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TR03-043 | 14th May 2003 00:00

On epsilon-Biased Generators in NC0



Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there is always a
distinguisher; in fact, they show that it is always
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>3, and conjecture that every NC0 generator can be
broken by a statistical test that simply XORs some bits
of the input. Equivalently, they conjecture that no
NC0 generator can sample an epsilon-biased space with
negligible epsilon.

We refute the conjecture for k>4, and we give a generator
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 exponentially
small in n/c^4.
For large values of k, we construct generators that map n
bits to n^{Omega(sqrt k})} bits and such that every XOR of
outputs has bias exponentially small in n^{1/(2 sqrt k)}.

We also present a polynomial-time distinguisher for the
case k=4, assuming the number of output bits is at least
24n. Our distinguisher has constant distinguishing probability.
For large values of k we show that a linear distinguisher with
a constant distinguishing probability exists once
the number of output bits is bigger than 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 pseudo random generator for which the XOR
of every subset of the outputs has exponentially small bias in
n, and which maps n bits to Omega(n^2) bits.

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