Generating safe primes

Journal of Mathematical Cryptology

Volumn 7, Issue 4, 2013, Pages 333-365

  Von Zur Gathen, Joachim ^a   Shparlinski, Igor E ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Hofheinz Kiltz Shoup cryptosystem; Safe prime; Sophie Germain prime

Indexed keywords

EID: 84890514001     PISSN: 18622976     EISSN: 18622984     Source Type: Journal    
DOI: 10.1515/jmc-2013-5011     Document Type: Article

References (35)

1. R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), 331-361.
2. R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes. II, Proc. Lond. Math. Soc. 83 (2001), 532-562.
3. P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
4. L. Blum, M. Blum and M. Shub, A simple unpredictable pseudo-random number generator, SIAM J. Comp. 15 (1986), 364-383.
5. R. P. Brent and P. Zimmermann, Modern Computer Arithmetic, Cambridge Monogr. Appl. Comput. Math. 18, Cambridge University Press, 2011.
6. R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, New York, 2005.
7. I. Damgård and M. Koprowski, Practical threshold RSA signatures without a trusted dealer, in: Advances in Cryptology (EUROCRYPT 2001), Lecture Notes in Comp. Sci. 2045, Springer, Berlin (2001), 152-165.
8. A. Decker and P. Moree, Counting RSA-integers, Results Math. 52 (2008), 35-39.
9. L. Devroye, Non-Uniform Random Variate Generation, Springer, New York, 1986.
10. P.-A. Fouque and J. Stern, Fully distributed threshold RSA under standard assumptions, in: Advances in Cryptology (ASIACRYPT 2001), Lecture Notes in Comp. Sci. 2248, Springer, Berlin (2001), 310-330.
11. S. Galbraith, Mathematics of Public Key Cryptography, Cambridge University Press, New York, 2012.
12. J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013.
13. R. Gennaro, S. Halevi and T. Rabin, Secure hash-and-sign signatures without the random oracle, in: Advances in Cryptology (EUROCRYPT '99), Lecture Notes in Comp. Sci. 1592, Springer, Berlin (1999), 123-139.
14. B.-N. Guo and F. Qi, Sharp bounds for harmonic numbers, Applied Math. Comp. 218 (2011), 991-995.
15. G. Harman, Prime-Detecting Sieves, Princeton University Press, Princeton, 2007.
16. D. R. Heath-Brown, Artin's conjecture for primitive roots, Quart. J. Math. 37 (1986), 27-38.
17. D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22-63.
18. D. Hofheinz, E. Kiltz and V. Shoup, Practical chosen ciphertext secure encryption from factoring, J. Cryptology 26 (2013), 375-441.
19. M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170.
20. M. N. Huxley, Exponential sums and lattice points III, Proc. London Math. Soc. 87 (2003) 591-609.
21. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, 2004.
22. M. Joye and P. Paillier, Fast generation of prime numbers on portable devices: An update, in: Cryptographic Hardware and Embedded Systems (CHES 2006), Lecture Notes in Comp. Sci. 4249, Springer, Berlin (2006), 160-173.
23. D. E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, Addison Wesly, Reading, 1981.
24. D. Loebenberger and M. Nüsken, Notions for RSA integers, Intern. J. Applied Crypto. (2013), 1753-053.
25. A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1997.
26. D. Naccache, Double-speed safe primes generation, preprint (2003), http://eprint.iacr.org/2003/175.
27. T. Nishide and K. Sakurai, Distributed Paillier cryptosystem without trusted dealer, in: Information Security Applications (WISA 2010), Lecture Notes in Comp. Sci. 6513, Springer, Berlin (2011), 44-60.
28. E. Ong and J. Kubiatowicz, Optimizing Robustness while generating shared secret safe primes, in: Public Key Cryptography (PKC 2005), Lecture Notes in Comp. Sci. 3386, Springer, Berlin (2005), 120-137.
29. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.
30. V. Shoup, Practical threshold signatures, in: Advances in Cryptology (EUROCRYPT 2000), Lecture Notes in Comp. Sci. 1807, Springer, Berlin (2000), 207-220.
31. T. Tao, E. Croot and H. Helfgott, Deterministic methods to find primes, Math. Comp. 81 (2012), 1233-1246.
32. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.
33. A. I. Vinogradov, On the remainder in Mertens' formula (in Russian), Dokl. Akad. Nauk SSSR 148, 262-263.
34. B. de Weger, Estimates for RSA moduli counting functions, preprint (2008).
35. K. Ziegler and J. Zollmann, Fast and uniform generation of safe RSA moduli - Extended abstract. in: WEWoRC 2013 - Book of Abstracts, Karlsruher Institut für Technologie (KIT), Karlsruhe (2013), 15-19.