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Functional Commitments for All Functions, with Transparent Setup and from SIS

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14006))

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Abstract

A functional commitment scheme enables a user to concisely commit to a function from a specified family, then later concisely and verifiably reveal values of the function at desired inputs. Useful special cases, which have seen applications across cryptography, include vector commitments and polynomial commitments.

To date, functional commitments have been constructed (under falsifiable assumptions) only for functions that are essentially linear, with one recent exception that works for arbitrarily complex functions. However, that scheme operates in a strong and non-standard model, requiring an online, trusted authority to generate special keys for any opened function inputs.

In this work, we give the first functional commitment scheme for nonlinear functions—indeed, for all functions of any bounded complexity—under a standard setup and a falsifiable assumption. Specifically, the setup is “transparent,” requiring only public randomness (and not any trusted entity), and the assumption is the hardness of the standard Short Integer Solution (SIS) lattice problem. Our construction also has other attractive features, including: stateless updates via generic composability; excellent asymptotic efficiency for the verifier, and also for the committer in important special cases like vector and polynomial commitments, via preprocessing; and post-quantum security, since it is based on SIS.

L. de Castro and C. Peikert—Some of this work was done while at Algorand, Inc.

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Notes

  1. 1.

    Some works consider a dual notion, in which the user commits to some data x and then can open various functions f of it. Our present formulation is a natural one for most specific purposes of interest. Moreover, assuming a sufficiently expressive scheme of either form, its dual notion can be achieved by swapping “code” and “data” using a universal function; see Sect. 4.3 for details.

  2. 2.

    Earlier works [IKO07, BC12] considered similar concepts, but allowing for interaction and private verifier randomness during the commitment and proving phases. Functional commitments require noninteractive commitments and proofs, and public verifiability.

  3. 3.

    The work of [PPS21] actually uses the dual formulation of functional commitments, where data x is committed and then functions f of the data are opened. However, the construction easily adapts to our present formulation of functional commitments.

  4. 4.

    We caution that some works on polynomial commitment schemes (e.g., [BDFG21]) consider additional requirements, e.g., that the committed function is indeed a polynomial of some bounded degree, or even that openings are proofs of knowledge of such a polynomial (which SNARK applications rely upon). These are much stronger properties than originally considered in [KZG10] and in this work, typically requiring strong (often unfalsifiable) assumptions, and our construction does not achieve them. In addition, many works like [BBB+18, VP19, BFS20, BDFG21, Lee21, BNO21, KPV22] allow openings to be interactive proofs, then heuristically (and unfalsifiably) make them noninteractive using a random oracle using Fiat–Shamir [FS86]. Our construction is natively noninteractive without any heuristics.

  5. 5.

    More generally, the scheme works for vector-valued (multi-output) functions, following Remark 3.

  6. 6.

    The norm bound \(\kappa \) should be set large enough so that the verifier accepts properly generated proofs for all functions in \(\mathcal {F}\) (see Lemma 1). Then, n and q should be set so that the SIS problem underlying the scheme’s evaluation binding is sufficiently hard (see Theorem 3).

  7. 7.

    We can make the commitment hide the function f by using circuit-private homomorphic computation, such as from [BPMW16], and retaining the randomness for use in opening.

  8. 8.

    In concert with a function-hiding commitment, we can make the opening reveal nothing more than the single input-output pair (x, f(x)) by giving a zero-knowledge proof (of knowledge) of some \(\textbf{S}_{f,x}\) that satisfies the verifier.

  9. 9.

    In some contexts, compressed commitments and proofs even be computed directly, without first computing uncompressed ones.

  10. 10.

    As mentioned in Sect. 3.2, security for the compressed variant extends to concatenations of \(k'\) functions with small integer outputs, where the additive n term in \(\beta \) is replaced by the maximum \(\ell _{1}\) norm of the difference between two such output vectors.

  11. 11.

    We use decreasing j here simply for consistency with the notation in Sect. 3.1, but any order can work.

  12. 12.

    For convenience of matrix operations, we have swapped the indices of \(\textbf{S}_{x,\bar{x}}\) from the usual \(\textbf{S}_{f,x}\) form.

  13. 13.

    Note that this makes 0 the implicit ‘default’ value for all such keys. If we wish to have a distinguished ‘undefined’ value \(\bot \) for such keys, we can replace \(\mathcal {Y}\) with a larger group like \(\mathcal {Y}' = \mathcal {Y} \times C\) for some nontrivial cycle C, representing each \(y \in \mathcal {Y}\) by \((y,1) \in \mathcal {Y}'\), and letting  \(\bot \) be represented by the identity element \((0,0) \in \mathcal {Y}'\). Note that this encoding requires some care regarding updates, because, for example, representing \(y-y = 0\) by \((y,1)-(y,1) = (0,0)\) yields \(\bot \), not the encoding of \(0 \in \mathcal {Y}\). A simple solution is for all insertions and deletions to use 1s in their second components, and for all modifications of existing values to use 0s.

  14. 14.

    See Footnote 13 for how insertions/deletions can be handled separately from additions/subtractions, when using an encoding that has a distinguished ‘undefined’ value \(\bot \) that is separate from 0.

  15. 15.

    More generally, the treatment is essentially the same for polynomials whose coefficients come from some \(\mathcal {R}\)-module.

  16. 16.

    Our treatment also adapts straightforwardly to polynomials of bounded total degree. We use individual degree because it follows more naturally from the univariate case.

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Acknowledgments

This material is based upon work supported by NSF Grant No. CCF-2006857 and by DARPA under Agreement No. HR00112020025. 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, the NSF, or DARPA.

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  1. EECS, Massachusetts Institute of Technology, Cambridge, USA

    Leo de Castro

  2. Computer Science and Engineering, University of Michigan, Ann Arbor, USA

    Chris Peikert

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de Castro, L., Peikert, C. (2023). Functional Commitments for All Functions, with Transparent Setup and from SIS. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_10

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