Abstract
Somewhere statistically binding (SSB) hashing allows us to sample a special hashing key such that the digest statistically binds the input at m secret locations. This hash function is said to be somewhere extractable (SE) if there is an additional trapdoor that allows the extraction of the input bits at the m locations from the digest.
Devadas, Goyal, Kalai, and Vaikuntanathan (FOCS 2022) introduced a variant of somewhere extractable hashing called rate-1 fully local SE hash functions. The rate-1 requirement states that the size of the digest is \(m + \textsf{poly}(\lambda )\) (where \(\lambda \) is the security parameter). The fully local property requires that for any index i, there is a “very short” opening showing that i-th bit of the hashed input is equal to b for some \(b \in \{0,1\}\). The size of this opening is required to be independent of m and in particular, this means that its size is independent of the size of the digest. Devadas et al. gave such a construction from Learning with Errors (LWE).
In this work, we give a construction of a rate-1 fully local somewhere extractable hash function from Decisional Diffie-Hellman (DDH) and BARGs. Under the same assumptions, we give constructions of rate-1 BARG and RAM SNARG with partial input soundness whose proof sizes are only matched by prior constructions based on LWE.
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Notes
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This framework can also be trivially adapted to use a \(\ell \)-ary tree, instead of a binary one. The resulting CRS has size \(\log _\ell (k)\cdot \textsf{poly}(\lambda ,\ell )\).
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Acknowledgements
Pedro Branco was partially funded by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038.
Nico Döttling and Riccardo Zanotto: Funded by the European Union (ERC, LACONIC, 101041207). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
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Branco, P., Döttling, N., Srinivasan, A., Zanotto, R. (2024). Rate-1 Fully Local Somewhere Extractable Hashing from DDH. In: Tang, Q., Teague, V. (eds) Public-Key Cryptography – PKC 2024. PKC 2024. Lecture Notes in Computer Science, vol 14603. Springer, Cham. https://doi.org/10.1007/978-3-031-57725-3_12
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