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.

翻译：本论文研究了统计密码学原语的扩展。具体而言，我们探讨了抗泄漏秘密共享、非延展性提取器以及免疫化理想单向函数。论文分为三个主要章节。在第一章中，我们证明2选2抗泄漏（及非延展性）秘密共享需要同时具备可提取性的随机源，这排除了使用最小熵源的可能性。第二章引入了抗碰撞种子提取器，证明任何种子提取器都能以较小的种子长度开销实现抗碰撞性。基于此，我们构建了熵率分别为0.81和多对数级别的双源非延展性提取器。该提取器催生了首个能抵御内存篡改攻击的统计隐私放大协议。最后一章研究了3SUM问题的数据结构变体计算难度，其研究动机源于近期提出的针对预计算攻击的随机预言机免疫构造。我们给出了3SUM问题的最坏情况数据结构硬度下界，该下界与自适应敌手场景下已知的数据结构障碍相匹配。针对非自适应场景，我们还给出了略强的下界。最后，我们在比特探测模型中提出了创新性结论。