On non-Abelian homomorphic public-key cryptosystems
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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.
- D. M. Barrington, H. Straubing, and D. Therien, “Nonuniform automata over groups, Inform. Comput., 132, 89–109 (1990).
- J. Benaloh, “Dense probabilistic encryption,” First Annual Workshop on Selected Areas in Cryptology (1994), pp. 120–128.
- S. Cook and R. A. Reckhow, “The relative efficiency of propositional proof systems,” J. Symbolic Logic, 44, 36–50 (1979).
- D. Coppersmith and I. Shparlinski, “On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping,” J. Cryptology, 13, 339–360 (2000).
- H. Davenport, Multiplicative Number Theory, Springer (1980).
- 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).
- J. Feigenbaum and M. Merritt, “Open questions, talk abstracts, and summary of discussions,” DIMACS Ser. Discrete Math. Theor. Comput. Sci., 2, 1–45 (1991).
- S. Goldwasser and M. Bellare, Lect. Notes Cryptography, http://www-cse.ucsd.edu/users/mihir/papers/gb.html (2001).
- S. Goldwasser and S. Micali, “Probabilistic encryption,” J. Comput. System. Sci., 28, 270–299 (1984).
- M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer-Verlag, New York (1979).
- 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).
- K. Koyama, U. Maurer, T. Okamoto, and S. Vanstone, “New public-key schemes based on elliptic curves over the ring ℤ n ,” Lect. Notes Comput. Sci., 576, 252–266 (1991).
- 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).
- U. Maurer and S. Wolf, “Lower bounds on generic algorithms in groups,” Lect. Notes Comput. Sci., 1403, 72–84 (1998).
- R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press (1995).
- 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.
- T. Okamoto and S. Uchiyama, “A new public-key cryptosystem as secure as factoring,” Lect. Notes Comput. Sci., 1403, 308–317 (1998).
- S.-H. Paeng, D. Kwon, K.-C. Ha, and J. H. Kim, “Improved public key cryptosystem using finite non-Abelian groups,” Preprint NSRI Korea.
- P. Paillier, “Public-key cryptosystem based on composite degree residuosity classes,” Lect. Notes Comput. Sci., 1592, 223–238 (1999).
- M. O. Rabin, “Probabilistic algorithms in finite fields,” SIAM J. Comput., 9, 273–280 (1980).
- D. K. Rappe, “Algebraisch homomorphe kryptosysteme,” Diplomarbeit, Fachbereich Mathematik der Universität Dortmund (2000).
- 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.
- 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.
- A. Yao, “How to generate and exchange secrets,” in: Proceedings of the 27th IEEE Symposium on Foundations of Computer Science (1986), pp. 162–167.
- On non-Abelian homomorphic public-key cryptosystems
Journal of Mathematical Sciences
Volume 126, Issue 3 , pp 1158-1166
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