Generating random braids

We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number of strands.Australian Research Council’s Discovery ProjectsMinisterio de Ciencia e InnovaciónJunta de AndalucíaFondo Europeo de Desarrollo Regiona

Computing growth functions of braid monoids and counting vertex-labelled bipartite graphs

We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$ with respect to the simple elements (permutation braids) as generators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we use matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of $n$.Comment: reference adde

Machine learning discovers invariants of braids and flat braids

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new convenient invariants of braids, including a complete invariant of flat braids.Comment: 24 page

Probabilidades de transición en diagramas de nudos

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Carles CasacubertaSome enzymes like condensins or type II topoisomerases are capable of changing the topology of cyclic DNA molecules. The first ones introduce positive superhelical tension into double-stranded DNA and the second ones perform strand passages that cut both strands of one DNA double helix, pass another unbroken DNA helix through it, and then religate the cut strands. Motivated by the action of topoisomerases in knotted DNA, we study transition probabilities between knot diagrams and compute stationary distributions of the strand-passage Markov process. First we give a brief introduction to knot theory, braid theory and random processes, particularly to Markov chains. After this, we propose a new model for strand passage in knot diagrams based on Artin’s braid group. This braid-based model allows us to generate a large set of braid diagrams with which we are able to simulate the action of type II topoisomerases

Towards Generating Secure Keys for Braid Cryptography

Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this paper, we first survey all the major attacks on the braid cryptosystem and conclude that the attacks were successful because the current ways of random key generation almost always result in weaker instances of the conjugacy problem. We then propose several alternate ways of generating hard instances of the conjugacy problem for use braid cryptography

Factoring Products of Braids via Garside Normal Form

Braid groups are infinite non-abelian groups naturally arising from geometric braids. For two decades they have been proposed for cryptographic use. In braid group cryptography public braids often contain secret braids as factors and it is hoped that rewriting the product of braid words hides individual factors. We provide experimental evidence that this is in general not the case and argue that under certain conditions parts of the Garside normal form of factors can be found in the Garside normal form of their product. This observation can be exploited to decompose products of braids of the form ABC when only B is known. Our decomposition algorithm yields a universal forgery attack on WalnutDSA™, which is one of the 20 proposed signature schemes that are being considered by NIST for standardization of quantum-resistant public-key cryptography. Our attack on WalnutDSA™ can universally forge signatures within seconds for both the 128-bit and 256-bit security level, given one random message-signature pair. The attack worked on 99.8% and 100% of signatures for the 128-bit and 256-bit security levels in our experiments. Furthermore, we show that the decomposition algorithm can be used to solve instances of the conjugacy search problem and decomposition search problem in braid groups. These problems are at the heart of other cryptographic schemes based on braid groups.SCOPUS: cp.kinfo:eu-repo/semantics/published22nd IACR International Conference on Practice and Theory of Public-Key Cryptography, PKC 2019; Beijing; China; 14 April 2019 through 17 April 2019ISBN: 978-303017258-9Volume Editors: Sako K.Lin D.Publisher: Springer Verla