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Annual International Cryptology Conference

CRYPTO 2019: Advances in Cryptology – CRYPTO 2019 pp 605–635Cite as

Public-Key Cryptography in the Fine-Grained Setting

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11694))

Abstract

Cryptography is largely based on unproven assumptions, which, while believable, might fail. Notably if \(P = NP\), or if we live in Pessiland, then all current cryptographic assumptions will be broken. A compelling question is if any interesting cryptography might exist in Pessiland.

A natural approach to tackle this question is to base cryptography on an assumption from fine-grained complexity. Ball, Rosen, Sabin, and Vasudevan [BRSV’17] attempted this, starting from popular hardness assumptions, such as the Orthogonal Vectors (OV) Conjecture. They obtained problems that are hard on average, assuming that OV and other problems are hard in the worst case. They obtained proofs of work, and hoped to use their average-case hard problems to build a fine-grained one-way function. Unfortunately, they proved that constructing one using their approach would violate a popular hardness hypothesis. This motivates the search for other fine-grained average-case hard problems.

The main goal of this paper is to identify sufficient properties for a fine-grained average-case assumption that imply cryptographic primitives such as fine-grained public key cryptography (PKC). Our main contribution is a novel construction of a cryptographic key exchange, together with the definition of a small number of relatively weak structural properties, such that if a computational problem satisfies them, our key exchange has provable fine-grained security guarantees, based on the hardness of this problem. We then show that a natural and plausible average-case assumption for the key problem Zero-k-Clique from fine-grained complexity satisfies our properties. We also develop fine-grained one-way functions and hardcore bits even under these weaker assumptions.

Where previous works had to assume random oracles or the existence of strong one-way functions to get a key-exchange computable in O(n) time secure against \(O(n^2)\) adversaries (see [Merkle’78] and [BGI’08]), our assumptions seem much weaker. Our key exchange has a similar gap between the computation of the honest party and the adversary as prior work, while being non-interactive, implying fine-grained PKC.

R. LaVigne—This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation. Research also supported in part by NSF Grants CNS-1350619 and CNS-1414119, and by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226 and W911NF-15-C-0236.

A. Lincoln—This work supported in part by NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF Grant BSF:2012338.

V. Williams—Partially supported by an NSF Career Award, a Sloan Fellowship, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF Grant BSF:2012338.

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Notes

  1. 1.

    Assuming the hardness of these problems for more general k will imply \(P \ne NP\), but that is not the focus of our work.

  2. 2.

    This reduction is done using color-coding [3], an example of this lemma exists in the paper “Tight Hardness for Shortest Cycles and Paths in Sparse Graphs” [40].

  3. 3.

    Thank you to Russell Impagliazzo for discussions related to the sizes of ranges R.

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Authors and Affiliations

  1. MIT CSAIL and EECS, Cambridge, USA

    Rio LaVigne, Andrea Lincoln & Virginia Vassilevska Williams

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  1. Rio LaVigne
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  2. Andrea Lincoln
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  3. Virginia Vassilevska Williams
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Correspondence to Rio LaVigne or Andrea Lincoln .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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LaVigne, R., Lincoln, A., Vassilevska Williams, V. (2019). Public-Key Cryptography in the Fine-Grained Setting. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_20

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