Fast arithmetic in unramified p-adic fields

TL;DR

This paper presents new quasi-linear time algorithms for fundamental operations in unramified p-adic fields, significantly improving computational efficiency by combining existing methods with recent advances in polynomial modular composition.

Contribution

It introduces fast algorithms for operations in unramified p-adic fields that leverage recent polynomial composition techniques, achieving near-linear time complexity.

Findings

Algorithms operate in quasi-linear time in parameters n and N

Operations like Galois conjugation and Teichmuller lifting are significantly faster

The approach combines existing methods with recent polynomial composition techniques

Abstract

Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmuller lifting and computing minimal polynomials.