Proving in zero-knowledge that a number is the product of two safe primes

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 1592, Issue , 1999, Pages 107-122

  Camenisch, Jan ^a   Michels, Markus ^b  

^a AARHUS UNIVERSITY   (Denmark)

^b r3 Security Engineering AG   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATION THEORY; CRYPTOGRAPHY;

BUILDING BLOCKES; COMPOSITE NUMBERS; MODULAR ADDITION; MODULAR EXPONENTIATION; MODULAR MULTIPLICATION; MODULAR POLYNOMIAL; STATISTICAL ZERO KNOWLEDGE; ZERO KNOWLEDGE;

POLYNOMIALS;

EID: 84957610863     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-48910-X_8     Document Type: Conference Paper

References (39)

1. E. Bach and J. Shallit. Factoring with cyclotomic polynomials. In 26th FOCS, IEEE, pp. 443-450, 1985.
2. J. Boyar, K. Friedl, and C. Lund. Practical zero-knowledge proofs: Giving hints and using defficiencies. Journal of Cryptology, 4(3):185-206, 1991.
3. S. Brands. Untraceable off-line cash in wallets with observers. In Advances in Cryptology - CRYPTO '93, volume 773 of LNCS, pp. 302-318, 1993.
4. S. Brands. Rapid demonstration of linear relations connected by boolean operators. In Advances in Cryptology ‑ EUROCRYPT '97, volume 1233 of LNCS, pp. 318- 333. Springer Verlag, 1997.
5. G. Brassard, D. Chaum, and C. Crépeau. Minimum disclosure proofs of knowledge. Journal of Computer and System Sciences, 37(2):156-189, Oct. 1988.
6. J. Camenisch and M. Michels. Proving in zero-knowledge that a number n is the product of two safe primes. Technical Report RS-98-29, BRICS, Departement of Computer Science, University of Aarhus, Nov. 1998.
7. J. Camenisch and M. Michels. A group signature scheme based on an RSA-variant. Tech. Rep. RS-98-27, BRICS, Departement of Computer Science, University of Aarhus, Nov. 1998. Preliminary version appeared in Advances in Cryptology ‑ ASIACRYPT '98, volume 1514 of LNCS, pages 160-174. Springer Verlag, 1998.
8. J. Camenisch and M. Stadler. Efficient group signature schemes for large groups. In Advances in Cryptology ‑ CRYPTO '97, volume 1296 of LNCS, pp. 410-424. Springer Verlag, 1997.
9. J. Camenisch and M. Stadler. Proof systems for general statements about discrete logarithms. Technical Report TR 260, Institute for Theoretical Computer Science, ETH Zürich, Mar. 1997.
10. J. L. Camenisch. Group Signature Schemes and Payment Systems Based on the Discrete Logarithm Problem. PhD thesis, ETH Zürich, 1998. Diss. ETH No. 12520.
11. A. Chan, Y. Frankel, and Y. Tsiounis. Easy come - easy go divisible cash. In Advances in Cryptology ‑ EUROCRYPT '98, volume 1403 of LNCS, pp. 561- 575. Springer Verlag, 1998. Revised version available as GTE Technical Report.
12. D. Chaum, J.-H. Evertse, and J. van de Graaf. An improved protocol for demonstrating possession of discrete logarithms and some generalizations. In Advances in Cryptology ‑ EUROCRYPT '87, volume 304 of LNCS, pp. 127-141. Springer- Verlag, 1988.
13. D. Chaum, J.-H. Evertse, J. van de Graaf, and R. Peralta. Demonstrating possession of a discrete logarithm without revealing it. In Advances in Cryptology ‑ CRYPTO '86, volume 263 of LNCS, pp. 200-212. Springer-Verlag, 1987.
14. D. Chaum and T. P. Pedersen. Wallet databases with observers. In Advances in Cryptology ‑ CRYPTO '92, volume 740 of LNCS, pp. 89-105. Springer-Verlag, 1993.
15. H. Cohen. A Course in Computational Algebraic Number Theory. Number 138 in Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993.
16. R. Cramer and I. Damgård. Zero-knowledge proof for finite field arithmetic, or: Can zero-knowledge be for free? In Advances in Cryptology ‑ CRYPTO '98, volume 1642 of LNCS, pp. 424-441, Berlin, 1998. Springer Verlag.
17. R. Cramer, I. Damg_ard, and B. Schoenmakers. Proofs of partial knowledge and simpli fied design of witness hiding protocols. In Advances in Cryptology ‑ CRYPTO '94, volume 839 of LNCS, pp. 174-187. Springer Verlag, 1994.
18. E. Fujisaki and T. Okamoto. Statistical zero knowledge protocols to prove modular polynomial relations. In Advances in Cryptology ‑ CRYPTO '97, volume 1294 of LNCS, pp. 16-30. Springer Verlag, 1997.
19. E. Fujisaki and T. Okamoto. A practical and provably secure scheme for publicly verifiable secret sharing and its applications. In Advances in Cryptology‑ EUROCRYPT '98, volume 1403 of LNCS, pp. 32-46. Springer Verlag, 1998.
20. R. Gennaro, S. Jarecki, H. Krawczyk, and T. Rabin. Robust and eficient sharing of RSA functions. In Advances in Cryptology ‑ CRYPT0 '96, volume 1109 of LNCS, pp. 157-172, Berlin, 1996. IACR, Springer Verlag.
21. R. Gennaro, H. Krawczyk, and T. Rabin. RSA-based undeniable signatures. In Advances in Cryptology ‑ CRYPTO '97, volume 1296 of LNCS, pp. 132-149. Springer Verlag, 1997.
22. R. Gennaro, D. Micciancio, and T. Rabin. An eficient non-interactive statistical zero-knowledge proof system for quasi-safe prime products. In 5rd ACM Conference on Computer and Communicatons Security, 1998.
23. O. Goldreich, S. Micali, and A. Wigderson. How to prove all NP statements in zero-knowledge and a methodology of cryptographic protocol design. In Advances in Cryptology ‑ CRYPTO '86, volume 263 of LNCS, pp. 171-185. Springer-Verlag, 1987.
24. J. Gordon. Strong RSA keys. Electronics Letters, 20(12):514-516, 1984.
25. K. Koyama, U. Maurer, T. Okamoto, and S. Vanstone. New public-key schemes based on elliptic curves over the ring Zn. In Advances in Cryptology ‑ CRYPTO '91, volume 576 of LNCS, pp. 252-266. Springer-Verlag, 1992.
26. E. Kranakis. Primality and Cryptography. Wiley-Teubner Series in Computer Science, 1986.
27. D. J. Lehmann. On primality tests. SIAM Journal of Computing, 11(2):374-375, May 1982.
28. M. Liskov and B. Silverman. A Statisical limited-knowledge proof for secure RSA keys. manuscript, (1998).
29. W. Mao. Verifable Partial Sharing of Integer Factors. to appear in Proc. SAC'98, 1998.
30. G. L. Miller. Riemann's hypothesis and tests for primality. Journal of Computer and System Sciences, 13:300-317, 1976.
31. T. P. Pedersen. Non-interactive and information-theoretic secure verifiable secret sharing. In Advances in Cryptology ‑ CRYPTO '91, volume 576 of LNCS, pp. 129-140. Springer Verlag, 1992.
32. J. M. Pollard. Theorems on factorization and primality testing. Proc. Cambridge Philosophical Society, 76:521-528, 1974.
33. M. O. Rabin. Probabilistic algorithm for testing primality. Journal of Number Theory, 12:128-138, 1980.
34. A. de Santis, L. di Crescenzo, G. Persiano, M. Yung. OnMonotone Formula Closure of SZK. 35th FOCS, IEEE, pp. 454-465, 1994.
35. C. P. Schnorr. Eficient signature generation for smart cards. Journal of Cryptology, 4(3):239-252, 1991.
36. R. Solovay and V. Strassen. A fast monte-carlo test for primality. SIAM Journal on Computing, 6(1):84-85, Mar. 1977.
37. J. van de Graaf and R. Peralta. A simple and secure way to show the validity of your public key. In Advances in Cryptology ‑ CRYPTO '87, volume 293 of LNCS, pp. 128-134. Springer-Verlag, 1988.
38. H. C. Williams. A p + 1 method of factoring. Mathematics of Computation, 39(159):225-234, 1982.
39. X9.31 - 1998 Digital Signatures using reversible public key cryptography for the financial services industry (rDSA). American National Standard, Working Draft, 59 pages, 1998.