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 $\mathbb{F}$. The Reed-Muller code $\mathrm{RM}_{\mathbb{F}}(n,d)$ is defined by $n$-variate degree-$d$ polynomials over $\mathbb{F}$. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field $\mathbb{F}=\mathbb{F}_p$, constant degree $d$ and large $n$. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by $\delta(d)$ the normalized minimal distance of $\mathrm{RM}_{\mathbb{F}}(n,d)$, then the number of codewords in any ball of radius $\delta(d)-\varepsilon$ is bounded by $c=c(p,d,\varepsilon)$ independent of $n$. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008], who among other results proved it in the special case of $\mathbb{F}=\mathbb{F}_2$; and extends the work of Gopalan [FOCS 2010] who proved the conjecture in the case of $d=2$. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$, we show that the number of codewords of $\mathrm{RM}_{\mathbb{F}}(n,d)$ in a ball of radius $\delta(e) - \varepsilon$ is bounded by $\exp(c \cdot n^{d-e})$, where $c=c(p,d,\varepsilon)$ is independent of $n$. The dependence on $n$ is tight. This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who proved similar bounds over $\mathbb{F}_2$. 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