# Perfect Local Randomness in Pseudo-random Sequences

## Abstract

The concept of provable cryptographic security for pseudo-random number generators that was introduced by Schnorr is investigated and extended. The cryptanalyst is assumed to have infinite computational resources and hence the security of the generators does not rely on any unproved hypothesis about the difficulty of solving a certain problem, but rather relies on the assumption that the number of bits of the generated sequence the enemy can access is limited. The concept of perfect local randomness of a sequence generator is introduced and investigated using some results from coding theory. The theoretical and practical cryptographic implications of this concept are discussed. Possible extensions of the concept of local randomness as well as some applications are proposed.

## Keywords

Linear Code Block Cipher Stream Cipher Dual Code Codeword Length## References

- [1]N. Alon, L. Babai and A. Itai,
*A fast and simple randomized parallel algorithm for the maximal independent set problem*, Journal of Algorithms, Vol. 7, pp. 567–583, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [2]L. Blum, M. Blum and M. Shub,
*A simple unpredictable pseudo-random number generator*, SIAM J. on Computing, Vol. 15, pp. 364–383, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [3]M. Blum and S. Micali,
*How to generate cryptographically strong sequences of pseudorandom bits*, SIAM J. on Computing, Vol. 13, pp. 850–864, 1984.zbMATHCrossRefMathSciNetGoogle Scholar - [4]B. Chor and O. Goldreich,
*On the power of two-point based sampling*, Journal of Complexity, Vol. 5, No. 1, pp. 96–106, 1989.zbMATHCrossRefMathSciNetGoogle Scholar - [5]B. Chor, O. Goldreich, J. Hastad, J. Freidmann, S. Rudich and R. Smolensky,
*The bit extraction problem or t-resilient functions*, Proc. 26th ann. Symp. on Foundations of Computer Science, pp. 396–407, 1985.Google Scholar - [6]P. Delsarte,
*An algebraic approach to the association schemes of coding theory*, Philips Research Reports Supplements, No. 10, 1973.Google Scholar - [7]A. Joffe,
*On a sequence of almost deterministic pairwise independent random variables*, Proc. Amer. Math. Soc, Vol. 29, No. 2, pp. 381–382, July 1971.zbMATHCrossRefMathSciNetGoogle Scholar - [8]A. Joffe,
*On a set of almost*deterministic κ-independent random variables, The Annals of Probability, Vol. 2, No. 1, pp. 161–162, 1974.zbMATHMathSciNetCrossRefGoogle Scholar - [9]E. Kranakis,
*Primality and cryptography*, Stuttgart and New York: Wiley-Teubner Series in Computer Science, 1986.zbMATHGoogle Scholar - [10]H.O. Lancaster,
*Pairwise statistical independence*, Ann. Math. Statist., Vol. 36, pp. 1313–1317, 1965.MathSciNetzbMATHCrossRefGoogle Scholar - [11]M. Luby,
*A simple parallel algorithm for the maximal independent set problem*, SIAM J. on Computing, Vol. 15, No. 4, pp. 1036–1053, Nov. 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [12]F.J. MacWilliams and N.J.A. Sloane,
*The theory of error-correcting codes*, Amsterdam, New York, Oxford: North-Holland Publishing Company, Fifth Printing, 1986.Google Scholar - [13]R.J. McEliece, E.R. Rodemich, H.C. Rumsey and L.R. Welch,
*New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities*, IEEE Trans. Info. Th., Vol. IT-23, pp. 157–166, 1977.CrossRefMathSciNetGoogle Scholar - [14]S. Micali and C.P. Schnorr,
*Efficient, perfect random number generators*, Preprint MIT, Universität Frankfurt, Nov. 1988.Google Scholar - [15]L.H. Ozarow and A. D. Wyuer, Wire-tap channel II, AT&T Bell Lab. Tech. J., Vol. 63, No. 10, pp. 2135–2157, Dec. 1984.zbMATHGoogle Scholar
- [16]J.-M. Piveteau,
*Local pseudorandom generators*, Preprint, ETH Zürich, 1989.Google Scholar - [17]D. Raghavarao,
*Constructions and combinatorial problems in Design of Experiments*, New York: Wiley, 1971.zbMATHGoogle Scholar - [18]C.P. Schnorr,
*On the construction of random number generators and random function generators*, Proc. EUROCRYPT’88, Lecture Notes in Computer Science, Vol. 330, Springer Verlag, pp. 225–232, 1988.Google Scholar - [19]C.E. Shannon,
*A mathematical theory of communication*, Bell Syst. Tech. J., Vol. 27, pp. 379–423 and 623–656, 1948.MathSciNetGoogle Scholar - [20]T. Verhoeff,
*An updated table of minimum-distance bounds for binary linear codes*, IEEE Trans. Info. Th., Vol. IT-33, pp. 665–680, 1987.CrossRefMathSciNetGoogle Scholar - [21]J.M. Wozencraft and B. Reiffen, Sequential Decoding, MIT Press, Cambridge, MA, 1961.zbMATHGoogle Scholar
- [22]G.Z. Xiao and J.L. Massey,
*A spectral characterization of correlation-immune combining functions*, IEEE Trans. Inform. Theory, Vol. 34, pp. 569–571, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - [23]A.C. Yao,
*Theory and applications of trapdoor functions*, Proc. 23rd IEEE Symposium on Foundations of Computer Science, pp. 80–91, 1982.Google Scholar