Predicting nonlinear pseudorandom number generators

Mathematics of Computation

Volumn 74, Issue 251, 2005, Pages 1471-1494

  Blackburn, Simon R ^a   Gomez Perez, Domingo ^b   Gutierrez, Jaime ^b   Shparlinski, Igor E ^c  

^a UNIVERSITY OF LONDON   (United Kingdom)

^b UNIVERSITY OF CANTABRIA   (Spain)

^c MACQUARIE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21644470546     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-04-01698-9     Document Type: Article

References (34)

1. M. Ajtai, R. Kumar and D. Sivakumar, 'A sieve algorithm for the shortest lattice vector problem', Proc. 33rd ACM Symp. on Theory of Comput., ACM, 2001, 601-610.
2. P. Beelen and J. Doumen, 'Pseudorandom sequences from elliptic curves', Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Springer-Verlag, Berlin, 2002, 37-52. MR 2004d:11066
3. S. R. Blackburn, D. Gomez-Perez, J. Gutierrez and I. E. Shparlinski, 'Predicting the inversive generator', Proc. 9th IMA Intern. Conf on Cryptography and Coding, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 2898 (2003), 264-275.
4. S. R. Blackburn, D. Gomez-Perez, J. Gutierrez and I. E. Shparlinski, 'Reconstructing noisy polynomial evaluation in residue rings', J. of Algorithms (to appear).
5. D. Boneh, S. Halevi and N. A. Howgrave-Graham, 'The modular inversion hidden number problem', Proc. Asiacrypt'2001, Lect. Notes in Comp. Sci., vol. 2248, Springer-Verlag, Berlin, 2001, 36-51. MR 2003h:94022
6. J. Boyar, 'Inferring sequences produced by pseudo-random number generators', J. ACM, 36 (1989), 129-141. MR 91g:68035
7. J. Boyar, 'Inferring sequences produced by a linear congruential generator missing low-order bits', J. Cryptology 1 (1989) 177-184. MR 90g:94012
8. E. F. Brickell and A. M. Odlyzko, 'Cryptanalysis: A survey of recent results', Contemp. Cryptology, IEEE Press, NY, 1992, 501-540. MR 93k:94009
9. W.-S. Chou, 'The period lengths of inversive pseudorandom vector generations', Finite Fields Appl., 1 (1995), 126-132. MR 961:11087
10. D. Coppersmith, 'Small solutions to polynomial equations, and low exponent RSA vulnerabilities', J. Cryptology, 10 (1997), 233-260. MR 99b:94027
11. D. Coppersmith, 'Small solutions of small degree polynomials', Proc. Intern. Conf. on Cryptography and Lattices, Lect. Notes in Comp. Sci., vol. 2146, Springer-Ver lag, Berlin, 2001, 20-31. MR 2003f:11034
12. M. Flahive and H. Niederreiter, 'On inversive congruential generators for pseudorandom numbers', Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993, 75-80. MR 94a:11117
13. A. M. Frieze, J. Håstad, R. Kannan, J. C. Lagarias and A. Shamir, 'Reconstructing truncated integer variables satisfying linear congruences', SIAM J. Comp., 17 (1988), 262-280. MR 89d:11115
14. G. Gong, T. A. Berson and D. A. Stinson, 'Elliptic curve pseudorandom sequence generators', Proc. 6th Workshop on Selected Areas in Cryptography, Lect. Notes in Comp. Sci., vol. 1758, Springer-Verlag, Berlin, 2000, 34-49. MR 2001j:94032
15. M. Griötschel, L. Lovász and A. Schrijver, Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin, 1993. MR 95e:90001
16. J. Gutierrez, I. E. Shparlinski and A. Winterhof, 'On the linear and nonlinear complexity profile of nonlinear pseudorandom number generators', IEEE Trans. on Information Theory, 49 (2003), 60-64.
17. F. Hess and I. E. Shparlinski, 'On the linear complexity and multidimensional distribution of congruential generators over elliptic curves', Designs, Codes and Cryptography (to appear).
18. N. A. Howgrave-Graham, 'Finding small roots of univariate modular equations revisited', Proc. 6th IMA Intern. Conf on Cryptography and Coding, Lect. Notes in Comp. Sci., vol. 1355, Springer-Verlag, Berlin, 1997, 131-142. MR 99j:94049
19. A. Joux and J. Stern, 'Lattice reduction: A toolbox for the crypt analyst', J. Cryptology, 11 (1998), 161-185. MR 99c:94031
20. R. Kannan, 'Algorithmic geometry of numbers', Annual Review of Comp. Sci., 2 (1987), 231-267. MR 89a:11131
21. R. Kannan, 'Minkowski's convex body theorem and integer programming', Math. Oper. Res., 12 (1987), 415-440. MR 89c:90078
22. D. E. Knuth, 'Deciphering a linear congruential encryption', IEEE Trans. Inf. Theory 31 (1985), 49-52. MR 87c:94040
23. S. V. Konyagin, 'On the number of solutions of a univariate congruence of nth degree', Matem. Sbornik, 102 (1979), 171-187 (in Russian). MR 80k:10013a
24. H. Krawczyk, 'How to predict congruential generators', J. Algorithms, 13 (1992), 527-545. MR 93g:65013
25. J. C. Lagarias, 'Pseudorandom number generators in cryptography and number theory', Proc. Symp. in Appl. Math., Amer. Math. Soc., Providence, RI, 42 (1990), 115-143. MR 92f:11109
26. A. K. Lenstra, H. W. Lenstra and L. Lovász, 'Factoring polynomials with rational coefficients', Mathematische Annalen, 261 (1982), 515-534. MR 84a:12002
27. D. Micciancio and S. Goldwasser, Complexity of lattice problems, Kluwer Acad. Publ., 2002.
28. P. Q. Nguyen and J. Stern, 'Lattice reduction in cryptology: An update', Proc. 4th Intern. Symp. on Algorithmic Number Theory, Lect. Notes in Comp. Sci., vol. 1838, Springer-Verlag, Berlin, 2000, 85-112. MR 2002h:94064
29. P. Q. Nguyen and J. Stern, 'The two faces of lattices in cryptology', Proc. Intern. Conf. on Cryptography and Lattices, Lect. Notes in Comp. Sci., vol. 2146, Springer-Verlag, Berlin, 2001, 146-180. MR 2003d:94082
30. H. Niederreiter, 'New developments in uniform pseudorandom number and vector generation', Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lect. Notes in Statistics, vol. 106, Springer-Verlag, Berlin, 1995, 87-120. MR 97k:65019
31. H. Niederreiter, 'Design and analysis of nonlinear pseudorandom number generators', Monte Carlo Simulation, A.A. Balkema Publishers, Rotterdam, 2001, 3-9.
32. H. Niederreiter and I. E. Shparlinski, 'Recent advances in the theory of nonlinear pseudorandom number generators', Proc. Conf. on Monte Carlo and Quasi-Monte Carlo Methods, 2000, Springer-Verlag, Berlin, 2002, 86-102. MR 2003k:65005
33. H. Niederreiter and I. E. Shparlinski, 'Dynamical systems generated by rational functions', Proc. the 15th Symp. on Appl. Algebra, Algebraic Algorithms, and Error-Correcting Codes, Lect. Notes in Comp. Sci., vol. 2643, Springer-Verlag, Berlin, 2003, 6-17.
34. V. Shoup, 'Number theory C++ library (NTL)', version 5.3.1, available at http://www.-shoup.net/ntl/.