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Subsets of groups in public-key cryptography (2311.15039v1)

Published 25 Nov 2023 in math.GR and cs.CR

Abstract: We suggest the usage of algebraic subsets instead of subgroups in public-key cryptography. In particular, we present the subset version of two protocols introduced by Shpilrain and Ushakov with some examples in ascending HNN-extensions of free-abelian groups and discuss their resistance to length and distance based attacks. We also introduce several new group theoretic problems arising from this work.

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References (39)
  1. Characterizations of the decidability of some problems for regular trace languages. Mathematical Systems Theory, 22:1–19, 1989.
  2. An algebraic method for public-key cryptography. Math. Res. Lett., 6:287–291, 1999.
  3. L. Bartholdi and P. V. Silva. Rational subsets of groups. In J.-E. Pin, editor, Handbook of Automata Theory, chapter 23. EMS Press, Berlin, 2021.
  4. J. Berstel. Transductions and Context-free Languages. Teubner, Stuttgart, 1979.
  5. The conjugacy problem is solvable in free-by-cyclic groups. Bull. London Math. Soc., 38:787–794, 2006.
  6. Orbit decidability and the conjugacy problem for some extensions of groups. Trans. Amer. Math. Soc., 362(4):2003–2036, 2010.
  7. P. Brinkmann. Detecting automorphic orbits in free groups. J. Algebra, 324(5):1083–1097, 2010.
  8. A. Carvalho. On generalized conjugacy and some related problems. Comm. Algebra, 51(8):3528–3542, 2022.
  9. A. Carvalho. Algebraic and context-free subsets of subgroups. Theor. Comput. Sci., 980:114229, 2023.
  10. A. Carvalho. On the dynamics of endomorphisms of the direct product of two free groups. arXiv:2307.13875, 2023.
  11. A. Carvalho. Quantifying Brinkmann’s problem: φ𝜑\varphiitalic_φ-order and φ𝜑\varphiitalic_φ-spectrum. arXiv: 2306.12563, 2023.
  12. A. Carvalho and J. Delgado. Decidability of the Brinkmann problems for endomorphisms of the free group. arXiv: 2310.10668, 2023.
  13. M. Casals-Ruiz and I. V. Kazachkov. Two remarks on first-order theories of Baumslag-Solitar groups. Sib Math J, 53:805–809, 2012.
  14. D. Collins. Baumslag-solitar group. In M. Hazewinkel, editor, Encyclopedia of Mathematics. Kluwer Academic Publishers, 2001.
  15. R. Gilman. Formal languages and infinite groups. In Geometric and computational perspectives on infinite groups, 1995.
  16. Public key exchange using semidirect product of (semi)groups. Applied Cryptography and Network Security ACNS 2013, pages 475–486, 2013.
  17. T. Herbst. On a subclass of context-free groups. RAIRO Inform. Théor. Appl., 25:255–272, 1991.
  18. Group-based cryptography in the quantum era. Notices Amer. Math. Soc., 70:752–763, 2023.
  19. R. Kannan and R. Lipton. Polynomial-time algorithm for the orbit problem. J. Assoc. Comput. Mach., 33(4), 1986.
  20. New public-key cryptosystem using braid groups. In Advances in cryptology - CRYPTO 2000 (Santa Barbara, CA), volume 1880 of Lecture Notes in Comput. Sci., pages 166–183. Springer, Berlin, 2000.
  21. Y. Kurt. A new key exchange primitive based on the triple decomposition problem. IACR Cryptology ePrint Archive, 378, 2006.
  22. M. Ladra and P. V. Silva. The generalized conjugacy problem for virtually free groups. Forum Math., 23:447–482, 2011.
  23. S. Lal and A. Chaturvedi. Authentication schemes using braid groups. arXiv: cs/0507066, 2005.
  24. J. C. Lennox. On soluble groups in which centralizers are finitely generated. Rend. Sem. Mat. Univ. Padova, 96:131–135, 1996.
  25. A. D. Logan. The conjugacy problem for ascending HNN-extensions of free groups. preprint, arXiv:2209.04357, 2022.
  26. M. Lohrey. The rational subset membership problem for groups: A survey. In C. M. Campbell, M. R. Quick, E. F. Robertson, and C. M. Roney-Dougal, editors, Groups St Andrews 2013, London Mathematical Society Lecture Note Series, pages 368–389. Cambridge University Press, Cambridge, 2015.
  27. M. Lohrey and B. Steinberg. The submonoid and rational subset membership problems for graph groups. J. Algebra, 320(2):728–755, 2008.
  28. F. Matucci. Cryptanalysis of the Shpilrain-Ushakov protocol in Thompson’s group. J. Cryptology, 21(3):458–468, 2008.
  29. Group-based cryptography. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2008.
  30. A. V. Rozhkov. Centralizers of elements in a group of tree automorphisms. Izv. Math., 43(3):471–492, 1994.
  31. Cryptanalysis of group-based key agreement protocols using subgroup distance functions. In T. Okamoto and X. Wang, editors, Public Key Cryptography – PKC 2007, pages 61–75, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg.
  32. V. Shpilrain. Problems in group theory motivated by cryptography. arXiv: 1802.07300, 2018.
  33. V. Shpilrain and A. Ushakov. Thompson’s group and public key cryptography. In Applied Cryptography and Network Security – ACNS 2005, volume 3531 of Lecture Notes in Computer Science, pages 151–164. Springer, 2005.
  34. V. Shpilrain and A. Ushakov. A new key exchange protocol based on the decomposition problem. In Algebraic Methods in Cryptography, volume 418 of Contemporary Mathematics, pages 161–167. American Mathematical Society, 2006.
  35. V. Shpilrain and G. Zapata. Using the subgroup membership search problem in public key cryptography. In Algebraic methods in cryptography, volume 418 of Contemp. Math., pages 169–178. Amer. Math. Soc., Providence, RI, 2006.
  36. P. V. Silva. On the rational subsets of the monogenic free inverse monoid. J. Algebra, 618:214–240, 2023.
  37. P. V. Silva and P. Weil. Automorphic orbits in free groups: words versus subgroups. Internat. J. Algebra Comput., 20.4:561–590, 2010.
  38. M. Valiunas. Negligibity of elliptic elements in ascending HNN-extensions of ℤmsuperscriptℤ𝑚\mathbb{Z}^{m}blackboard_Z start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT. Comm. Algebra, 47(12):5101–5120, 2019.
  39. E. Ventura. Group-theoretic orbit decidability. Groups Complex. Cryptol., 6(2):133–148, 2014.

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