Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6 √ n. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is "truly" constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

We study the question of designing cryptographic schemes which are secure even if an arbitrary function f (sk) of the secret key is leaked, as long as the secret key sk is still (exponentially) hard to compute from this auxiliary input. This setting of auxiliary input is more general than the more traditional setting, which assumes that some of information about the secret key sk may be leaked, but sk still has high min-entropy left. In particular, we deal with situations where f (sk) information-theoretically determines the entire secret key sk.As our main result, we construct CPA/CCA secure symmetric encryption schemes that remain secure with exponentially hard-to-invert auxiliary input. We give several applications of such schemes.• We construct an average-case obfuscator for the class of point functions, which remains secure with exponentially hard-to-invert auxiliary input, and is reusable. • We construct a reusable and robust extractor that remains secure with exponentially hard-to-invert auxiliary input.Our results rely on a new cryptographic assumption, Learning Subspace-with-Noise (LSN), which is related to the well known Learning Parity-with-Noise (LPN) assumption.

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman [4] on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for all distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura [8] on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to all distances.

A degree-d polynomial p in n variables over a field F is equidistributed if it takes on each of its |F| values close to equally often, and biased otherwise. We say that p has a low rank if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [GT07] have shown that bias imply low rank over large fields (i.e. for the case d < |F|). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be approximated by few polynomials of bounded degree, can be computed by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.

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" F is completely unrestricted, they are known to exist for many broad tampering families F. One such natural family is the family of tampering functions in the so called 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 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 constructed either (1) in the random oracle model, or (2) relied on advanced cryptographic assumptions (such as non-interactive zero-knowledge proofs and leakageresilient encryption), or (3) could only encode 1-bit messages. 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 L, R over the vector space F n p (for a prime p and large enough dimension n): if L and R are uniformly random over F n p , and f, g : F n p → F n p are two arbitrary functions on L and R , then the joint distribution (L, R , f (L), g(R)) is "close" to the convex combination of "affine distributions" {(U, aU + b) | a, b ∈ 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 Quasi-polynomial Freiman-Ruzsa Theorem which was recently established by Sanders [San12] as a step towards resolving the Polynomial Freiman-Ruzsa Conjecture [Gre05].