Abstract
We call a distribution on n-bit strings (ɛ, e)-locally random, if for every choice of e ≤ n positions the induced distribution on e-bit strings is in the L 1-norm at most ɛ away from the uniform distribution on e-bit strings. We establish local randomness in polynomial random number generators (RNG) that are candidate one-way functions. Let N be a squarefree integer and let f 1, ..., f ℓ be polynomials with coefficients in ℤN = ℤ/Nℤ. We study the RNG that stretches a random x ∈ ℤN into the sequence of least significant bits of f 1(x), ..., f ℓ(x). We show that this RNG provides local randomness if for every prime divisor p of N the polynomials f 1, ..., f ℓ are linearly independent modulo the subspace of polynomials of degree ≤ 1 in ℤp[x]. We also establish local randomness in polynomial random function generators. This yields candidates for cryptographic hash functions. The concept of local randomness in families of functions extends the concept of universal families of hash functions by Carter and Wegman (1979). The proofs of our results rely on upper bounds for exponential sums.
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Alexi, W., Chor, B., Goldreich, O. and Schnorr, C.P.: RSA and Rabin Functions: certain parts are as hard as the whole. SIAM J. Comput., 17,2 (1988), pp. 194–208.
Alon, N., Babai, L. and Itai, A.: A fast and simple randomised parallel algorithm for the maximal independent set problem. J. of Alg. 7 (1986), pp. 567–583.
Alon, N., Goldreich, O., Hastad, J. and Peralta, R.: Simple constructions of almost k-wise independent random variables. Proceedings of the 31st IEEE Symposium on Foundations of Computer Science (1990) pp. 544–552.
Blum, L, Blum, M., and Shub, M.: A simple unpredictable pseudo-random number generator. SIAM J. Comput. 15 (1986), pp. 364–383.
Blum, M. and Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, IEEE, New York (1982); also SIAM J. Comput. 13 (1984), pp. 850–864.
Carlitz, L. and Uchiyama, S.: Bounds for exponential sum. Duke Math. J. 24, (1957), pp. 37–41.
Carter, L. and Wegman, M.: Universal hash functions. J. Comp. and Syst. Sci. 18, (1979) pp. 143–154.
Goldreich, O., Goldwasser, S. and Micali, S.: How to Construct Random Functions. Proceedings of the 25th IEEE Symposium on Foundations of Computer Science, New York (1984); also Journal ACM 33, 4 (1986), pp. 792–807.
Lidl, R. and Niederreiter, H.: Finite Fields. Reading: Addison-Wesley 1983.
Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput., 15 (1986), pp. 1036–1053.
Maurer, U. M., and Massey, J.L.: Perfect local randomness in pseudo-random sequences. Proceedings Crypto’ 89, Lecture Notes in Computer Science, Vol. 435, Springer-Verlag 1990, pp. 100–112.
Micali, S. and Schnorr, C.P.: Efficient, perfect polynomial random number generators. J. of Cryptology 3, (1991), pp. 157–172.
Naor, J. and Naor, M: Small-bias Probability Spaces: Efficient Constructions and Applications. Proceedings of the 22nd ACM Symposium on Theory of Computing (1990), pp. 213–223.
Niederreiter, H.: Pseudo-random numbers and optimal coefficients. Advances in Math. 26, (1977) pp. 99–181.
Nisan, N.: Pseudorandom generators for space-bounded computation. Proceedings of the 22nd ACM Symposium on Theory of Computing (1990), pp. 204–208.
Schnorr, C.P.: On the construction of random number generators and random function generators. Proc. EUROCRYPT’ 88, Lecture Notes in Computer Science, Vol. 330, Springer-Verlag 1988, pp. 225–232.
Wil, A.: On some exponential sums. Proc. Nat. Acad. Sci. USA 34, (1948), pp. 204–207.
Yao, A.C.: Theory and applications of trapdoor functions. Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, IEEE, New York (1982), pp. 80–91.
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Niederreiter, H., Schnorr, C.P. (1993). Local Randomness in Candidate One-Way Functions. In: Rueppel, R.A. (eds) Advances in Cryptology — EUROCRYPT’ 92. EUROCRYPT 1992. Lecture Notes in Computer Science, vol 658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47555-9_33
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DOI: https://doi.org/10.1007/3-540-47555-9_33
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