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On One-Way Functions and Sparse Languages

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14369))

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Abstract

We show equivalence between the existence of one-way functions and the existence of a sparse language that is hard-on-average w.r.t. some efficiently samplable “high-entropy” distribution. In more detail, the following are equivalent:

  • The existence of a \(S(\cdot )\)-sparse language L that is hard-on-average with respect to some samplable distribution with Shannon entropy \(h(\cdot )\) such that \(h(n)-\log (S(n)) \ge 4\log n\);

  • The existence of a \(S(\cdot )\)-sparse language \(L \in \textsf{NP}\), that is hard-on-average with respect to some samplable distribution with Shannon entropy \(h(\cdot )\) such that \(h(n)-\log (S(n)) \ge n/3\);

  • The existence of one-way functions.

where a language L is said to be \(S(\cdot )\)-sparse if \(|L \cap \{0,1\}^n| \le S(n)\) for all \(n \in \mathbb {N}\). Our results are inspired by, and generalize, results from the elegant recent paper by Ilango, Ren and Santhanam (IRS, STOC’22), which presents similar connections for specific sparse languages.

Supported by a JP Morgan fellowship.

Supported in part by NSF Award CNS 2149305, NSF Award CNS-2128519, NSF Award RI-1703846, AFOSR Award FA9550-18-1-0267, FA9550-23-1-0312, a JP Morgan Faculty Award, the Algorand Centres of Excellence programme managed by Algorand Foundation, and DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government, DARPA or the Algorand Foundation.

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Notes

  1. 1.

    It appears that a similar generalization was concurrently and independently obtained by IRS in the proceedings version [12]; see Sect. 5 for more details.

  2. 2.

    The only-if direction is a direct consequence of [10].

  3. 3.

    As mentionned above, it appears that a similar generalization was concurrently and independently obtained by IRS in the proceedings version [12]; see Sect. 5 for more details.

  4. 4.

    The naive approach to try to prove such a result would be to simply try running the decision heuristic on different thresholds. There are several problems with this approach. First, for every threshold \(t=(t_0,t_1)\), there may exist a different heuristic \(H_t\) that solves the decision problem for that threshold, so it’s not clear how to get a uniform search heuristic. Next, its not even clear how to define efficient threshold functions as we require n/Gap thresholds to approximate within an additive term of Gap. Finally, it is not a-prior clear how to use a Gap-K heuristic to approximate K given that the Gap-K heuristic only works on average.

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Authors and Affiliations

  1. Cornell Tech, New York, USA

    Yanyi Liu

  2. Tel-Aviv University & Cornell Tech, Tel Aviv-Yafo, Israel

    Rafael Pass

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  1. Yanyi Liu
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Correspondence to Yanyi Liu .

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Editors and Affiliations

  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Liu, Y., Pass, R. (2023). On One-Way Functions and Sparse Languages. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14369. Springer, Cham. https://doi.org/10.1007/978-3-031-48615-9_8

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  • DOI: https://doi.org/10.1007/978-3-031-48615-9_8

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