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Inner Product Functional Commitments with Constant-Size Public Parameters and Openings

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Security and Cryptography for Networks (SCN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13409))

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Abstract

Functional commitments (Libert et al. [ICALP’16]) allow a party to commit to a vector \(\boldsymbol{v}\) of length n and later open the commitment at functions of the committed vector succinctly, namely with communication logarithmic or constant in n. Existing constructions of functional commitments rely on trusted setups and have either O(1) openings and O(n) parameters, or they have short parameters generatable using public randomness but have \(O(\log n)\)-size openings. In this work, we ask whether it is possible to construct functional commitments in which both parameters and openings can be of constant size. Our main result is the construction of the first FC schemes matching this complexity. Our constructions support the evaluation of inner products over small integers; they are built using groups of unknown order and rely on succinct protocols over these groups that are secure in the generic group and random oracle model.

The full version of the paper can be found at https://eprint.iacr.org/2022/524.pdf

Dimitris Kolonelos—The third author is the contact author.

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Notes

  1. 1.

    It is also easy to note that it is possible to construct an FC for polynomials from one for linear functions, by linearizing the polynomial.

  2. 2.

    One could use Bulletproofs arithmetic circuit protocol in order to simulate \(\mod p\) algebra over \(\mathbb {Z}_q\), at the price of a prover’s overhead.

  3. 3.

    In fact over the quotient group \(\mathbb {G}/\{1,-1\}\) of an RSA group, since we need to exclude the element \(-1 \in \mathbb {G}\) whose order is known.

  4. 4.

    In these types of proofs though one should set \(\ell \) to be of size \(2\lambda \) for the non-interactive case [8].

  5. 5.

    As we discuss next and in more detail in Sect. 3.3, the choice of \(\textsf{Hprime}\) depends on n.

  6. 6.

    We can securely fix the range of a hash function (as SHA512) by fixing some of its bits and truncating others.

  7. 7.

    We do not consider optimizations for the arguments of knowledge with which we could reduce the size of |C| by 1 and \(|\varLambda |\) by 6 group elements respectively.

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Acknowledgements

The second and third authors received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under project PICOCRYPT (grant agreement No. 101001283), by the Spanish Government under projects SCUM (ref. RTI2018-102043-B-I00) and RED2018-102321-T, and by the Madrid Regional Government under project BLOQUES (ref. S2018/TCS-4339). This work is also supported by a research grant (ref. PL-RGP1-2021-051) from Protocol Labs.

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Authors and Affiliations

  1. Friedrich Alexander Universität Erlangen-Nürnberg, Erlangen, Germany

    Hien Chu & Dominique Schröder

  2. IMDEA Software Institute, Madrid, Spain

    Dario Fiore & Dimitris Kolonelos

  3. Universidad Politenica de Madrid, Madrid, Spain

    Dimitris Kolonelos

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  1. Hien Chu
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  2. Dario Fiore
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Correspondence to Dimitris Kolonelos .

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  1. Università degli Studi di Salerno, Fisciano, Italy

    Clemente Galdi

  2. University of California at Irvine, Irvine, CA, USA

    Stanislaw Jarecki

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Chu, H., Fiore, D., Kolonelos, D., Schröder, D. (2022). Inner Product Functional Commitments with Constant-Size Public Parameters and Openings. In: Galdi, C., Jarecki, S. (eds) Security and Cryptography for Networks. SCN 2022. Lecture Notes in Computer Science, vol 13409. Springer, Cham. https://doi.org/10.1007/978-3-031-14791-3_28

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