Cryptographic assumptions and security goals are fundamentally distributional. As a result, statistical techniques are ubiquitous in cryptographic constructions and proofs. In this thesis, we build upon existing techniques and seek to improve both theoretical and practical constructions in three fundamental primitives in cryptography: blockciphers, hash functions, and encryption schemes. First, we present a tighter hybrid argument via collision probability that is more general than previously known, allowing applications to blockciphers. We then use our result to improve the bound of the Swap-or-Not cipher. We also develop a new blockcipher composition theorem that is both class and security amplifying. Second, we prove a variant of Leftover Hash Lemma for joint leakage, inspired by the Universal Computational Extractor (UCE) assumption. We then apply this technique to construct various standard-model UCE- secure hash functions. Third, we survey existing "lossy primitives" in cryptography, in particular Lossy Trapdoor Functions (LTDF) and Lossy Encryptions (LE); we pro- pose a generalized primitive called Lossy Deterministic Encryption (LDE). We show that LDE is equivalent to LTDFs. This is in contrast with the block-box separation of trapdoor functions and public-key encryption schemes in the computational case. One common theme in our methods is the focus on statistical techniques. Another theme is that the results obtained are in contrast with their computational counterparts---the corresponding computational results are implausible or are know to be false.