On the Bogolyubov–Ruzsa lemma

Sanders, Tom

doi:10.2140/apde.2012.5.627

Cited by 140 publications

(174 citation statements)

References 45 publications

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“…What may be called the core of additive combinatorics is the study of the behavior of arbitrary sets under addition (as opposed to, say, the primes or kth powers). In this sense, the subject originated from at least two streams, one coursing through work on arithmetic progressions by Schur, van der Waerden, Roth [Rot53], Szemerédi [Sze69], Furstenberg [Fur77] (leading to the ergodic work of Host and Kra [HK05] and Ziegler [Zie07]; see also Szegedy [Sze]), Gowers [Gow01], and Green and Tao [GT08], among many others, and another one based on the study of growth in abelian groups, starting with Freiman [Fre73], Erdős and Szemerédi [ES83], and Ruzsa [Ruz94], [Ruz99] and continuing in [GR05], [Cha02], [San12]. There has also been a vein of a more geometrical flavor (e.g., [ST83]).…”

Section: Methodological Background: Arithmetic Combinatoricsmentioning

confidence: 99%

“…The strength of this result must be underlined: A is growing by a factor of |A| δ , where δ > 0 is moreover independent of p. In contrast, even after impressive recent improvements ( [San12], see also [CS10]), the main additive-combinatorial result for abelian groups (Freiman's theorem) gives growth by smaller factors.…”

Section: And Thus Of a Hence μ(Hg) K −O(1)mentioning

confidence: 99%

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2015

Bull. Amer. Math. Soc.

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Section: Methodological Background: Arithmetic Combinatoricsmentioning

confidence: 99%

Section: And Thus Of a Hence μ(Hg) K −O(1)mentioning

confidence: 99%

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2015

Bull. Amer. Math. Soc.

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“…The main step is to exhibit a large monochromatic affine subspace for f if the communication complexity of F is small. To this end, we adapt the quasipolynomial Bogolyubov-Ruzsa lemma [14], which says that 4A…”

Section: Our Techniquesmentioning

confidence: 99%

“…Readers may refer to the excellent textbook [16]. The following is the famous quasi-polynomial Bogolyubov-Ruzsa lemma due to Sanders [14]. It asserts that 4A contains a large subspace if A ⊆ F n 2 is large.…”

Section: Fact 25 Let F : {0 1}mentioning

confidence: 99%

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Yao¹

2016

Chicago J. of Theoretical Comp. Sci.

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“…The last partition we need to analyze is L a × R 1 , corresponding to the case where f is far from linear and g is far from constant. For this, we need the following result that can be seen as a generalization of the linearity test from [31] that was proved in [3] using results from [6,21,32].…”

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Optimal Computational Split-state Non-malleable Codes

Aggarwal

Agrawal

Gupta

et al. 2015

Theory of Cryptography

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Abstract. Non-malleable codes are a generalization of classical errorcorrecting codes where the act of "corrupting" a codeword is replaced by a "tampering" adversary. Non-malleable codes guarantee that the message contained in the tampered codeword is either the original message m, or a completely unrelated one. In the common split-state model, the codeword consists of multiple blocks (or states) and each block is tampered with independently. The central goal in the split-state model is to construct high rate nonmalleable codes against all functions with only two states (which are necessary). Following a series of long and impressive line of work, constant rate, two-state, non-malleable codes against all functions were recently achieved by Aggarwal et al. (STOC 2015). Though constant, the rate of all known constructions in the split state model is very far from optimal (even with more than two states). In this work, we consider the question of improving the rate of split-state non-malleable codes. In the "information theoretic" setting, it is not possible to go beyond rate 1/2. We therefore focus on the standard computational setting. In this setting, each tampering function is required to be efficiently computable, and the message in the tampered codeword is required to be either the original message m or a "computationally" independent one. In this setting, assuming only the existence of one-way functions, we present a compiler which converts any poor rate, two-state, (sufficiently strong) non-malleable code into a rate-1, two-state, computational nonmalleable code. These parameters are asymptotically optimal. Furthermore, for the qualitative optimality of our result, we generalize the result of Cheraghchi and Guruswami (ITCS 2014) to show that the existence of one-way functions is necessary to achieve rate > 1/2 for such codes. Our compiler requires a stronger form of non-malleability, called augmented non-malleability. This notion requires a stronger simulation guarantee for non-malleable codes and simplifies their modular usage in cryptographic settings where composition occurs. Unfortunately, this form of non-malleability is neither straightforward nor generally guaranteed by known results. Nevertheless, we prove this stronger form of nonmalleability for the two-state construction of Aggarwal, Dodis, and Lovett (STOC 14). This result is of independent interest.

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On the Bogolyubov–Ruzsa lemma

Abstract: Our main result is that if A is a finite subset of an abelian group with jA C Aj 6 KjAj, then 2A 2A contains an O.log O.1/ 2K/-dimensional coset progression M of size at least exp. O.log O

Cited by 140 publications

References 45 publications

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Optimal Computational Split-state Non-malleable Codes

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