Related papers: Braid Group Cryptography
This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups…
The paper analyzes a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system…
This article addresses code-based cryptography and is designed to depict the complete outline of a code based public key cryptosystem. This report includes basic mathematics and fundamentals of coding theory which are useful for studying…
We present a novel approach to gaining insight into the structure of cryptographic migration problems which are classic problems in applied cryptography. We use a formal model to capture the inherent dependencies and complexities of such…
Conjugacy is not the only possible primitive for designing braid-based protocols. To illustrate this principle, we describe a Fiat--Shamir-style authentication protocol that be can be implemented using any binary operation that satisfies…
The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make…
We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,\xi_std), we associate a corresponding multi-pointed nice…
In recent years cyber-attacks are continuously developing. This means that hackers can find their way around the traditional cryptosystems. This calls for new and more secure cryptosystems to take their place. This paper outlines a new…
We study an efficient algorithm to hash any single qubit gate (or unitary matrix) into a braid of Fibonacci anyons represented by a product of icosahedral group elements. By representing the group elements by braid segments of different…
Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory…
Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of…
We propose modifications of known quasigroup based stream ciphers. Systems of orthogonal n-ary groupoids are used.
Recently, Pareek et al. proposed a symmetric key block cipher using multiple one-dimensional chaotic maps. This paper reports some new findings on the security problems of this kind of chaotic cipher: 1) a number of weak keys exists; 2)…
In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…
We suggest the usage of algebraic subsets instead of subgroups in public-key cryptography. In particular, we present the subset version of two protocols introduced by Shpilrain and Ushakov with some examples in ascending HNN-extensions of…
We consider an application to the discrete log problem using completely regular semigroups which may provide a more secure symmetric cryptosystem than the classic system based on groups. In particular we describe a scheme that would appear…
Although recent advances of blockchain systems, notably in the form of cryptocurrency, have drawn tremendous interests from both researchers and practitioners, limited studies existed toward the theoretical foundation of blockchains. This…
The conjugacy problem in braid groups has been extensively studied, particularly from an algorithmic perspective. Established methods based on Garside structures, such as initial summit sets and super summit sets, provide effective…
In this paper, we obtain Groebner-Shirshov (non-commutative Gr\"obner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word" $\delta$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form…
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive…