Quantum Learning Algorithms and Post-Quantum Cryptography
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Quantum algorithms have demonstrated promising speed-ups over classical algorithms in the context of computational learning theory - despite the presence of noise. In this work, we give an overview of recent quantum speed-ups, revisit the Bernstein-Vazirani algorithm in a new learning problem extension over an arbitrary cyclic group and discuss recent applications in cryptography, such as the Learning with Errors problem. We turn to post-quantum cryptography and investigate attacks in which an adversary is given quantum access to a classical encryption scheme. In particular, we consider new notions of security under non-adaptive quantum chosen-ciphertext attacks and propose symmetric-key encryption schemes based on quantum-secure pseudorandom functions that fulfil our definitions. In order to prove security, we introduce a novel relabeling game and show that, in an oracle model, no quantum algorithm making superposition queries can reliably distinguish between the class of functions that are randomly relabeled at a small subset of the domain. Finally, we discuss current progress in quantum computing technology, particularly with regard to the ion-trap architecture, as well as the implementation of quantum algorithms. Moreover, we shed light on the relevance and effectiveness of common noise models adopted in computational learning theory.
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