The key agreement protocol based on infinite non-commutative group presentation and representation levels is proposed. Two simultaneous problems in group representation level are used: the conjugator search problem (CSP) and modified discrete logarithm problem (DLP). The modified DLP in our approach is a matrix DLP and is different from that's used in other publications. The algorithm construction does not allow to perform a crypto-analysis by replacing the existing CSP solution to the decomposition problem (DP) solution. The group presentation level serves for two commuting subgroups and invertible group's word image matrix construction. The group representation level allows reliable factors disguising in the initial word. The word equivalence problem (WEP) solution is transformed from the group presentation level to the group representation level. Hence there are not necessary to solve WEP in the group presentation level and hence there are no restrictions on the group complexity in this sense. The construction of irreducible representation of group is required. The presented protocol is a modernization of protocol declared in (Sakalauskas et al., 2005).

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

A new enhanced matrix power function (MPF) is presented for the construction of cryptographic primitives. According to the definition in previously published papers, an MPF is an action of two matrices powering some base matrix on the left and right. The MPF inversion equations, corresponding to the MPF problem, are derived and have some structural similarity with classical multivariate quadratic (MQ) problem equations. Unlike the MQ problem, the MPF problem seems to be more complicated, since its equations are not defined over the field, but are represented as left–right action of two matrices defined over the infinite near-semiring on the matrix defined over the certain infinite, additive, noncommuting semigroup. The main results are the following: (1) the proposition of infinite, nonsymmetric, and noncommuting algebraic structures for the construction of the enhanced MPF, satisfying associativity conditions, which are necessary for cryptographic applications; (2) the proof that MPF inversion is polynomially equivalent to the solution of a certain kind of generalized multivariate quadratic (MQ) problem which can be reckoned as hard; (3) the estimation of the effectiveness of direct MPF value computation; and (4) the presentation of preliminary security analysis, the determination of the security parameter, and specification of its secure value. These results allow us to make a conjecture that enhanced MPF can be a candidate one-way function (OWF), since the effective (polynomial-time) inversion algorithm for it is not yet known. An example of the application of the proposed MPF for the Key Agreement Protocol (KAP) is presented. Since the direct MPF value is computed effectively, the proposed MPF is suitable for the realization of cryptographic protocols in devices with restricted computation resources.

In this paper we present a cryptographic primitive based on non-commutative cryptography. This primitive is used for key exchange protocol (KEP) construction. We prove that the security of this primitive relies on a nondeterministic polynomial complete (NP-Complete) decisional problem. Recently there are no known quantum cryptanalysis algorithms effectively solving NP-Complete problems. So far, KEPs are widely used in secure communication channel creation, e.g., in hypertext transfer protocol secure (https://) and are based on traditional cryptographic primitives representing commutative cryptography. However, the security of these protocols does not rely on NP-Complete problems and hence, according to P. W. Shorr, they are vulnerable to quantum cryptanalysis. We use one of seven non-commuting groups of order 16 which is not isomorphic to any other group to define a platform group for a key exchange protocol based on previously considered matrix power function (MPF). By investigating basic properties on the group ${\mathbf{M}}_{16}$ and their implementation for our goals we fix the order of actions in MPF from left to right. Furthermore, we define a special form of the base matrix and separate templates for left and right power matrices. Using properties of the specified templates and Schaeffer criteria we prove that the security of the proposed key exchange relies on an NP-Complete decisional problem.