Post-quantum security in the presence of superposition queries

As we are going toward the quantum era, the need to revisit the security of cryptographic constructions against a quantum adversary is getting more appreciable. Post-quantum cryptography is an emerging discipline that deals with classical cryptographic constructions that remain secure against a quantum adversary. In this setting, the honest parties are willing to communicate using only classical devices while the adversary may have a quantum computing device. Due to the breaking of many public-key cryptosystems based on the factoring and discrete logarithm problems by Shor’s quantum algorithm, to achieve the post-quantum security, it is necessary to substitute those hard problems with some quantumhard problems. Then we need to design the cryptographic construction based on quantum hard problems. Finally, we have to prove mathematically the security of the new constructions. The classical security proof techniques may fail against a quantum adversary due to some strange properties of a quantum adversary, therefore, even if we design a cryptographic construction based on a quantum-hard problem, proving the security of the construction remains a challenging task. In this thesis, we focus on proving the quantum security of some cryptographic constructions. We also present some quantum attacks to argue the insecurity of some constructions. We prove the post-quantum security of a slightly modified version of the FujisakiOkamoto (FO) construction that transforms two weakly secure encryption schemes into a strongly secure one using three hash functions in the quantum random oracle model. In the quantum random oracle model the quantum adversary has superposition (quantum) access to the random oracles. In order to prove the security of the FO construction, we need to study the properties of hash functions in the quantum setting. The Indifferentiability Framework has been used to prove the soundness of some hash function constructions when underlying primitive used in the construction is modeled as an ideal primitive. In the quantum setting, a quantum adversary can evaluate an ideal primitive in superposition, therefore we need to redefine the Indifferentiability Framework for a quantum adversary. We define the quantum indifferentiability and show that most of classical constructions of hash functions are not perfectly-quantum indifferentiable from a random oracle under some conjecture. Our quantum definition of indifferentiability allows superposition access to the underlying primitive used in the construction, however, since we claim the impossibility of quantum indifferentiability, we consider a classical access to the construction and this leads to a stronger impossibility result. In other words, our impossibility result will hold if we consider a quantum indifferentiability definition in which the adversary has superposition access to both the construction and the underlying primitive because this definition is stronger than ours. Also, we study the collision-resistance property of a function whose outputs are chosen according to some non-uniform distribution against a quantum adversary that has quantum access to the function. We obtain some upper and lower bounds