Hardness of Learning Problems over Burnside Groups of Exponent 3

In this work we investigate the hardness of a computational problem introduced in the recent work of Baumslag et al. In particular, we study the $B_n$-LHN problem, which is a generalized version of the learning with errors (LWE) problem, instantiated with a particular family of non-abelian groups (free Burnside groups of exponent 3). In our main result, we demonstrate a random self-reducibility property for $B_n$-LHN. Along the way, we also prove a sequence of lemmas regarding homomorphisms of free Burnside groups of exponent 3 that may be of independent interest

ALGORITHMIC PROBLEMS IN ENGEL GROUPS AND CRYPTOGRAPHIC APPLICATIONS

The theory of Engel groups plays an important role in group theory since they are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptog- raphy

Generalized Learning Problems and Applications to Non-commutative Cryptography

Abstract. We propose a generalization of the learning parity with noise (LPN) and learning with errors (LWE) problems to an abstract class of group-theoretic learning problems that we term learning homomorphisms with noise (LHN). This class of problems contains LPN and LWE as spe-cial cases, but is much more general. It allows, for example, instantiations based on non-abelian groups, resulting in a new avenue for the applica-tion of combinatorial group theory to the development of cryptographic primitives. We then study a particular instantiation using relatively free groups and construct a symmetric cryptosystem based upon it

Some applications of noncommutative groups and semigroups to information security

We present evidence why the Burnside groups of exponent 3 could be a good candidate for a platform group for the HKKS semidirect product key exchange protocol. We also explore hashing with matrices over SL2(Fp), and compute bounds on the girth of the Cayley graph of the subgroup of SL2(Fp) for specific generators A, B. We demonstrate that even without optimization, these hashes have comparable performance to hashes in the SHA family

О некоторых подгруппах бернсайдовой группы B0(2, 5)

Пусть B0(2, 5) = {x,y) — наибольшая конечная двупорождённая бернсайдова группа периода 5, порядок которой равен 534. В работе изучена серия подгрупп Hi = {ai,bi) группы Bo(2, 5), где ao = x, bo = y, ai = Oi-ibi-i и bi = bi-iOi-i-1 для i G N. Получено, что группа H4 является абелевой, поэтому H5 — циклическая группа, и серия подгрупп прерывается. Показано, что элементы = = xy^xyx^y^x^yxy^x и b4 = yx^yxy^x^y^xyx^y длины 16 порождают в Bo(2, 5) абелеву подгруппу порядка 25, и никакие другие два групповых слова, длины которых меньше 16, не порождают нециклическую абелеву подгруппу в Bo(2, 5). Let В0(2,5) = (x,y) be the largest finite two generator Burnside group of exponent five and order 534. We study a series of subgroups Hi = (ai,bi) of the group B0(2, 5), where a0 = x, b0 = y, ai = ai-ibi-i and bi = bi-iai-i for i E N. It has been found that H4 is a commutative group. Therefore, H5 is a cyclyc group and the series of subgroups is broken. The elements a4 = xy2xyx2y2x2yxy2x and b4 = yx2yxy2x2y2xyx2y of length 16 generate an abelian subgroup of order 25 in B0(2, 5). Using computer calculations, we have found that there is no other pair of group words of length less than 16 that generate a noncyclic abelian subgroup in B0(2, 5)

Smoothening Functions and the Homomorphism Learning Problem

This thesis is an exploration of certain algebraic and geometrical aspects of the Learning With Errors (LWE) problem introduced in Reg05. On the algebraic front, we view it as a Learning Homomorphisms with Noise problem, and provide a generic construction of a public-key cryptosystem based on this generalization. On the geometric front, we explore the importance of the Gaussian distribution for the existing relationships between LWE and lattice problems. We prove that their smoothing properties does not make them special, but rather, the fact that it is infinitely divisible and l2 symmetric are important properties that make the Gaussian unique

Accessing numeric data via flags and tags: A final report on a real world experiment

An experiment is reported which: extended the concepts of data flagging and tagging to the aerospace scientific and technical literature; generated experience with the assignment of data summaries and data terms by documentation specialists; and obtained real world assessments of data summaries and data terms in information products and services. Inclusion of data summaries and data terms improved users' understanding of referenced documents from a subject perspective as well as from a data perspective; furthermore, a radical shift in document ordering behavior occurred during the experiment toward proportionately more requests for data-summarized items

Rheology

This book contains a wealth of useful information on current rheology research. By covering a broad variety of rheology-related topics, this e-book is addressed to a wide spectrum of academic and applied researchers and scientists but it could also prove useful to industry specialists. The subject areas include, polymer gels, food rheology, drilling fluids and liquid crystals among others

Degree bounds for fields of rational invariants of Z/pZ\mathbb{Z}/p\mathbb{Z} and other finite groups

Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite group, focusing on abelian groups and especially the case of $\mathbb{Z}/p\mathbb{Z}$. The inquiry is motivated by an application to signal processing. We give new lower and upper bounds depending on the number of distinct nontrivial characters in the representation. We obtain additional detailed information in the case of two distinct nontrivial characters. We conjecture a sharper upper bound in the $\mathbb{Z}/p\mathbb{Z}$ case, and pose questions for further investigation.Comment: 39 pages, 1 tabl