Volume 12, no. 1Pages 66 - 81
Finite Non-Commutative Associative Algebras as Carriers of Hidden Discrete Logarithm ProblemN.A. Moldovyan, A.A. Moldovyan
The article introduces new finite algebras attractive as carriers of the discrete logarithm problem in a hidden group. In particular new 4-dimensional and 6-dimensional finite non-commutative algebras with associative multiplication operation and their properties are described. It is also proposed a general method for defining finite non-commutative associative algebras of arbitrary even dimension mge 2. Some of the considered algebras contain a global unit, but the other ones include no global unit element. In the last case the elements of the algebra are invertible locally relatively local bi-side units that act in the frame of some subsets of elements of algebra. For algebras of the last type there have been derived formulas describing the sets of the (right-side, left-side, and bi-side) local units. Algebras containing a large set of the global single-side (left-side and right-side) units and no global bi-side unit are also introduced. Since the known form of defining the hidden discrete logarithm problem uses invertibility of the elements of algebra relatively global unit, there are introduced new forms of defining this computationally difficult problem. The results of the article can be applied for designing public-key cryptographic algorithms and protocols, including the post-quantum ones. For the first time it is proposed a digital signature scheme based on the hidden discrete logarithm problem. Full text
- finite associative algebra; non-commutative algebra; global unit; left-side units; local unit; local invertibility; discrete logarithm problem; public-key cryptoscheme; digital signature; post-quantum cryptography.
- 1. Sirwan A., Majeed N. New Algorithm for Wireless Network Communication Security. International Journal on Cryptography and Information Security, 2016, vol. 6, no. 3, pp. 1-8.
2. Yiteng Feng, Guomin Yang, Joseph K.Liu. A New Public Remote Integrity Checking Scheme with User and Data Privacy. International Journal of Applied Cryptography, 2017, vol. 3, no. 3, pp. 196-209. DOI: 10.1504/IJACT.2017.086232
3. Chiou S.Y. Novel Digital Signature Schemes Based on Factoring and Discrete Logarithms. International Journal of Security and Its Applications, 2016, vol. 10, no. 3, pp. 295-310. DOI: 10.14257/ijsia.2016.10.3.26
4. Yan S.Y. Quantum Computational Number Theory. N.Y., Springer, 2015. DOI: 10.1007/978-3-319-25823-2
5. Yan S.Y. Quantum Attacks on Public-Key Cryptosystems. N.Y., Springer, 2014.
6. Proceedings of the 7th International Workshop on Post-Quantum Cryptography, PQCrypto 2016. Fukuoka, Springer, 2016.
7. Post-Quantum Cryptography. 9th International Conference, PQCrypto 2018. Fort Lauderdale, Springer, 2018.
8. Hiranvanichakorn P. Provably Authenticated Group Key Agreement based on Braid Groups. The Dynamic Case. International Journal of Network Security, 2017, vol. 19, no. 4, pp. 517-527.
9. Verma G.K. Probable Security Proof of a Blind Signature Scheme over Braid Groups. International Journal of Network Security, 2011, vol. 1, no. 2, pp. 118-120.
10. Myasnikov A., Shpilrain V., Ushakov A. A Practical Attack on a Braid Group Based Cryptographic Protocol. 2005. Springer, vol. 3621, pp. 86-96.
11. Moldovyan D.N., Moldovyan N.A. A New Hard Problem over Non-Commutative Finite Groups for Cryptographic Protocols. Conference on Mathematical Methods, Models and Architectures for Computer Network Security. 2010, Springer, vol. 6258, pp. 183-194. DOI: 10.1007/978-3-642-14706-7_14
12. Sakalauskas E., Tvarijonas P., Raulynaitis A. Key Agreement Protocol Using Conjugacy and Discrete Logarithm Problems in Group Representation Level. Informatica, 2007, vol. 18, no. 1, pp. 115-124.
13. Moldovyan D.N. Non-Commutative Finite Groups as Primitive of Public-Key Cryptoschemes. Quasigroups and Related Systems. 2010, vol. 18, no. 2, pp. 165-176.
14. Moldovyan D.N., Moldovyan N.A. Cryptoschemes Over Hidden Conjugacy Search Problem and Attacks Using Homomorphisms. Quasigroups Related Systems, 2010, vol. 18, no. 2, pp. 177-186.
15. Kuzmin A.S., Markov V.T., Mikhalev A.A., Mikhalev A.V., Nechaev A.A. Cryptographic Algorithms on Groups and Algebras. Journal of Mathematical Sciences, 2017, vol. 223, no. 5, pp. 629-641. DOI: 10.1007/s10958-017-3371-y
16. Moldovyan A.A., Moldovyan N.A., Shcherbacov V.A. Non-Commutative Finite Associative Algebras of 2-Dimension Vectors. Computer Science Journal of Moldova, 2017, vol. 25, no. 3, pp. 344-356.
17. Moldovyan D.N., Moldovyan N.A., Shcherbacov V.A. Non-Commutative Finite Associative Algebras of 3-Dimensional Vectors. Quasigroups and Related Systems, 2018, vol. 26, no. 1, pp. 109-120.
18. Moldovyan N.A., Moldovyan P.A. Vector Form of the Finite Fields GF(p^m). Bulletinul Academiei de stiinte a Republicii Moldova. Matematica, 2009, no. 3, pp. 57-63.
19. Schnorr C.P. Efficient Signature Generation by Smart Cards. Journal of Cryptology, 1991, vol. 4, pp. 161-174. DOI: 10.1007/BF00196725