Journal of Mathematical Sciences

, Volume 126, Issue 3, pp 1158–1166

On non-Abelian homomorphic public-key cryptosystems

  • D. Grigoriev
  • I. Ponomarenko
Article

Abstract

An important problem of modern cryptography concerns secret public-key computations in algebraic structures. We construct homomorphic cryptosystems, which are (secret) epimorphisms f : G → H, where G and H are (publically known) groups and H is finite. A letter of a message to be encrypted is an element h ∈ H, while its encryption is an element g ∈ G such that f(g) = h. A homomorphic cryptosystem allows one to perform computations (in the group G) with encrypted information (without knowing the original message over H).

In this paper, homomorphic cryptosystems are constructed for the first time for non-Abelian groups H (earlier, homomorphic cryptosystems were known only in the Abelian case). In fact, we present such a system for any (fixed) solvable group H. Bibliography: 24 titles.

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REFERENCES

  1. 1.
    D. M. Barrington, H. Straubing, and D. Therien, “Nonuniform automata over groups, Inform. Comput., 132, 89–109 (1990).Google Scholar
  2. 2.
    J. Benaloh, “Dense probabilistic encryption,” First Annual Workshop on Selected Areas in Cryptology (1994), pp. 120–128.Google Scholar
  3. 3.
    S. Cook and R. A. Reckhow, “The relative efficiency of propositional proof systems,” J. Symbolic Logic, 44, 36–50 (1979).Google Scholar
  4. 4.
    D. Coppersmith and I. Shparlinski, “On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping,” J. Cryptology, 13, 339–360 (2000).Google Scholar
  5. 5.
    H. Davenport, Multiplicative Number Theory, Springer (1980).Google Scholar
  6. 6.
    Do Long Van, A. Jeyanthi, R. Siromony, and K. Subramanian, “Public key cryptosystems based on word problems,” in: ICOMIDC Symp. Math. of Computations, Ho Chi Minh City (1988).Google Scholar
  7. 7.
    J. Feigenbaum and M. Merritt, “Open questions, talk abstracts, and summary of discussions,” DIMACS Ser. Discrete Math. Theor. Comput. Sci., 2, 1–45 (1991).Google Scholar
  8. 8.
    S. Goldwasser and M. Bellare, Lect. Notes Cryptography, http://www-cse.ucsd.edu/users/mihir/papers/gb.html (2001).Google Scholar
  9. 9.
    S. Goldwasser and S. Micali, “Probabilistic encryption,” J. Comput. System. Sci., 28, 270–299 (1984).Google Scholar
  10. 10.
    M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer-Verlag, New York (1979).Google Scholar
  11. 11.
    K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J. Kang, and C. Park, “New public-key cryptosystem using braid groups,” Lect. Notes Comput. Sci., 1880, 166–183 (2000).Google Scholar
  12. 12.
    K. Koyama, U. Maurer, T. Okamoto, and S. Vanstone, “New public-key schemes based on elliptic curves over the ring ℤn,” Lect. Notes Comput. Sci., 576, 252–266 (1991).Google Scholar
  13. 13.
    W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Interscience Publishers, New York-London-Sydney (1966).Google Scholar
  14. 14.
    U. Maurer and S. Wolf, “Lower bounds on generic algorithms in groups,” Lect. Notes Comput. Sci., 1403, 72–84 (1998).Google Scholar
  15. 15.
    R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press (1995).Google Scholar
  16. 16.
    D. Naccache and J. Stern, “A new public key cryptosystem based on higher residues,” Proceedings of the 5th ACM Conference on Computer and Communication Security (1998), pp. 59–66.Google Scholar
  17. 17.
    T. Okamoto and S. Uchiyama, “A new public-key cryptosystem as secure as factoring,” Lect. Notes Comput. Sci., 1403, 308–317 (1998).Google Scholar
  18. 18.
    S.-H. Paeng, D. Kwon, K.-C. Ha, and J. H. Kim, “Improved public key cryptosystem using finite non-Abelian groups,” Preprint NSRI Korea.Google Scholar
  19. 19.
    P. Paillier, “Public-key cryptosystem based on composite degree residuosity classes,” Lect. Notes Comput. Sci., 1592, 223–238 (1999).Google Scholar
  20. 20.
    M. O. Rabin, “Probabilistic algorithms in finite fields,” SIAM J. Comput., 9, 273–280 (1980).Google Scholar
  21. 21.
    D. K. Rappe, “Algebraisch homomorphe kryptosysteme,” Diplomarbeit, Fachbereich Mathematik der Universität Dortmund (2000).Google Scholar
  22. 22.
    R. L. Rivest, L. Adleman, and M. Dertouzos, “On data banks and privacy homomorphisms,” in: Foundation of Secure Computation, Academic Press (1978), pp. 169–177.Google Scholar
  23. 23.
    T. Sander, A. Young, and M. Young, “Noninteractive cryptocomputing for NC1,” in: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (1999), pp. 554–566.Google Scholar
  24. 24.
    A. Yao, “How to generate and exchange secrets,” in: Proceedings of the 27th IEEE Symposium on Foundations of Computer Science (1986), pp. 162–167.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • D. Grigoriev
  • I. Ponomarenko

There are no affiliations available

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