Abstract
Trapdoor Claw-free Functions (TCFs) are two-to-one trapdoor functions where it is computationally hard to find a claw, i.e., a colliding pair of inputs. TCFs have recently seen a surge of renewed interest due to new applications to quantum cryptography: as an example, TCFs enable a classical machine to verify that some quantum computation has been performed correctly. In this work, we propose a new family of (almost two-to-one) TCFs based on conjectured hard problems on isogeny-based group actions. This is the first candidate construction that is not based on lattice-related problems and the first scheme (from any plausible post-quantum assumption) with a deterministic evaluation algorithm. To demonstrate the usefulness of our construction, we show that our TCF family can be used to devise a computational test of a qubit, which is the basic building block used in the general verification of quantum computations.
G. Malavolta—Research partially supported by the German Federal Ministry of Education and Research BMBF (grant 16K15K042, project 6GEM) and partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.
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Notes
- 1.
The reduction is entirely classical, so if correlated pseudorandomness holds with respect to all classical PPT adversaries, then the proposition holds for the same class of adversaries as well.
- 2.
Note that knowledge of \(\textbf{t}\) is enough to recover \(\textbf{w}\) even if \(\textbf{w}\) is non-binary but with short entries, i.e., if each entry of \(\textbf{w}\) is polynomially bounded.
- 3.
Although we present our results in terms of EGA, one can also obtain the same results from a restricted EGA assuming a one-time quantum preprocessing, since EGA and restricted EGA are quantumly equivalent [ADMP20].
- 4.
- 5.
Note that one needs to explicitly check for the domain membership of the preimages, similar to what is done for the qubit test protocol.
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Alamati, N., Malavolta, G., Rahimi, A. (2022). Candidate Trapdoor Claw-Free Functions from Group Actions with Applications to Quantum Protocols. In: Kiltz, E., Vaikuntanathan, V. (eds) Theory of Cryptography. TCC 2022. Lecture Notes in Computer Science, vol 13747. Springer, Cham. https://doi.org/10.1007/978-3-031-22318-1_10
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