On the Complexity of Computing Units in a Number Field

Abstract

Given an algebraic number field K, such that [K : Q] is constant, we show that the problem of computing the units group O * K is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in OK is in SPP. Furthermore, assuming the GRH, the class number of K, and a presentation for the class group of K can also be computed in SPP. A corollary of our result is that solving PELL S EQUATION, recently shown by Hallgren [12] to have a quantum polynomial-time algorithm, is also in SPP.

Quantum Algorithms for One-Dimensional Infrastructures

“…The incorrectness in h cannot be avoided if the evaluation points r are chosen arbitrarily. As already stated in Lemma 8, we assume that the evaluation points are of the form k/L for some large integer L and bounded k. Even so, we cannot evaluate always h correctly for every k. Therefore, we further restrict the evaluation of h to a subset of the points which are 1 N uniformly spaced along a bounded interval, where N divides L. We choose N ≥ ⌈2/d min ⌉ so that there are at least two evaluation points i N and i+1 N between any two adjacent elements of I. This is shown in the figure below.…”

“…The (constant) factor δ := ǫ h − ηζ in front of a is less than 3/4 in absolute value (ǫ h ≤ 1 2 , ζ < 1 and η < 1 2 ). Assume we measure k ≤ ⌊B/64⌋ − 1.…”

“…Schmidt and Vollmer [20,21] and Hallgren [12] presented a polynomial time quantum algorithm for determining a hidden lattice in R n for fixed n. They showed that computing the unit group and solving the principal ideal problem in number fields of fixed unit rank can be solved efficiently with this algorithm. 1 In stark contrast to Z n , the success probability of the above quantum algorithms for finding a hidden lattice in R n decreases exponentially with the dimension, making them inefficient with respect to the dimension. Thus, an important open problem is to determine whether there exist quantum algorithms whose success probability decrease less rapidly with the dimension.…”

Ρητές Καμπύλες Επί Αλγεβρικών Σωμάτων Αριθμών Και Διοφαντικές Εξισώσεις

“…Με την ίδια υπόθεση µπορούµε να ϐρούµε κάποιους υποεκθετικούς στοχαστικούς αλγορίθµους στα [1,52]. Στο [5] αποδεικνύεται ότι το πρόβληµα επίλυσης της εξίσωσης Pell ανήκει στην τάξη πολυπλοκότητας SPP. Επιπλέον, στο [19] περιγράφεται ένας κβαντικός αλγόριθµος που τρέχει σε πολυωνυµικό χρόνο.…”