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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A secret-sharing scheme allows to distribute a secret s among n parties such that only some predefined "authorized" sets of parties can reconstruct the secret s, and all other "unauthorized" sets learn nothing about s. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size 2^{n-o(n)} and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to 2^{0.994n+o(n)}, and this was further improved by several follow-ups accumulating in an upper bound of 1.5^{n+o(n)} (Applebaum and Nir, CRYPTO 2021). Following these advances, it is natural to ask whether these new approaches can lead to a truly sub-exponential upper-bound of 2^{n^{1-ε}} for some constant ε > 0, or even all the way down to polynomial upper-bounds.
In this paper, we relate this question to the complexity of computing monotone Boolean functions by monotone real circuits (MRCs) - a computational model that was introduced by Pudlák (J. Symb. Log., 1997) in the context of proof complexity. We introduce a new notion of "separable" MRCs that lies between monotone real circuits and monotone real formulas (MRFs). As our main results, we show that recent constructions of general secret-sharing schemes implicitly give rise to separable MRCs for general monotone functions of similar complexity, and that some monotone functions (in monotone NP) cannot be computed by sub-exponential size separable MRCs. Interestingly, it seems that proving similar lower-bounds for general MRCs is beyond the reach of current techniques.
We use this connection to obtain lower-bounds against a natural family of secret-sharing schemes, as well as new non-trivial upper-bounds for MRCs. Specifically, we conclude that recent approaches for secret-sharing schemes cannot achieve sub-exponential share size and that every monotone function can be realized by an MRC (or even MRF) of complexity 1.5^{n+o(n)}. To the best of our knowledge, this is the first improvement over the trivial 2^{n-o(n)} upper-bound. Along the way, we show that the recent constructions of general secret-sharing schemes implicitly give rise to Boolean formulas over slice functions and prove that such formulas can be simulated by separable MRCs of similar size. On a conceptual level, our paper continues the rich line of study that relates the share size of secret-sharing schemes to monotone complexity measures.

Benny Applebaum, Amos Beimel, Oded Nir, Naty Peter, and Toniann Pitassi. Secret Sharing, Slice Formulas, and Monotone Real Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{applebaum_et_al:LIPIcs.ITCS.2022.8, author = {Applebaum, Benny and Beimel, Amos and Nir, Oded and Peter, Naty and Pitassi, Toniann}, title = {{Secret Sharing, Slice Formulas, and Monotone Real Circuits}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {8:1--8:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.8}, URN = {urn:nbn:de:0030-drops-156046}, doi = {10.4230/LIPIcs.ITCS.2022.8}, annote = {Keywords: Secret Sharing Schemes, Monotone Real Circuits} }

Document

**Published in:** LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

We define and study a new cryptographic primitive, named One-One Constrained Pseudorandom Functions. In this model there are two parties, Alice and Bob, that hold a common random string K, where Alice in addition holds a predicate f:[N] → {0,1} and Bob in addition holds an input x ∈ [N]. We then let Alice generate a key K_f based on f and K, and let Bob evaluate a value K_x based on x and K. We consider a third party that sees the values (x,f,K_f) and the goal is to allow her to reconstruct K_x whenever f(x)=1, while keeping K_x pseudorandom whenever f(x)=0. This primitive can be viewed as a relaxation of constrained PRFs, such that there is only a single key query and a single evaluation query.
We focus on the information-theoretic setting, where the one-one cPRF has perfect correctness and perfect security. Our main results are as follows.
1) A Lower Bound. We show that in the information-theoretic setting, any one-one cPRF for punctured predicates is of exponential complexity (and thus the lower bound meets the upper bound that is given by a trivial construction). This stands in contrast with the well known GGM-based punctured PRF from OWF, which is in particular a one-one cPRF. This also implies a similar lower bound for all NC1.
2) New Constructions. On the positive side, we present efficient information-theoretic constructions of one-one cPRFs for a few other predicate families, such as equality predicates, inner-product predicates, and subset predicates. We also show a generic AND composition lemma that preserves complexity.
3) An Amplification to standard cPRF. We show that all of our one-one cPRF constructions can be amplified to a standard (single-key) cPRF via any key-homomorphic PRF that supports linear computations. More generally, we suggest a new framework that we call the double-key model which allows to construct constrained PRFs via key-homomorphic PRFs.
4) Relation to CDS. We show that one-one constrained PRFs imply conditional disclosure of secrets (CDS) protocols. We believe that this simple model can be used to better understand constrained PRFs and related cryptographic primitives, and that further applications of one-one constrained PRFs and our double-key model will be found in the future, in addition to those we show in this paper.

Naty Peter, Rotem Tsabary, and Hoeteck Wee. One-One Constrained Pseudorandom Functions. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{peter_et_al:LIPIcs.ITC.2020.13, author = {Peter, Naty and Tsabary, Rotem and Wee, Hoeteck}, title = {{One-One Constrained Pseudorandom Functions}}, booktitle = {1st Conference on Information-Theoretic Cryptography (ITC 2020)}, pages = {13:1--13:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-151-1}, ISSN = {1868-8969}, year = {2020}, volume = {163}, editor = {Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.13}, URN = {urn:nbn:de:0030-drops-121188}, doi = {10.4230/LIPIcs.ITC.2020.13}, annote = {Keywords: Constrained pseudorandom functions, function secret-sharing, conditional disclosure of secrets} }

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