Acceleration of Euclidean Algorithm and Rational Number Reconstruction

Acceleration of Euclidean Algorithm and Rational Number Reconstruction

We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexity bound for the integer gcd, which our algorithm computes as a special case.

Eecient Rational Number Reconstruction

The paper presents an eecient algorithm for reconstructing a rational number from its residue modulo a given integer. The algorithm is based on a double-digit version of Lehmer's multi-precision extended Euclidean algorithm. While asymptotic complexity remains quadratic in the length of the input, experiments with an implementation show that for small inputs the new algorithm is more than 3 tim...

Efficient Rational Number Reconstruction

Vector Rational Number Reconstruction

The final step of some algebraic algorithms is to reconstruct the common denominator d of a collection of rational numbers (ni/d)1≤i≤n from their images (ai)1≤i≤n modulo M , subject to the condition 0 < d ≤ N and |ni| ≤ N for a given magnitude bound N . Using elementwise rational reconstruction requires that M > 2N2. We present an algorithm, based on lattice basis reduction, which can perform t...

The Euclidean Algorithm in Cubic Number Fields

In this note we present algorithms for computing Euclidean minima of cubic number fields; in particular, we were able to find all normEuclidean cubic number fields with discriminants −999 < d < 104.