Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem
Abstract
:1. Introduction
1.1. Early Days of Asymmetric Cryptography
1.2. NP-Complete Problems and Post-Quantum Cryptography
1.3. Our Previous Contributions and Novelty of This Paper
- By applying properties of we construct a executable KEP despite the fact that in general MPF defined over considered group is not associative;
- The constrains of private session parameters come naturally from predefined templates. These constrains are used to limit the choice of private session parameters to non-invertible matrices only, thus preventing any attempts of a linear algebra attack.
- The security of our proposed key exchange protocol is based on NP-Complete decisional problem and satisfies the generalized decisional Diffie–Hellman assumption. The proof of NP-Completeness of the considered problem is the main goal of this article.
1.4. Application of Our Protocol in Real Life
1.5. Organization of the Paper
2. Preliminaries
2.1. Description of the Modular Group of Order 16
2.2. Description of MPF and Its Basic Properties
3. Key Exchange Protocol
3.1. Definition of Publicly Known Data
3.2. Description of Our KEP
- Base matrix W defined over and having the structure (12);
- Power matrix L defined over and satisfying Template 1;
- Power matrix R defined over and satisfying Template 2.
- She chooses at random a vector of entries .
- Alice uses vector of scalars to calculate two matrices as polynomials of L and R respectively:
- She then uses the obtained values of X and Y to calculate matrix as follows:
- Bob generates a random vector of coefficients .
- He then uses these coefficients to calculate matrices U and V in a following way:
- He calculates matrix as follows:
- Alice calculates ;
- Bob calculates .
3.3. Proof of Validity of Our KEP
- If (or ), then no extra terms are needed;
- If (or ), then an extra term needs to be added.
4. Complexity of LRMPF Problem
4.1. Definition of the LRMPF Decisional Problem
4.2. Construction of an Homomorphism
4.3. Reduction of LRMPF Problem to Binary Matrix Multivariate Quadratic Problem
4.4. Proof of NP-Completeness of the LRMPF Decisional Problem
- (a)
- Every relation in S is satisfied when all the variables are 0 (0-valid clause);
- (b)
- Every relation in S is satisfied when all the variables are 1 (1-valid clause);
- (c)
- Every relation in S is definable by a CNF formula in which each conjunct has at most one negated variable (dual Horn clause);
- (d)
- Every relation in S is definable by a CNF formula in which each conjunct has at most one unnegated variable (Horn clause);
- (e)
- Every relation in S is definable by a CNF formula having at most two literals in each conjunct (bijunctive clause);
- (f)
- Every relation in S is the set of solutions of a system of linear equation over the two element field (affine clause).
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Diffie, W.; Hellman, M. New directions in cryptography. IEEE Trans. Inf. Theory 1976, 22, 644–654. [Google Scholar] [CrossRef] [Green Version]
- Koblitz, N. Elliptic curve cryptosystems. Math. Comput. 1987, 48, 203–209. [Google Scholar] [CrossRef]
- Shor, P.W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 1999, 41, 303–332. [Google Scholar] [CrossRef]
- Gawiejnowicz, S. -complete problems. In Models and Algorithms of Time-Dependent Scheduling; Springer: Berlin/Heidelberg, Germany, 2020; pp. 35–44. [Google Scholar]
- Rivest, R.L.; Shamir, A.; Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 1978, 21, 120–126. [Google Scholar] [CrossRef]
- Patarin, J. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms. In International Conference on the Theory and Applications of Cryptographic Techniques; Springer: Berlin/Heidelberg, Germany, 1996; pp. 33–48. [Google Scholar]
- Kipnis, A.; Shamir, A. Cryptanalysis of the HFE public key cryptosystem by relinearization. In Annual International Cryptology Conference; Springer: Berlin/Heidelberg, Germany, 1999; pp. 19–30. [Google Scholar]
- Courtois, N.T. The security of hidden field equations (HFE). In Cryptographers’ Track at the RSA Conference; Springer: Berlin/Heidelberg, Germany, 2001; pp. 266–281. [Google Scholar]
- Faugere, J.C.; Joux, A. Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases. In Annual International Cryptology Conference; Springer: Berlin/Heidelberg, Germany, 2003; pp. 44–60. [Google Scholar]
- Micciancio, D.; Regev, O. Lattice-based cryptography. In Post-Quantum Cryptography; Springer: Berlin/Heidelberg, Germany, 2009; pp. 147–191. [Google Scholar]
- Bindel, N.; Buchmann, J.; Krämer, J. Lattice-based signature schemes and their sensitivity to fault attacks. In Proceedings of the 2016 Workshop on Fault Diagnosis and Tolerance in Cryptography (FDTC), Santa Barbara, CA, USA, 16 August 2016; pp. 63–77. [Google Scholar]
- Ko, K.H.; Lee, S.J.; Cheon, J.H.; Han, J.W.; Kang, J.S.; Park, C. New public-key cryptosystem using braid groups. In Annual International Cryptology Conference; Springer: Berlin/Heidelberg, Germany, 2010; pp. 166–183. [Google Scholar]
- Anshel, I.; Anshel, M.; Goldfeld, D. An algebraic method for public-key cryptography. Math. Res. Lett. 1999, 6, 287–292. [Google Scholar] [CrossRef]
- Shpilrain, V.; Ushakov, A. The conjugacy search problem in public key cryptography: Unnecessary and insufficient. Appl. Algebra Eng. Commun. Comput. 2006, 17, 285–289. [Google Scholar] [CrossRef] [Green Version]
- Sakalauskas, E.; Listopadskis, N.; Tvarijonas, P. Key agreement protocol (KAP) based on matrix power function. In Advanced Studies in Software and Knowledge Engineering; Information Science and Computing; Institute of Information Theories and Applications FOI ITHEA: Sofia, Bulgaria, 2008; pp. 92–96. [Google Scholar]
- Sakalauskas, E.; Luksys, K. Matrix power function and its application to block cipher s-box construction. Int. J. Inn. Comp. Inf. Contr. 2012, 8, 2655–2664. [Google Scholar]
- Mihalkovich, A.; Sakalauskas, E. Asymmetric cipher based on MPF and its security parameters evaluation. Proc. Lith. Math. Soc. Ser. A 2012, 53, 72–77. [Google Scholar]
- Mihalkovich, A.; Sakalauskas, E.; Venckauskas, A. New asymmetric cipher based on matrix power function and its implementation in microprocessors efficiency investigation. Elektronika ir Elektrotechnika 2013, 19, 119–122. [Google Scholar] [CrossRef]
- Sakalauskas, E.; Mihalkovich, A. New asymmetric cipher of non-commuting cryptography class based on matrix power function. Informatica 2014, 25, 283–298. [Google Scholar] [CrossRef] [Green Version]
- Liu, J.; Zhang, H.; Jia, J. A linear algebra attack on the non-commuting cryptography class based on matrix power function. In International Conference on Information Security and Cryptology; Springer: Cham, Switzerland, 2016; pp. 343–354. [Google Scholar]
- Sakalauskas, E.; Mihalkovich, A. Improved Asymmetric Cipher Based on Matrix Power Function Resistant to Linear Algebra Attack. Informatica 2017, 28, 517–524. [Google Scholar] [CrossRef] [Green Version]
- Mihalkovich, A.; Levinskas, M. Investigation of Matrix Power Asymmetric Cipher Resistant to Linear Algebra Attack. In International Conference on Information and Software Technologies 2019; Springer: Cham, Switzerland, 2019; pp. 197–208. [Google Scholar]
- Sakalauskas, E. Enhanced matrix power function for cryptographic primitive construction. Symmetry 2018, 10, 43. [Google Scholar] [CrossRef] [Green Version]
- Sakalauskas, E.; Mihalkovich, A. MPF Problem over Modified Medial Semigroup Is NP-Complete. Symmetry 2018, 10, 571. [Google Scholar] [CrossRef] [Green Version]
- Mihalkovich, A. On the associativity property of MPF over M16. Proc. Lith. Math. Soc. Ser. A 2018, 59, 7–12. [Google Scholar] [CrossRef]
- Grundman, H.; Smith, T. Automatic realizability of Galois groups of order 16. Proc. Am. Math. Soc. 1996, 124, 2631–2640. [Google Scholar] [CrossRef] [Green Version]
- Grundman, H.G.; Smith, T.L.; Swallow, J.R. Groups of order 16 as Galois groups. Expo. Math 1995, 13, 289–319. [Google Scholar]
- Inassaridze, N.; Kandelaki, T.; Ladra, M. Categorical interpretations of some key agreement protocols. J. Math. Sci. 2013, 195, 439–444. [Google Scholar] [CrossRef] [Green Version]
- Garey, M.R.; Johnson, D.S. Computers and Intractability; Freeman: San Francisco, CA, USA, 1979; Volume 74. [Google Scholar]
- Schaefer, T.J. The complexity of satisfiability problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA, 1–3 May 1978; ACM: New York, NY, USA, 1978; pp. 216–226. [Google Scholar]
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Mihalkovich, A.; Sakalauskas, E.; Luksys, K. Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem. Symmetry 2020, 12, 1389. https://doi.org/10.3390/sym12091389
Mihalkovich A, Sakalauskas E, Luksys K. Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem. Symmetry. 2020; 12(9):1389. https://doi.org/10.3390/sym12091389
Chicago/Turabian StyleMihalkovich, Aleksejus, Eligijus Sakalauskas, and Kestutis Luksys. 2020. "Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem" Symmetry 12, no. 9: 1389. https://doi.org/10.3390/sym12091389
APA StyleMihalkovich, A., Sakalauskas, E., & Luksys, K. (2020). Key Exchange Protocol Defined over a Non-Commuting Group Based on an NP-Complete Decisional Problem. Symmetry, 12(9), 1389. https://doi.org/10.3390/sym12091389