Abstract
Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation \(U_{n+1}\equiv au{^{-1}_{n}}+b {\rm mod} p\).We show that if b and sufficiently many of the most significant bits of three consecutive values u n of the ICG are given, one can recover in polynomial time the initial value u 0 (even in the case where the coefficient a is unknown) provided that the initial value u 0 does not lie in a certain small subset of exceptional values.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proc. 33rd ACM Symp. on Theory of Comput (STOC 2001), Association for Computing Machinery, pp. 601–610 (2001)
Blackburn, S.R., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Predicting nonlinear pseudorandom number generators (2003) (Preprint)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer, Berlin (1993)
Joux, A., Stern, J.: Lattice reduction: A toolbox for the cryptanalyst. J. Cryptology 11, 161–185 (1998)
Kannan, R.: Algorithmic geometry of numbers. Annual Review of Comp. Sci. 2, 231–267 (1987)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)
Micciancio, D., Goldwasser, S.: Complexity of lattice problems. Kluwer Acad. Publ., Dordrecht (2002)
Nguyen, P.Q., Stern, J.: Lattice reduction in cryptology: An update. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 85–112. Springer, Heidelberg (2000)
Nguyen, P.Q., Stern, J.: The two faces of lattices in cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146–180. Springer, Heidelberg (2001)
Niederreiter, H.: New developments in uniform pseudorandom number and vector generation. In: Niederreiter, H., Shiue, P.J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lect. Notes in Statistics, vol. 106, pp. 87–120. Springer, Berlin (1995)
Niederreiter, H.: Design and analysis of nonlinear pseudorandom number generators. In: Schueller, G.I., Spanos, P.D. (eds.) Monte Carlo Simulation, pp. 3–9. A.A. Balkema Publishers, Rotterdam (2001)
Niederreiter, H., Shparlinski, I.E.: Recent advances in the theory of nonlinear pseudorandom number generators. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Proc. Conf. on Monte Carlo and Quasi-Monte Carlo Methods, pp. 86–102. Springer, Berlin (2000)
Niederreiter, H., Shparlinski, I.E.: Dynamical systems generated by rational functions. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 6–17. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Blackburn, S.R., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E. (2003). Predicting the Inversive Generator. In: Paterson, K.G. (eds) Cryptography and Coding. Cryptography and Coding 2003. Lecture Notes in Computer Science, vol 2898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40974-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-40974-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20663-7
Online ISBN: 978-3-540-40974-8
eBook Packages: Springer Book Archive