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.
New asymmetric cipher based on matrix power function is presented. Cipher belongs to the class of recently intensively evolving non-commuting cryptography due to expectation of its resistance to potential quantum cryptanalysis. The algebraic structures for proposed cipher construction are defined. Security analysis was performed and security parameters are defined. On the base of this research the secure parameters values are determined. The comparison of efficiency of microprocessor realization of proposed algorithm with different security parameters values is presented.
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.
Abstract:The improved version of the author's previously declared asymmetric cipher protocol based on matrix power function (MPF) is presented. Proposed modification avoids discrete logarithm attack (DLA) which could be applied to the previously declared protocol. This attack allows us to transform the initial system of MPF equations to so-called matrix multivariate quadratic (MMQ) system of equations, which is a system representing a subclass of multivariate quadratic (MQ) systems of equations. We are making a conjecture that avoidance of DLA in protocol, presented here, should increase its security, since an attempt to solve the initial system of MPF equations would appear to be no less complex than solving the system of MMQ equations. No algorithms are known to solve such a system of equations. Security parameters and their secure values are defined. Security analysis against chosen plaintext attack (CPA) and chosen ciphertext attack (CCA) is presented. Measures taken to prevent DLA attack increase the security of this protocol with respect to the previously declated protocol.
The objective of this paper is to find suitable non-commuting algebraic structure to be used as a platform structure in the so-called matrix power function (MPF). We think it is non-trivial and interesting problem could be useful for candidate one-way function (OWF) construction with application in cryptography. Since the cornerstone of OWF construction using non-commuting algebraic structures is the satisfiability of certain associativity conditions, we consider one of the possible choices, i.e. the group M16, explore its basic properties and construct templates to use in our future work.
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