In this thesis, we study extensions of statistical cryptographic primitives. In particular we study leakage-resilient secret sharing, non-malleable extractors, and immunized ideal one-way functions. The thesis is divided into three main chapters. In the first chapter, we show that 2-out-of-2 leakage resilient (and also non-malleable) secret sharing requires randomness sources that are also extractable. This rules out the possibility of using min-entropic sources. In the second, we introduce collision-resistant seeded extractors and show that any seeded extractor can be made collision resistant at a small overhead in seed length. We then use it to give a two-source non-malleable extractor with entropy rate 0.81 in one source and polylogarithmic in the other. The non-malleable extractor lead to the first statistical privacy amplification protocol against memory tampering adversaries. In the final chapter, we study the hardness of the data structure variant of the 3SUM problem which is motivated by a recent construction to immunise random oracles against pre-processing adversaries. We give worst-case data structure hardness for the 3SUM problem matching known barriers in data structures for adaptive adversaries. We also give a slightly stronger lower bound in the case of non-adaptivity. Lastly, we give a novel result in the bit-probe setting.

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