Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem

Abstract

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 M16 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.

Sigma Identification Protocol Construction Based on MPF

“…This assumption can be supported by our previous results presented in [11][12][13][14]. The MPF is constructed using similar algebraic structures as in [14].…”

“…MPF, presented here, has some similarity to the certain MPF problem which was proven to be NP-complete in our previous publications [12,13]. It is believed so far that NP-complete problems cannot be effectively solved by quantum computers.…”

“…The MPF is constructed using similar algebraic structures as in [14]. In [12,13] the NPcompleteness of the similar MPF problem is proven. MPF function introduced there is defined in finite modified medial platform semigroup and finite power near-semiring.…”

Matrix Power Function Based Block Cipher Operating in CBC Mode

“…The main goal is to propose a symmetric cipher based on the conjectured one-way function (OWF). During our previous research, we proved that a certain realization of asymmetric encryption based on our function is NP-complete [13]. It is conjectured that such cryptographic primitives could be resistant to algebraic analysis and quantum cryptanalysis.…”

PURA-SCIS Protocol: A Novel Solution for Cloud-Based Information Sharing Protection for Sectoral Organizations

“…The data exchange protocol, such as keys between entities with accountable optimistic fair exchange protocol, was also utilized in [14]. Data exchange protocols (e.g., cryptographic keys) are protected by other methods, including Non-Commuting Group based on an NP-complete decisional problem as described in [15] and Multi-Stage Quantum in [16]. However, the previous studies do not validate the security requirements of their research by utilizing the formal security validation tools (e.g., Scyther, AVISPA, Tamarin Proofer).…”