Public Key Cryptography based on Semigroup Actions

A generalization of the original Diffie-Hellman key exchange in $(\Z/p\Z)^*$ found a new depth when Miller and Koblitz suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general. In Section 2 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.Comment: 20 pages. To appear in Advances in Mathematics of Communication

Sound and complete axiomatizations of coalgebraic language equivalence

Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

Asymmetric Cipher Protocol Using Decomposition Problem

The asymmetric cipher protocol based on decomposition problem in matrix semiring M over semiring of natural numbers N is presented. The security parameters are defined and preliminary security analysis is presented

Key Agreement Protocol (KAP) Based on Matrix Power Function

* Work is partially supported by the Lithuanian State Science and Studies Foundation.The key agreement protocol (KAP) is constructed using matrix power functions. These functions are based on matrix ring action on some matrix set. Matrix power functions have some indications as being a one- way function since they are linked with certain generalized satisfiability problems which are potentially NP- Complete. A working example of KAP with guaranteed brute force attack prevention is presented for certain algebraic structures. The main advantage of proposed KAP is considerable fast computations and avoidance of arithmetic operations with long integers

Key Agreement Protocol Using Elliptic Curve Matrix Power Function

* Work is partially supported by the Lithuanian State Science and Studies Foundation.The key agreement protocol (KAP) using elliptic curve matrix power function is presented. This function pretends be a one-way function since its inversion is related with bilinear equation solution over elliptic curve group. The matrix of elliptic curve points is multiplied from left and right by two matrices with entries in Zn. Some preliminary security considerations are presented

Matrix Power S-box Analysis

* Work supported by the Lithuanian State Science and Studies Foundation.Construction of symmetric cipher S-box based on matrix power function and dependant on key is analyzed. The matrix consisting of plain data bit strings is combined with three round key matrices using arithmetical addition and exponent operations. The matrix power means the matrix powered by other matrix. This operation is linked with two sound one-way functions: the discrete logarithm problem and decomposition problem. The latter is used in the infinite non-commutative group based public key cryptosystems. The mathematical description of proposed S-box in its nature possesses a good “confusion and diffusion” properties and contains variables “of a complex type” as was formulated by Shannon. Core properties of matrix power operation are formulated and proven. Some preliminary cryptographic characteristics of constructed S-box are calculated

Rakto apsikeitimo protokolas begalinės pusgrupės įvaizdžio lygmenyje

Matrix decomposition problem over integer ring is presented. Solving methods are discussed and it is showed, that this problem is hard computational problem regard to computer memory resources. A key agreement protocol based on matrix decomposition problem is presented