Related papers: Algorithmic problems in Engel groups and cryptogra…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…
Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them.…
In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our…
We propose a new cryptosystem based on polycyclic groups. The cryptosystem is based on the fact that the word problem can be solved effectively in polycyclic groups, while the known solutions to the conjugacy problem are far less efficient.
The secure instantiation of the random oracle is one of the major open problems in modern cryptography. We investigate this problem using concepts and methods of algorithmic randomness. In modern cryptography, the random oracle model is…
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate…
In 1991 the first public key protocol involving automaton groups has been proposed. In this paper we give a survey about algorithmic problems around automaton groups which may have potential applications in cryptography. We then present a…
This paper investigates the Terwilliger algebra of some group association schemes related to codes. In addition, it also shows the generators of invariant rings appearing by E-polynomials.
We give an exposition of the hidden subgroup problem for dihedral groups from the point of view of the standard hidden subgroup quantum algorithm for finite groups. In particular, we recall the obstructions for strong Fourier sampling to…
We give a survey of results on the structure of right and left Engel elements of a group. We also present some new results in this topic.
Several cryptographic protocols constructed based on less-known algorithmic problems, such as those in non-commutative groups, group rings, semigroups, etc., which claim quantum security, have been broken through classical reduction methods…
We address a cryptanalysis of two protocols based on the supposed difficulty of discrete logarithm problem on (semi) groups of matrices over a group ring. We can find the secret key and break entirely the protocols.
Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The…
After some excitement generated by recently suggested public key exchange protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of…
We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional…
For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
General cryptographic schemes are presented where keys can be one-time or ephemeral. Processes for key exchange are derived. Public key cryptographic schemes based on the new systems are easily established. Authentication and signature…
Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in…
As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…