The celebrated LKS algorithm by Lehmer, Knuth, and Schönhage, combined with the product tree technique, enables an equivalently rapid alternative to our recent modification of the extended Euclidean algorithm for the reconstruction of a rational number from its modular as well as numerical approximations.

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.

We accelerate the extended Euclidean algorithm for integers, the rational number reconstruction, and consequently, the stage of the recovery of the solution of a nonsingular integer system of linear equations via Hensel's lifting. The acceleration is by the order of magnitude and yields nearly optimal randomized algorithms. In the highly important case of Toeplitz, Hankel, and Toeplitz/Hankel-like linear systems, the accleration is potentially practical.