20 research outputs found
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
On a conjecture of Marton
We prove a conjecture of K. Marton, widely known as the polynomial
Freiman--Ruzsa conjecture, in characteristic . The argument extends to odd
characteristic, with details to follow in a subsequent paper.Comment: 33 pages, updated version now to be submitted for publicatio
The Kelley--Meka bounds for sets free of three-term arithmetic progressions
We give a self-contained exposition of the recent remarkable result of Kelley
and Meka: if has no non-trivial three-term
arithmetic progressions then for
some constant .
Although our proof is identical to that of Kelley and Meka in all of the main
ideas, we also incorporate some minor simplifications relating to Bohr sets.
This eases some of the technical difficulties tackled by Kelley and Meka and
widens the scope of their method. As a consequence, we improve the lower bounds
for finding long arithmetic progressions in , where .Comment: 20 page
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
Fourier sparsity, spectral norm, and the Log-rank conjecture
We study Boolean functions with sparse Fourier coefficients or small spectral
norm, and show their applications to the Log-rank Conjecture for XOR functions
f(x\oplus y) --- a fairly large class of functions including well studied ones
such as Equality and Hamming Distance. The rank of the communication matrix M_f
for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree
of f and D^CC(f) stand for the deterministic communication complexity for
f(x\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\hat f||_1). In
particular, the Log-rank conjecture holds for XOR functions with constant
F2-degree. 2. D^CC(f) = O(d ||\hat f||_1) = O(\sqrt{rank(M_f)}\logrank(M_f)).
We obtain our results through a degree-reduction protocol based on a variant of
polynomial rank, and actually conjecture that its communication cost is already
\log^{O(1)}rank(M_f). The above bounds also hold for the parity decision tree
complexity of f, a measure that is no less than the communication complexity
(up to a factor of 2).
Along the way we also show several structural results about Boolean functions
with small F2-degree or small spectral norm, which could be of independent
interest. For functions f with constant F2-degree: 1) f can be written as the
summation of quasi-polynomially many indicator functions of subspaces with
\pm-signs, improving the previous doubly exponential upper bound by Green and
Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having
a small parity decision tree complexity; 3) f depends only on polylog||\hat
f||_1 linear functions of input variables. For functions f with small spectral
norm: 1) there is an affine subspace with co-dimension O(||\hat f||_1) on which
f is a constant; 2) there is a parity decision tree with depth O(||\hat f||_1
log ||\hat f||_0).Comment: v2: Corollary 31 of v1 removed because of a bug in the proof. (Other
results not affected.
Structures linéaires dans les ensembles à faible densité
Réalisé en cotutelle avec l'Université Paris-Diderot.Nous présentons trois résultats
en combinatoire additive,
un domaine récent à la croisée
de la combinatoire, l'analyse harmonique
et la théorie analytique des nombres.
Le thème unificateur de notre thèse
est la détection de structures additives
dans les ensembles arithmétiques à faible densité,
avec un intérêt particulier pour les aspects quantitatifs.
Notre première contribution est une estimation
de densité améliorée pour le problème,
initié entre autres par Bourgain,
de trouver une longue progression arithmétique
dans un ensemble somme triple.
Notre deuxième résultat consiste en une généralisation
des bornes de Sanders pour le théorème de Roth,
du cas d'un ensemble dense dans les entiers à
celui d'un ensemble à faible croissance additive
dans un groupe abélien arbitraire.
Finalement, nous étendons
les meilleures bornes quantitatives
connues pour le théorème de Roth dans les premiers,
à tous les systèmes d'équations linéaires
invariants par translation et de
complexité un.We present three results in additive combinatorics,
a recent field at the interface of
combinatorics, harmonic analysis and analytic number theory.
The unifying theme in our thesis
is the detection of additive structure
in arithmetic sets of low density,
with an emphasis on quantitative aspects.
Our first contribution is an improved density estimate
for the problem, initiated by Bourgain and others,
of finding a long arithmetic progression in a triple sumset.
Our second result is a generalization of
Sanders' bounds for Roth's theorem
from the dense setting,
to the setting of small doubling in an arbitrary abelian group.
Finally, we extend the best known quantitative results
for Roth's theorem in the primes,
to all translation-invariant systems
of equations of complexity one
Non-malleable Codes from Additive Combinatorics
Non-malleable codes provide a useful and meaningful security guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely unrelated value. Although such codes do not exist if the family of tampering functions \cF is completely unrestricted, they are known to exist for many broad tampering families \cF. One such natural family is the family of tampering functions in the so called {\em split-state} model. Here the message m is encoded into two shares L and R, and the attacker is allowed to arbitrarily tamper with L and R {\em individually}. The split-state tampering arises in many realistic applications, such as the design of non-malleable secret sharing schemes, motivating the question of designing efficient non-malleable codes in this model.
Prior to this work, non-malleable codes in the split-state model received considerable attention in the literature, but were either (1) constructed in the random oracle model [DPW10], or (2) relied on advanced cryptographic assumptions (such as non-interactive zero-knowledge proofs and leakage-resilient encryption) [LL12], or (3) could only encode 1-bit messages [DKO13]. As our main result, we build the first efficient, multi-bit, information-theoretically-secure non-malleable code in the split-state model.
The heart of our construction uses the following new property of the inner-product function over the vector space F_p^n (for any prime p and large enough dimension n): if L and R are uniformly random over F_p^n, and f,g: F_p^n \rightarrow F_p^n are two arbitrary functions on L and R, the joint distribution (,) is ``close\u27\u27 to the convex combination of affine distributions {(U,c U+d)| c,d \in F_p}, where U is uniformly random in F_p. In turn, the proof of this surprising property of the inner product function critically relies on some results from additive combinatorics, including the so called {\em Quasi-polynomial Freiman-Ruzsa Theorem} (which was recently established by Sanders [San12] as a step towards resolving the Polynomial Freiman-Ruzsa conjecture [Gre05])