50 research outputs found
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Revisiting the Sanders-Freiman-Ruzsa Theorem in and its Application to Non-malleable Codes
Non-malleable codes (NMCs) protect sensitive data against degrees of
corruption that prohibit error detection, ensuring instead that a corrupted
codeword decodes correctly or to something that bears little relation to the
original message. The split-state model, in which codewords consist of two
blocks, considers adversaries who tamper with either block arbitrarily but
independently of the other. The simplest construction in this model, due to
Aggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bit
messages to -bit codewords. It is conjectured, however, that the
construction allows linear-length codewords. Towards resolving this conjecture,
we show that the construction allows for code-length . This is achieved
by analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for general
Abelian groups. Closely following the excellent exposition of this result for
the group by Lovett, we expose its dependence on for the
group , where is a prime
Lower bounds for adaptive linearity tests
Linearity tests are randomized algorithms which have oracle access to the
truth table of some function f, and are supposed to distinguish between linear
functions and functions which are far from linear. Linearity tests were first
introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP
theorem, among other applications. The quality of a linearity test is described
by its correctness c - the probability it accepts linear functions, its
soundness s - the probability it accepts functions far from linear, and its
query complexity q - the number of queries it makes. Linearity tests were
studied in order to decrease the soundness of linearity tests, while keeping
the query complexity small (for one reason, to improve PCP constructions).
Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the
Complete Graph Test, and prove that no Hyper Graph Test can perform better than
the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove,
among other results, that no non-adaptive linearity test can perform better
than the Complete Graph Test. Their proof uses the algebraic machinery of the
Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to
generalize this lower bound also to adaptive linearity tests. We also prove the
same optimal lower bound for adaptive linearity test, but our proof technique
is arguably simpler and more direct than the one used in (Samorodnitsky and
Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the
behavior of linearity tests on quadratic functions. However, instead of
analyzing the Gowers Norm of certain functions, we provide a more direct
combinatorial proof, studying the behavior of linearity tests on random
quadratic functions..
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
We establish a correspondence between inverse sumset theorems (which can be
viewed as classifications of approximate (abelian) groups) and inverse theorems
for the Gowers norms (which can be viewed as classifications of approximate
polynomials). In particular, we show that the inverse sumset theorems of
Freiman type are equivalent to the known inverse results for the Gowers U^3
norms, and moreover that the conjectured polynomial strengthening of the former
is also equivalent to the polynomial strengthening of the latter. We establish
this equivalence in two model settings, namely that of the finite field vector
spaces F_2^n, and of the cyclic groups Z/NZ.
In both cases the argument involves clarifying the structure of certain types
of approximate homomorphism.Comment: 23 page