Abstract
We present an improved version of the one-way to hiding (O2H) Theorem by Unruh, J ACM 2015. Our new O2H Theorem gives higher flexibility (arbitrary joint distributions of oracles and inputs, multiple reprogrammed points) as well as tighter bounds (removing square-root factors, taking parallelism into account). The improved O2H Theorem makes use of a new variant of quantum oracles, semi-classical oracles, where queries are partially measured. The new O2H Theorem allows us to get better security bounds in several public-key encryption schemes.
Keywords
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- 1.
Which allows to reprogram the random oracle at a location that is influenced by the adversary.
- 2.
In Game 7 in [30], a secret \(\delta ^*\) is encrypted using a one-time secure encryption scheme, and the final step in the proof concludes that therefore \(\delta ^*\) cannot be guessed. However, Game 7 contains an oracle \( Dec ^{**}\) that in turn accesses \(\delta ^*\) directly, invalidating that argument.
- 3.
Theorem 1 gives us different options how to define the right game. Conceptually simplest is variant (1) (it does not involve a semi-classical oracle in the right game), but it does not apply in all situations. The basic idea behind all variants is the same, namely that the adversary gets access to an oracle G that behaves differently on the set S of marked elements.
In the present proof, we use specifically variant (4) because then Game 4 will be of a form that is particularly easy to analyze (the adversary has winning probability 0 there).
- 4.
Choosing a different variant here would slightly change the formula below but lead to the same problems.
- 5.
The reason for choosing this particular variant is that same as in footnote 3.
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Acknowledgements
Thanks to Daniel Kane, Eike Kiltz, and Kathrin Hövelmanns for valuable discussions. Ambainis was supported by the ERDF project 1.1.1.5/18/A/020. Unruh was supported by institutional research funding IUT2-1 of the Estonian Ministry of Education and Research, the United States Air Force Office of Scientific Research (AFOSR) via AOARD Grant “Verification of Quantum Cryptography” (FA2386-17-1-4022), the Mobilitas Plus grant MOBERC12 of the Estonian Research Council, and the Estonian Centre of Exellence in IT (EXCITE) funded by ERDF.
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A Optimality of Corollary 1
A Optimality of Corollary 1
Lemma 8
If \(S=\{x\}\) where , then there is a q-query algorithm such that
Proof
The algorithm is as follows:
-
Make the first query with amplitude \(1/\sqrt{N}\) in all positions.
-
Between queries, transform the state by the unitary \(U:=2E/N-I\) where E is the matrix containing 1 everywhere. That U is unitary follows since \(U^\dagger U=4E^2/N^2-4E/N+I=I\) using \(E^2=NE\).
One may calculate by induction that the final non-normalized state has amplitude
in all positions except for the xth one (where the amplitude is 0), so its squared norm is
As a function of 1 / N, this expression’s derivatives alternate on [0, 1 / 2], so it is below its second-order Taylor expansion:
This completes the proof. \(\square \)
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Ambainis, A., Hamburg, M., Unruh, D. (2019). Quantum Security Proofs Using Semi-classical Oracles. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_10
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