48 research outputs found
Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory
Let be a polynomial of degree in variables over a finite field
. The polynomial is said to be unbiased if the distribution of
for a uniform input is close to the uniform
distribution over , and is called biased otherwise. The polynomial
is said to have low rank if it can be expressed as a composition of a few lower
degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and
Lovett [FOCS 2008] showed that bias implies low rank for fixed degree
polynomials over fixed prime fields. This lies at the heart of many tools in
higher order Fourier analysis. In this work, we extend this result to all prime
fields (of size possibly growing with ). We also provide a generalization to
nonprime fields in the large characteristic case. However, we state all our
applications in the prime field setting for the sake of simplicity of
presentation.
As an immediate application, we obtain improved bounds for a suite of
problems in effective algebraic geometry, including Hilbert nullstellensatz,
radical membership and counting rational points in low degree varieties.
Using the above generalization to large fields as a starting point, we are
also able to settle the list decoding radius of fixed degree Reed-Muller codes
over growing fields. The case of fixed size fields was solved by Bhowmick and
Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman
[STOC 2008]. Here, we show that the list decoding radius is equal the minimum
distance of the code for all fixed degrees, even when the field size is
possibly growing with
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree
Bounds on the leakage of the input's distribution in information-hiding protocols
International audienceIn information-hiding, an adversary that tries to infer the secret information has a higher probability of success if it knows the distribution on the secrets. We show that if the system leaks probabilistically some information about the secrets, (that is, if there is a probabilistic correlation between the secrets and some observables) then the adversary can approximate such distribution by repeating the observations. More precisely, it can approximate the distribution on the observables by computing their frequencies, and then derive the distribution on the secrets by using the correlation in the inverse direction. We illustrate this method, and then we study the bounds on the approximation error associated with it, for various natural notions of error. As a case study, we apply our results to Crowds, a protocol for anonymous communication