Braid Group Cryptography

Abstract

In the last decade, a number of public key cryptosystems based on combinatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them.This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of cryptosystems which are based on other non-commutative groups).

The Cracking of WalnutDSA: A Survey

“…Unfortunately, many problems were brought to light after a thorough scrutiny carried out by pure mathematicians and cryptographers. In this section, we briefly review two of the most prominent proposals within this area and refer the interested reader to the survey on the topic by David Garber [5].…”

“…Let us focus on how to solve CSP and CDP. As explained in detail in [5], the basic idea that has proven more fruitful towards a solution for the CSP and CDP problems involves a set I x for each braid x (typically a subset of the conjugacy class of A), which characterizes the conjugacy class (i.e., A and B are conjugates if and only if I A = I B ). Furthermore, there should be an efficient algorithm to compute a representative ∈ I A and a witness X ∈ B n , such that X −1 AX =Â.…”

“…However, not only full theoretical solutions for the conjugacy problems have been of interest in the cryptographic context; indeed, heuristic algorithms with a significant success rate suffice to thwart the security of a scheme that is based on one of the above problems (we refer again to [5] for details). As a consequence, all cryptographic proposals built around the above problems are currently considered problematic.…”

Double Cosets for and Outer Automorphisms of Free Groups

“…In particular, we get the map Br k → Out(F k ). It is not injective (see, e.g., [8]), the kernel is generated by…”

Factoring Products of Braids via Garside Normal Form

“…Current approaches to attain quantum-resistance include cryptography based on codes, isogenies, lattices and multivariate polynomials over finite fields [19,36,38,44]. Another approach are cryptographic systems based on non-abelian groups [22]. Indeed no quantum algorithm to solve the hidden subgroup problem (the core problem solved by Shor's algorithm for finite abelian groups) is known for general non-abelian groups.…”