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Security notions for the random oracle model in classical and quantum settings

Jeeun Lee, Seunghyun Lee, Kwangjo Kim

Published Jan 25, 2018• 1 citation
OpenAlex

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quant-pharXiv:2605.09957•May 12, 2026•0 citations
In cryptography, generic transformations are often used to strengthen simpler but weaker schemes, into more sophisticated and stronger ones. This thesis aims to establish rigorous, provable notions of security for various such transformations that are relevant in the scope of post-quantum cryptography. We achieve this via formal mathematical proofs, and in some cases, via proposing new security definitions, when existing ones are inadequate. Most analyses are treated in an idealized setting known as the random oracle model.
We consider the task of constructing pseudorandom unitaries (PRUs) with scalable security, i.e. families in which the security parameter may vary independently of the dimension (or input bit-length). It is not known whether scalable PRUs can be constructed. In this work we show that if scalable PRUs can be constructed via the prevailing paradigm for analyzing PRUs, then there would be a positive solution to the Aaronson-Kuperberg unitary synthesis problem, a longstanding question in quantum complexity theory about whether implementing arbitrary unitaries can be efficiently reduced to computing a Boolean function. Specifically, we formalize the notion of ROM-PRUs, which are statistically secure PRUs in the random oracle model (ROM). All prior known constructions of cryptographically secure PRUs are based on a ROM-PRU construction. We prove novel connections between ROM-PRUs, approximate unitary designs, epsilon-nets over the unitary group, and the unitary synthesis problem. In particular, we prove that any unitary synthesis algorithm (and thus any ROM-PRU) must use a classical oracle with input length (2 - o(1)) log d bits, where d is the dimension of the unitary to be implemented. This bound rules out all existing candidates for scalable PRUs in the literature. Together, these connections indicate that ROM-PRUs provide a fruitful idealized model for studying pseudorandom unitaries.