A review of symmetric key cryptosystem using group representations

D. Samaila^1*, G. N. Shu’aibu², B. A. Madu²

¹Department of Mathematics, Adamawa State University, Mubi, Nigeria

²Mathematical Sciences Department, University of Maiduguri, Borno State, Nigeria

Abstract

The use of group representations in crypto-analysis has been a long-standing problem in group theory. Due to the nebulous presentations by few authors, this paper aimed at simplifying the procedure and to make it more secure than the existing literatures. We successfully presented the most radical techniques of symmetric key crypto-system using group theoretic approach. The procedure appears to be more secured and the result shows that different techniques can be used within the limit of representation theory for key selection, encryption and decryption. The difficulty in factorizing each   as a product   for large n makes it hard to break, and high flexibility for key selection also makes the DLP highly resistant to attacks. There is also an added advantage of easy and fast implementation.

Keywords

conjugacy; commutators; representation; symmetric group; cryptography.

References

1. Javad N. D., Ehsan M. and Ali Z.: A Cryptosystem Based on the Symmetric Group S_n. International Journal of Computer Science and Network Security, 2008; VOL. 8(2), pp226-234.
2. Peter Mayr: Group Action and applications; 2019.
3. Maurer U. and Wolf S.: The relationship between breaking the Diffe-Hellman protocol and computing discrete logarithms, SIAM Journal on Computing,1999; 28(5) pp.1689-1731.
4. Boneh D. and Lipton R.: Algorithms for black-box fields and their applications to cryptography, Advances in Cryptology-CRYPTO ’96, Lecture Notes in Computer Science, Springer-Verlag,1996; (1109) pp.283-297.
5. El-Gamal T.: A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Transactions on Information Theory, 1985;(31) 469-472.
6. Rivest R., Shamir A. and Adleman L.: A Method for Obtaining Digital Signatures and Public Key Cryptosystems, Comm. ACM,1978;(21) pp.120-126.
7. Hellman M. E. and Merkle R. C.: Hiding information and signatures in trapdoor knapsacks, IEEE Transactions on Information Theory, 1978;(24) pp.525-530.
8. Shamir A.: A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem, Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press,1982; pp.145-152.
9. Chor B. and Rivest R.: A knapsack-type public key cryptosystem based on arithmetic in finite fields, IEEE Transactions on Information Theory,1988;(34) pp.901-909.
10. Vaudenay S.: Cryptanalysis of the Chor-Rivest Cryptosystem, Advances in Cryptology-CRYPTO ’98, Lecture Notes in Computer Science, 1462 Springer Verlag, 1998; pp.243-256.
11. Koblitz N.: Elliptic curve cryptosystems, Mathematics of Computation, 1987;(48) pp.203-209.
12. Miller V.: Uses of elliptic curves in cryptography, Advances in Cryptology- CRYPTO ’85, Lecture Notes in Computer Science, Springer-Verlag, 1986;(218) pp.417-426.
13. Ayan M.: The MOR Cryptosystems and Finite p-groups, Contemporary Mathematics,2015; Volume 633, pp.:81-95.
14. McCurley K.: The discrete logarithm problem, Cryptology and Computational Number Theory, volume 42 of Proceedings of Symposia in Applied Mathematics, American Mathematical Society,1990; pp.49-74.
15. Simon R. B., Carlos C. and Ciaran M.: Group theory in Cryptography, Egham, Surrey TW20 0EX, United Kingdom, 2010.
16. Andre N. and Katrin T.: Describing Finite Groups by Short First-Order Sentences, Israel Journal of Mathematics, 2014; pp.1-13.
17. Robert C.: Codes and Ciphers; Julius Caesar, the Enigma and the Internet, Cambridge University Press, Cambridge, 2002.
18. Michal S.: New Results in Group Theoretic Cryptology. Ph.D. Thesis, Florida Atlantic University, Boca Raton, 2006.
19. Douglas R. S.: Cryptography; Theory and Practice, 2nd ed, CRC Press,New York, NY, 2002.