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In this talk, we will explore various lifting theorems establishing security in the quantum idealized models. The high-level theme of these lifting theorems is to relate the success probability of an arbitrary quantum adversary to that of a classical algorithm making only a small number of classical queries. We will see some example applications and limitations of the lifting theorems, as well as some open problems.

Several well-known beyond-birthday-bound PRFs are built by XORing outputs of permutations. In this talk, I will describe some of the main ideas used in recent Fourier-analytic tight security proofs of such PRFs. I will then discuss some open problems in the application of Fourier analysis in provable security of symmetric-key primitives.