## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

8-2006

#### Publication Source

Random Structures & Algorithms

#### Volume

29

#### Issue

1

#### Start Page

56

#### Last Page

81

#### DOI

10.1002/rsa.20112

#### Abstract

Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC^{0}, that is, a pseudorandom generator that maps *n*-bit strings to *m*-bit strings such that every bit of the output depends on a constant number *k* of bits of the seed.

They show that for *k* = 3, if *m* ≥ 4*n* + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a *linear test*, that is, there is a subset of bits of the output whose XOR has a noticeable bias.

They leave the question open for *k* ≥ 4. In fact, they ask whether every NC^{0} generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC^{0} generator can sample an ε-biased space with negligible ε?

We give a generator for *k* = 5 that maps *n* bits into *cn* bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of *k*, we construct generators that map *n* bits to bits such that every XOR of outputs has bias .

We also present a polynomial-time distinguisher for *k* = 4,*m* ≥ 24*n* having constant distinguishing probability. For large values of *k* we show that a linear distinguisher with a constant distinguishing probability exists once *m* ≥ Ω(2^{k}*n*^{⌈k/2⌉}).

Finally, we consider a variant of the problem where each of the output bits is a degree *k* polynomial in the inputs. We show there exists a degree *k* = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^{−Ω(n)} and which maps *n* bits to Ω(*n*^{2}) bits.

#### Copyright/Permission Statement

This is the peer reviewed version of the following article: Mossel, E., Shpilka, A. and Trevisan, L. (2006), On ε-biased generators in NC^{0}. Random Struct. Alg., 29: 56–81., which has been published in final form at doi: 10.1002/rsa.20112. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving http://olabout.wiley.com/WileyCDA/Section/id-820227.html#terms.

#### Recommended Citation

Mossel, E.,
Shpilka, A.,
&
Trevisan, L.
(2006).
On ε-biased generators in NC^{0}.
*Random Structures & Algorithms,*
*29*
(1),
56-81.
http://dx.doi.org/10.1002/rsa.20112

**Date Posted:** 27 November 2017

This document has been peer reviewed.