Calculating the Power Residue Symbol and Ibeta

Abstract

In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number eld completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications.e rst application is the computation of an important invariant of number eld completions, called ibeta. e algorithm that computes ibeta is deterministic and runs in polynomial time.e second application uses Hilbert norm residue symbols and yields a probabilistic algorithm that computes the m-th power residue symbol α b m in arbitrary number elds K.is probabilistic algorithm relies on LLL-reduction and sampling of nearprimes. Using heuristics, we analyse its complexity to be polynomial expected time in n = [K : Q], log |∆ K | and the input bit size -a tentative conclusion corroborated by timing experiments. An implementation of the algorithm in Magma will be available at h ps://github.com/kodebro/powerresiduesymbol.

Random Self-reducibility of Ideal-SVP via Arakelov Random Walks

“…One such application is the object of another work in progress. Namely, we note that many algorithms [5,4,8] rely on nding elements a in an ideal I such that aI −1 is easy to factor (e.g. prime, near-prime, or B-smooth).…”

Jump sets in local fields

“…The pair pI A , β A q satisfies the following three conditions. pC.1q 1 One has that I A Ď T ρ (resp. I A Ď T ρ ), pC.2q 1 the map β A is a strictly decreasing map β : I A Ñ Z ě1 , pC.3q 1 the map i Þ Ñ ρ βpiq piq from I A to Z ě1 is strictly increasing.…”

“…Another instance of this phenomenon will emerge at the end of this sub-section in Theorem 1.13, in the context of Eisenstein polynomials. We mention that this way of computing filt-ord is used in [1] where, among other things, algorithmic problems of this subject are explored.…”

“…Moreover in that Section we provide a procedure that allows one to compute pI gpxq , β gpxq q very quickly, even by hand. See [1] for an actual implementation of this as well.…”

“…Clearly J K is determined by the structure of U 1 pKq as a filtered Z p -module. So one can use the answer to question (1) in order to answer question (3). The first step is answering the same problem for free filtered modules.…”