International Journal of Computer Science & Network Security

국제컴퓨터통신보호논문지학회 (International Journal of Computer Science & Network Security)
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A Survey of Public-Key Cryptography over Non-Abelian Groups

  • Lanel, G.H.J. (Department of Mathematics, University of Sri Jayewardenepura) ;
  • Jinasena, T.M.K.K. (Department of Computer Science, University of Sri Jayewardenepura) ;
  • Welihinda, B.A.K. (Department of Mathematics, University of Sri Jayewardenepura)
  • 투고 : 2021.04.27
  • 발행 : 2021.04.30
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초록

Non-abelian group based Cryptography is a field which has become a latest trend in research due to increasing vulnerabilities associated with the abelian group based cryptosystems which are in use at present and the interesting algebraic properties associated that can be thought to provide higher security. When developing cryptographic primitives based on non-abelian groups, the researchers have tried to extend the similar layouts associated with the traditional underlying mathematical problems and assumptions by almost mimicking their operations which is fascinating even to observe. This survey contributes in highlighting the different analogous extensions of traditional assumptions presented by various authors and a set of open problems. Further, suggestions to apply the Hamiltonian Cycle/Path Problem in a similar direction is presented.

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과제정보

We would like to express our warm thanks to the Editor and the Reviewers of the open access journal IJCSNS, for their kind comments and support during the publication of this manuscript.

참고문헌

  1. G. H. J. Lanel, H. K. Pallage, J. K. Ratnayake, S. Thevasha, and B. A. K. Welihinda, "A survey on Hamiltonicity in Cayley graphs and digraphs on different groups," Discrete Math. Algorithms Appl., vol. 11, no. 05, p. 1930002, 2019, doi: 10.1142/s1793830919300029.
  2. G. H. J. Lanel, T. M. K. K. Jinasena, and B. A. K. Welihinda, "Hamiltonian Cycles in Cayley Graphs of Semidirect Products of Finite Groups," Eur. Mod. Stud. J., vol. 04, no. 03, pp. 1-19, 2020.
  3. H. Hong, J. Shao, L. Wang, H. Ahmad, and Y. Yang, "Public Key Encryption in Non-Abelian Groups," ArXiv Prepr. ArXiv160506608, 2016.
  4. B. Fine, M. Habeeb, D. Kahrobaei, and G. Rosenberger, "Aspects of nonabelian group based cryptography: a survey and open problems," JP J. Algebra Number Theory Appl., 2011.
  5. T. C. Lin, "A study of non-abelian public key cryptography," Int. J. Netw. Secur., vol. 20, no. 2, pp. 278-290, 2018.
  6. R. Cramer and V. Shoup, "Signature schemes based on the strong RSA assumption," ACM Trans. Inf. Syst. Secur. TISSEC, vol. 3, no. 3, pp. 161-185, 2000, doi: 10.1145/357830.357847.
  7. I. Ilic, "The Discrete Logarithm Problem in Non-abelian Groups," Computing, vol. 1, p. 1, 2010. https://doi.org/10.1007/BF02235848
  8. I. Anshel, M. Anshel, B. Fisher, and D. Goldfeld, "New key agreement protocols in braid group cryptography," 2001, pp. 13-27, doi: 10.1007/3-540-45353-9_2.
  9. I. Anshel, M. Anshel, and D. Goldfeld, "Non-abelian key agreement protocols," Discrete Appl. Math., vol. 130, no. 1, pp. 3-12, 2003, doi: 10.1016/s0166-218x(02)00585-1.
  10. D. Grigoriev and I. Ponomarenko, "Constructions in public-key cryptography over matrix groups," ArXiv Prepr. Math0506180, 2005, doi: 10.1090/conm/418/07949.
  11. I. Ilic and S. S. Magliveras, "Weak discrete logarithms in non-abelian groups," J. Comb. Math. Comb. Comput., vol. 74, p. 3, 2010.
  12. L. C. Klingler, S. S. Magliveras, F. Richman, and M. Sramka, "Discrete logarithms for finite groups," Computing, vol. 85, no. 1-2, p. 3, 2009, doi: 10.1007/s00607-009-0032-0.
  13. A. Mahalanobis, "The Diffie-Hellman key exchange protocol and non-abelian nilpotent groups," Isr. J. Math., vol. 165, no. 1, pp. 161-187, 2008, doi: 10.1007/s11856-008-1008-z.
  14. I. Anshel, M. Anshel, and D. Goldfeld, "An algebraic method for public-key cryptography," Math. Res. Lett., vol. 6, no. 3, pp. 287-291, 1999, doi: 10.4310/mrl.1999.v6.n3.a3.
  15. K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J. S. Kang, and C. Park, "New public-key cryptosystem using braid groups," 2000, pp. 166-183, doi: 10.1007/3-540-44598-6_10.
  16. J. Birman, "Braids, links, and mapping class groups, volume 82 of Annals of Math," Stud. Princet. Univ. Press, 1974, doi: 10.1515/9781400881420.
  17. P. Dehornoy, "Braid-based cryptography," Contemp Math, vol. 360, pp. 5-33, 2004, doi: 10.1090/conm/360/06566.
  18. V. Shpilrain and G. Zapata, "Combinatorial group theory and public key cryptography," Appl. Algebra Eng. Commun. Comput., vol. 17, no. 3-4, pp. 291-302, 2006, doi: 10.1007/s00200-006-0006-9.
  19. I. S. Lee, W. H. Kim, D. Kwon, S. Nahm, N. S. Kwak, and Y. J. Baek, "On the security of MOR public key cryptosystem," 2004, pp. 387-400.
  20. S. H. Paeng, "On the security of cryptosystem using automorphism groups," Inf. Process. Lett., vol. 88, no. 6, pp. 293-298, 2003, doi: 10.1016/j.ipl.2003.09.001.
  21. C. Tobias, "Security analysis of the MOR cryptosystem," 2003, pp. 175-186.
  22. A. Mahalanobis, "A simple generalization of the ElGamal cryptosystem to non-abelian groups," Commun. Algebr., vol. 36, no. 10, pp. 3878-3889, 2008, doi: 10.1080/00927870802160883.
  23. A. Mahalanobis, "A note on using finite non-abelian pgroups in the MOR cryptosystem," ArXiv Prepr. Cs0702095, 2007.
  24. A. Mahalanobis, "A simple generalization of the ElGamal cryptosystem to non-abelian groups II," Commun. Algebra, vol. 40, no. 9, pp. 3583-3596, 2012, doi: 10.1080/00927872.2011.602998.
  25. W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations. Courier Corporation, 2004.
  26. G. Baumslag, Topics in combinatorial group theory. Birkhauser, 2012.
  27. A. I. S. Moldenhauer and G. Rosenberger, "Cryptosystems using automorphisms of finitely generated free groups," ArXiv Prepr. ArXiv160302328, 2016.
  28. S. H. Paeng, K. C. Ha, J. H. Kim, S. Chee, and C. Park, "New public key cryptosystem using finite non Abelian groups," 2001, pp. 470-485, doi: 10.1007/3-540-44647-8_28.
  29. C. Bates, N. Meyer, and T. Pulickal, "Cryptographic applications of nonabelian groups," Math Ariz. Edu Asp2008crypto Pdf, 2008.
  30. M. Cohen, S. Flannery, and D. Flannery, "In Code: A Mathematical Journey, by Sarah Flannery and David Flannery," Am. Math. Mon., vol. 109, no. 10, p. 929, 2002, doi: 10.2307/3072480.
  31. R. Alvarez, L. Tortosa, J. Vicent, and A. Zamora, "A nonabelian group based on block upper triangular matrices with cryptographic applications," 2009, pp. 117-126, doi: 10.1007/978-3-642-02181-7_13.
  32. R. Alvarez, F. M. Martinez, J. F. Vicent, and A. Zamora, "A new public key cryptosystem based on matrices," WSEAS Inf. Secur. Priv., vol. 3639, 2007.
  33. R. Alvarez, L. Tortosa, J. F. Vicent, and A. Zamora, "Analysis and design of a secure key exchange scheme," Inf. Sci., vol. 179, no. 12, pp. 2014-2021, 2009, doi: 10.1016/j.ins.2009.02.008.
  34. J. J. Climent, E. Gorla, and J. Rosenthal, "Cryptanalysis of the CFVZ cryptosystem," Adv. Math. Commun., vol. 01, no. 01, pp. 1-11, 2007, doi: 10.3934/amc.2007.1.1.
  35. H. K. Pathak and M. Sanghi, "Public key cryptosystem and a key exchange protocol using tools of non-abelian group," IJCSE Int. J. Comput. Sci. Eng., vol. 2, no. 04, pp. 1029-1033, 2010.
  36. A. J. Menezes and Y. H. Wu, "The discrete logarithm problem in GL (n, q)," Ars Comb., vol. 47, pp. 23-32, 1997.
  37. E. Stickel, "A new public-key cryptosystem in non abelian groups," 2004, pp. 70-80.
  38. V. Shpilrain, "Cryptanalysis of Stickel's key exchange scheme," in Computer Science - Theory and Applications, 2008, pp. 283-288, doi: 10.1007/978-3-540-79709-8_29.
  39. L. Wang, L. Wang, Z. Cao, E. Okamoto, and J. Shao, "New constructions of public-key encryption schemes from conjugacy search problems," in Information Security and Cryptology, 2010, pp. 1-17, doi: 10.1007/978-3-642-21518-6_1.
  40. S. Baba, S. Kotyad, and R. Teja, "A non-Abelian factorization problem and an associated cryptosystem.," IACR Cryptol EPrint Arch, vol. 2011, p. 48, 2011.
  41. L. Gu, L. Wang, K. Ota, M. Dong, Z. Cao, and Y. Yang, "New public key cryptosystems based on non-Abelian factorization problems," Secur. Commun. Netw., vol. 6, no. 7, pp. 912-922, 2013, doi: 10.1002/sec.710.
  42. L. Gu and S. Zheng, "Conjugacy systems based on nonabelian factorization problems and their applications in cryptography," J. Appl. Math., vol. 2014, 2014, doi: 10.1155/2014/630607.
  43. V. Roman'kov, "Two general schemes of algebraic cryptography," Groups Complex. Cryptol., vol. 10, no. 2, pp. 83-98, 2018, doi: 10.1515/gcc-2018-0009.
  44. V. Shpilrain and G. Zapata, "Using the subgroup membership search problem in public key cryptography," Contemp. Math., vol. 418, p. 169, 2006, doi: 10.1090/conm/418/07955.
  45. N. R. Wagner and M. R. Magyarik, "A public-key cryptosystem based on the word problem," 1984, pp. 19-36, doi: 10.1007/3-540-39568-7_3.
  46. M. Dehn, "Over infinite discontinuous groups," Math. Ann., vol. 71, no. 1, pp. 116-144, 1911. https://doi.org/10.1007/BF01456932
  47. M. Garzon and Y. Zalcstein, "The complexity of Grigorchuk groups with application to cryptography," Theor. Comput. Sci., vol. 88, no. 1, pp. 83-98, 1991, doi: 10.1016/0304-3975(91)90074-c.
  48. M. I. G. Vasco, D. Hofheinz, C. Martinez, and R. Steinwandt, "On the security of two public key cryptosystems using non-abelian groups," Des. Codes Cryptogr., vol. 32, no. 1, pp. 207-216, 2004, doi: 10.1023/b:desi.0000029223.76665.7e.
  49. J. C. Birget, S. S. Magliveras, and W. Wei, "Trap doors from subgroup chains and recombinant bilateral transversals," Proc. RECSI, vol. 7, pp. 31-48, 2002.
  50. R. I. Grigorchuk, "Degrees of growth of finitely generated groups, and the theory of invariant means," Izv. Ross. Akad. Nauk Seriya Mat., vol. 48, no. 5, pp. 939-985, 1984, doi: 10.1070/im1985v025n02abeh001281.
  51. T. Van Trung, Magliveras, and Stinson, "New approaches to designing public key cryptosystems using one-way functions and trapdoors in finite groups," J. Cryptol., vol. 15, no. 4, pp. 285-297, 2002, doi: 10.1007/s00145-001-0018-3.
  52. W. Lempken, T. Van Tran, S. S. Magliveras, and W. Wei, "A public key cryptosystem based on non-abelian finite groups," J. Cryptol., vol. 22, no. 1, pp. 62-74, 2009, doi: 10.1007/s00145-008-9033-y.
  53. D. Kahrobaei and M. Anshel, "Decision and search in nonabelian Cramer-Shoup public key cryptosystem," Groups Complex. Cryptol., vol. 1, no. 2, pp. 217-225, 2009, doi: 10.1515/gcc.2009.217.