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Vector Commitments with Proofs of Smallness: Short Range Proofs and More

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Public-Key Cryptography – PKC 2024 (PKC 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14602))

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Abstract

Vector commitment schemes are compressing commitments to vectors that make it possible to succinctly open a commitment for individual vector positions without revealing anything about other positions. We describe vector commitments enabling constant-size proofs that the committed vector is small (i.e., binary, ternary, or of small norm). As a special case, we obtain range proofs featuring the shortest proof length in the literature with only 3 group elements per proof. As another application, we obtain short pairing-based NIZK arguments for lattice-related statements. In particular, we obtain short proofs (comprised of 3 group elements) showing the validity of ring LWE ciphertexts and public keys. Our constructions are proven simulation-extractable in the algebraic group model and the random oracle model.

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Notes

  1. 1.

    For our applications, we will assume that the commitment is in \(\hat{\mathbb {G}}\) rather than \(\mathbb {G}\) in order for the proof of knowledge-soundness to work out.

  2. 2.

    More precisely, [35] proceeds by proving a relation \(\sum _{i=1}^M \boldsymbol{a}_i \cdot s_i - \boldsymbol{r}_1 \cdot q - \boldsymbol{r}_2 \cdot \varPhi = \boldsymbol{t}\) over \(\mathbb {Z}[X]\), where \(\boldsymbol{r}_1\) and \(\boldsymbol{r}_2\) contain polynomials of degree \(2(d-1)\) and \(d-2\), respectively.

  3. 3.

    Committing x to a different, pairing-free, group \(\mathbb {G}\) would not strengthen security since an adversary that would be able to compute \(\alpha \) from \(\textsf{pp}\) would still break knowledge-soundness. The construction of [9] similarly assumes that the integer x is committed as a polynomial f[X] such that \(f(1)=x\).

  4. 4.

    These representations are supplied by \(\mathcal {A}\) at the first query involving the corresponding group elements.

  5. 5.

    In randomized versions of the KZG commitment (described in [60, Section 3.3], [8, Appendix B.2] and [83]), each evaluation proof consists of an element of \(\mathbb {G}\) and at least one scalar or an additional element of \(\mathbb {G}\).

  6. 6.

    In order to prove the knowledge soundness of the range proof of [9] when the polynomial commitment of [65] is used, it is necessary to rely on the latter’s knowledge soundness in the AGM (as defined in [8, Appendix C.1.3]) but we believe this property holds under the (2n, n)-DLOG assumption.

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  1. Zama, Paris, France

    Benoît Libert

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  1. Benoît Libert
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Correspondence to Benoît Libert .

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  1. The University of Sydney, Sydney, NSW, Australia

    Qiang Tang

  2. The Australian National University, Acton, ACT, Australia

    Vanessa Teague

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© 2024 International Association for Cryptologic Research

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Libert, B. (2024). Vector Commitments with Proofs of Smallness: Short Range Proofs and More. In: Tang, Q., Teague, V. (eds) Public-Key Cryptography – PKC 2024. PKC 2024. Lecture Notes in Computer Science, vol 14602. Springer, Cham. https://doi.org/10.1007/978-3-031-57722-2_2

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  • DOI: https://doi.org/10.1007/978-3-031-57722-2_2

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