Unitary Complexity and the Uhlmann Transformation Problem

State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann's theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.Comment: 126 pages, comments welcom

Probabilistic Proof Systems

Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems -- interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs -- stressing the essential role of randomness in each of them. This exposition is an expanded version of a survey written for the proceedings of the International Congress of Mathematicians (ICM94) held in Zurich in 1994. It is hope that this exposition may be accessible to a broad audience of computer scientists and mathematians

Indistinguishability Obfuscation: From Approximate to Exact

We show general transformations from subexponentially-secure approximate indistinguishability obfuscation (IO) where the obfuscated circuit agrees with the original circuit on a 1/2+ϵ fraction of inputs on a certain samplable distribution, into exact indistinguishability obfuscation where the obfuscated circuit and the original circuit agree on all inputs. As a step towards our results, which is of independent interest, we also obtain an approximate-to-exact transformation for functional encryption. At the core of our techniques is a method for “fooling” the obfuscator into giving us the correct answer, while preserving the indistinguishability-based security. This is achieved based on various types of secure computation protocols that can be obtained from different standard assumptions. Put together with the recent results of Canetti, Kalai and Paneth (TCC 2015), Pass and Shelat (TCC 2016), and Mahmoody, Mohammed and Nemathaji (TCC 2016), we show how to convert indistinguishability obfuscation schemes in various ideal models into exact obfuscation schemes in the plain model.National Science Foundation (U.S.) (Grant CNS-1350619)National Science Foundation (U.S.) (Grant CNS-1414119

Multilinear Maps from Obfuscation

International audienceWe provide constructions of multilinear groups equipped with natural hard problems from in-distinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the DDH assumption hold for them. Our first construction is symmetric and comes with a κ-linear map e : G κ −→ G T for prime-order groups G and G T. To establish the hardness of the κ-linear DDH problem, we rely on the existence of a base group for which the (κ − 1)-strong DDH assumption holds. Our second construction is for the asymmetric setting, where e : G 1 × · · · × G κ −→ G T for a collection of κ + 1 prime-order groups G i and G T , and relies only on the standard DDH assumption in its base group. In both constructions the linearity κ can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: (probabilistic) indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness indistinguishability and zero knowledge), and additively homomorphic encryption for the group Z + N. At a high level, we enable " bootstrapping " multilinear assumptions from their simpler counterparts in standard cryptographic groups, and show the equivalence of IO and multilinear maps under the existence of the aforementioned primitives

Post-quantum security of hash functions

The research covered in this thesis is dedicated to provable post-quantum security of hash functions. Post-quantum security provides security guarantees against quantum attackers. We focus on analyzing the sponge construction, a cryptographic construction used in the standardized hash function SHA3. Our main results are proving a number of quantum security statements. These include standard-model security: collision-resistance and collapsingness, and more idealized notions such as indistinguishability and indifferentiability from a random oracle. All these results concern quantum security of the classical cryptosystems. From a more high-level perspective we find new applications and generalize several important proof techniques in post-quantum cryptography. We use the polynomial method to prove quantum indistinguishability of the sponge construction. We also develop a framework for quantum game-playing proofs, using the recently introduced techniques of compressed random oracles and the One-way-To-Hiding lemma. To establish the usefulness of the new framework we also prove a number of quantum indifferentiability results for other cryptographic constructions. On the way to these results, though, we address an open problem concerning quantum indifferentiability. Namely, we disprove a conjecture that forms the basis of a no-go theorem for a version of quantum indifferentiability

Transfinite Extension of the Mu-Calculus

In [1] Bradfield found a link between finite differences formed by Sigma(2)(0) sets and the mu-arithmetic introduced by Lubarski [7]. We extend this approach into the transfinite: in allowing countable disjunctions we show that this kind of extended mu-calculus matches neatly to the transfinite difference hierarchy of E-2(0) sets. The difference hierarchy is intimately related to parity games. When passing to infinitely many priorities, it might not longer be true that there is a positional winning strategy. However, if such games are derived from the difference hierarchy, this property still holds true