The Root Extraction Problem for Generic Braids

María Cumplido,Juan González-Meneses,Marithania Silvero

doi:10.3390/sym11111327

Abstract

We show that, generically, finding the k-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid x on n strands and canonical length l, and an integer k > 1 , computes a k-th root of x, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is O ( l ( l + n ) n 3 log n ) . The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee. This algorithm uses the fact that the ultra summit set of a braid is, generically, very small and symmetric (through conjugation by the Garside element Δ ), consisting of either a single orbit conjugated to itself by Δ or two orbits conjugated to each other by Δ .

Highlights

Group theory is ‘the language of symmetry’, as it is beautifully explained by Marcus du Sautoy in his book Symmetry
There are several computational problems in braid groups that have been proposed for their potential applications in cryptography [1]
The conjugacy problem in the braid group Bn was proposed as a non-commutative alternative to the discrete logarithm problem [2,3]

Summary

Introduction

Group theory is ‘the language of symmetry’, as it is beautifully explained by Marcus du Sautoy in his book Symmetry. While the future of braid-cryptography depends on finding a good key-generation procedure, there are some other problems in braid groups whose generic-case complexity is still to be studied. This is the case of the k-th root (extraction) problem. Symmetry 2019, 11, 1327 case, the algorithm in this paper shows that root extraction in braid groups is generically very fast, but can be used by those mathematicians needing a simple algorithm for finding a k-th root of a braid (or proving that it does not exist), which works in most cases. The generic-case complexity is O(l (l + n)n3 log n) (Theorem 6)

Garside Structure of Bn

Summit Sets

Generic Braids

An Algorithm to Find the k-th Root of a Braid