Technical Report PHD-2014-10

Title: Additive Combinatorics Methods in Computational Complexity
Authors: Noga Ron-Zewi
Supervisors: Eli Ben-Sasson
Abstract: This thesis focuses on applications of methods and techniques from the mathematical field of additive combinatorics in computational complexity, the sub-area of theoretical computer science that studies the inherent limitations on efficient computation.

Additive combinatorics is the branch of discrete mathematics aimed at quantifying the amount of additive structure in subsets of additive groups. Over the last decade, additive combinatorics has become a successful and active area of mathematics with many remarkable results. In this thesis we show novel applications of additive combinatorics methods in computational complexity, in the sub-fields of pseudorandomness, communication complexity, public key cryptography and local ecoding.

More specifically, in the field of pseudorandomness, we use additive combinatorics methods for the design of randomness extractors. These are procedures that distill (almost) perfect randomness,required for the performance of randomized algorithms and protocols, from weak sources of randomness that exist in nature. Unfortunately, it is impossible to extract randomness from a single weak source of randomness, therefore further assumptions on the structure of the source are needed. One of the most natural assumptions is that we have in hand a pair of independent weak sources of randomness. Another common assumption is that the source has an “algebraic structure”, for example the source is distributed uniformly over an affine subspace (such a source is called an “affine source”). We show how affine extractors, which distill randomness from affine sources, can be converted in a black-box manner to two-source extractors, which distill randomness from a pair of independent weak sources of randomness.

In order to show the above, we introduce a new conjecture in additive combinatorics that we call the approximate duality conjecture and we justify this conjecture by showing its tight connections with the polynomial Freiman-Ruzsa conjecture, a central conjecture in additive combinatorics which attempts to classify ”approximate subgroups” of abelian groups. Since its introduction, the approximate duality conjecture has found a variety of other applications in computational complexity,to communication complexity (see below), to public key cryptography (see below) and to showing limitations on the performance of locally decodable codes [Bhowmick, Dvir and Lovett, STOC 2013].

In the field of communication complexity, we use the approximate duality conjecture mentioned above for the design of communication protocols that minimize the amount of communication needed for performing computational tasks, jointly, by multiple parties. More precisely,the rank of the task serves as a measure of how complex the task is. A fundamental 25-year old conjecture in the area of communication, known as the log-rank conjecture, suggests that one can design communication protocols in which the amount of communication is only polylogarithmic in the rank of the task. However, till very recently there has been essentially no improvement on the trivial protocol in which the amount of communication is linear in the rank. We propose the first non-trivial such protocol in which the amount of communication is sub-linear in the rank.

In the field of public key cryptography, we use the approximate duality conjecture to show limitations on public key encryption from noisy codewords. Public key encryption schemes, considered one of the greatest achievements of modern cryptography, are ’magical’ cryptographic protocols that enable two parties to communicate securely over a public channel without having to agree on a secret key in advance. Several well-known public key encryption schemes, including those of Alekhnovich [STOC 2003], Regev [STOC 2005] and Gentry, Peikert and Vaikuntanathan [STOC 2008], rely on the hardness of inverting a noisy linear encoding. We show that, assuming the approximate duality conjecture, instances of all these schemes over the binary field can be attacked in time sub-exponential in n where n is the maximum of the ciphertext size and the public key size.

Finally, in the field of local decoding, we show applications of additive combinatorics methods to the design of local decoding procedures. These are procedures that allow an extremely efficient detection and correction of errors in transmission in a local manner by examining only a few bits of the corrupted codeword. Our first result in this direction uses almost-periodicity results from additive combinatorics to obtain an improved local decoding procedure for the well-known class of Reed-Muller codes in the scenario of an highly erroneous transmission channel. Our procedure improves on previous such procedures in that its running time and performance guarantee depend only quasipolynomially on the error parameter instead of exponentially. Our second result uses the Waring’s problem over finite fields from additive combinatorics for the design of local decoding procedures for the class of sparse affine-invairant linear codes.

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