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Compressing Unit-Vector Correlations via Sparse Pseudorandom Generators

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Advances in Cryptology – CRYPTO 2024 (CRYPTO 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14927))

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Abstract

A unit-vector (UV) correlation is an additive secret-sharing of a vector of length B that contains 1 in a secret random position and 0’s elsewhere. UV correlations are a useful resource for many cryptographic applications, including low-communication secure multiparty computation and multi-server private information retrieval. However, current practical methods for securely generating UV correlations involve a significant communication cost per instance, and become even more expensive when requiring security against malicious parties.

In this work, we present a new approach for constructing a pseudorandom correlation generator (PCG) for securely generating n independent instances of UV correlations of any polynomial length B. Such a PCG compresses the n UV instances into correlated seeds whose length is sublinear in the description size \(n\cdot \log B\). Our new PCGs apply in both the honest-majority and dishonest-majority settings, and are based on a variety of assumptions. In particular, in the honest-majority case they only require “unstructured” assumptions. Our PCGs give rise to secure end-to-end protocols for generating n instances of UV correlations with o(n) bits of communication. This applies even to an authenticated variant of UV correlations, which is useful for security against malicious parties. Unlike previous theoretical solutions, some instances of our PCGs offer good concrete efficiency.

Our technical approach is based on combining a low-degree sparse pseudorandom generator, mapping a sparse seed to a pseudorandom sparse output, with homomorphic secret sharing for low-degree polynomials. We then reduce such sparse PRGs to local PRGs over large alphabets, and explore old and new approaches for maximizing the stretch of such PRGs while minimizing their locality.

Finally, towards further compressing the PCG seeds, we present a new PRG-based construction of a multiparty distributed point function (DPF), whose outputs are degree-1 Shamir-shares of a secret point function. This result is independently motivated by other DPF applications.

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Notes

  1. 1.

    Intuitively, a compression ratio of s suggests that our PCG approach is \(s \times \) better compared to the baseline approach for compressing a given number of unit vectors (more details are included in the full version).

  2. 2.

    Composition between the PRG and even simple distribution sampling not only increases the computation degree, but also ruins “restricted-multiplication friendliness” of the computation. Several HSS schemes support “restricted” multiplications, where one multiplicand must be an original input. Thus for each multiplication of outputs from the PRG, one must be completely recomputed from scratch.

  3. 3.

    We describe a natural generalization of our construction in the full version which allows for Local PRGs with different input and output alphabet. Here we restrict to Local PRGs with same input and output alphabet for simplicity.

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Acknowledgements

We thank Benny Applebaum, Andrej Bogdanov, Geoffroy Couteau, Itai Dinur, Aayush Jain and Vinod Vaikuntanathan for helpful discussions and pointers. E. Boyle was supported in part by AFOSR Award FA9550-21-1-0046 and ERC Project HSS (852952). Y. Ishai was supported by ERC grant NTSC (742754), BSF grant 2022370, ISF grant 2774/20, and ISF-NSFC grant 3127/23. Y. Ma was supported by a Microsoft Research PhD Fellowship.

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Authors and Affiliations

  1. UIUC, Champaign, USA

    Amit Agarwal

  2. Reichman University, Herzliya, Israel

    Elette Boyle

  3. NTT Research, Sunnyvale, USA

    Elette Boyle

  4. Ben-Gurion University, Beersheba, Israel

    Niv Gilboa

  5. Technion, Haifa, Israel

    Yuval Ishai

  6. Cornell University, Ithaca, USA

    Mahimna Kelkar

  7. University of Pennsylvania, Philadelphia, USA

    Yiping Ma

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  1. Amit Agarwal
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  1. Boston University, Boston, MA, USA

    Leonid Reyzin

  2. University of Waterloo, Waterloo, ON, Canada

    Douglas Stebila

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Agarwal, A., Boyle, E., Gilboa, N., Ishai, Y., Kelkar, M., Ma, Y. (2024). Compressing Unit-Vector Correlations via Sparse Pseudorandom Generators. In: Reyzin, L., Stebila, D. (eds) Advances in Cryptology – CRYPTO 2024. CRYPTO 2024. Lecture Notes in Computer Science, vol 14927. Springer, Cham. https://doi.org/10.1007/978-3-031-68397-8_11

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