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Locally Verifiable Signature and Key Aggregation

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13508))

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Abstract

Aggregate signatures (Boneh, Gentry, Lynn, Shacham, Eurocrypt 2003) enable compressing a set of N signatures on N different messages into a short aggregate signature. This reduces the space complexity of storing the signatures from linear in N to a fixed constant (that depends only on the security parameter). However, verifying the aggregate signature requires access to all N messages, resulting in the complexity of verification being at least \(\varOmega (N)\).

In this work, we introduce the notion of locally verifiable aggregate signatures that enable efficient verification: given a short aggregate signature \(\sigma \) (corresponding to a set \(\mathcal {M}\) of N messages), the verifier can check whether a particular message m is in the set, in time independent of N. Verification does not require knowledge of the entire set \(\mathcal {M}\). We demonstrate many natural applications of locally verifiable aggregate signature schemes: in the context of certificate transparency logs; in blockchains; and for redacting signatures, even when all the original signatures are produced by a single user.

We provide two constructions of single-signer locally verifiable aggregate signatures, the first based on the RSA assumption and the second on the bilinear Diffie-Hellman inversion assumption, both in the random oracle model.

As an additional contribution, we introduce the notion of compressing cryptographic keys in identity-based encryption (IBE) schemes, show applications of this notion, and construct an IBE scheme where the secret keys for N identities can be compressed into a single aggregate key, which can then be used to decrypt ciphertexts sent to any of the N identities.

R. Goyal—Research supported by grants listed under the second author.

V. Vaikuntanathan—Research supported in part by NSF CNS Award #1718161, an IBM-MIT grant, and by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR00112020023. 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 or DARPA.

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Notes

  1. 1.

    Incidentally, we mention that the problem of constructing a multi-signer aggregate signature scheme from RSA has been a long-standing open problem, although constructions of relaxed variants such as sequential or synchronized (multi-signer) aggregate signature schemes based on RSA exist [LMRS04, HW18].

  2. 2.

    We point out that we use deterministic primality testing only for the ease of exposition, and this is not necessary as our scheme is secure even if we rely on efficient randomized primality testing. Such an approach was already outlined in [MRV99] where the idea is to generate a sequence of random coins as part of the setup, and use those random coins to run the randomized primality test deterministically on all those random coins. The proof relies on the fact that, with all but negligible probability over the choice of random coins sampled during setup, randomized primality test will fail on at least one random coins for a non-prime.

  3. 3.

    We point out that this does not contradict our unforgeability property with adversarial openings. Since, irrespective of whether the adversary is maliciously aggregating signature or generating hints in a malicious way, the adversary is never allowed to make a sign query for the message associated with a forged signature. While it seems like since local verifier is independent of the aggregate signature \(\widehat{\sigma }\), thus a verifier might supply any arbitrary string and still pass local verification. The point is in order for the local verification to accept, it must be provided with a valid signature (as a hint), thus an attacker can not forge by supplying only malformed aggregated signatures \(\widehat{\sigma }\).

  4. 4.

    For simplicity, we ignore the possibility that \(\alpha + h_m = 0\) as that could be easily handled as a special case by outputting the identity group element, but keeping it as part of the scheme description makes it cumbersome.

  5. 5.

    Note that the verification algorithm does not the entire verification key, but the local portion of verification key would be sufficient.

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  1. MIT, Cambridge, MA, USA

    Rishab Goyal & Vinod Vaikuntanathan

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  1. Rishab Goyal
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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Goyal, R., Vaikuntanathan, V. (2022). Locally Verifiable Signature and Key Aggregation. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13508. Springer, Cham. https://doi.org/10.1007/978-3-031-15979-4_26

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