We obtain a very precise analysis of the algorithm Eα, and describe the behaviour of main parameters (number of iterations, bit complexity) as a function of parameter α. Since the Lehmer– Euclid Algorithm LEμ when running on n-bit integers can be viewed as a sequence of executions of the Interrupted Euclid Algorithm E1/2 on μn-bit integers, we then come back to the analysis of the LEμ algorithm and obtain our results.