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Concurrently Secure Blind Schnorr Signatures

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Advances in Cryptology – EUROCRYPT 2024 (EUROCRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14652))

Abstract

Many applications of blind signatures, e.g. in blockchains, require compatibility of the resulting signatures with the existing system. This makes blind issuing of Schnorr signatures (now being standardized and supported by major cryptocurrencies) desirable. Concurrent security of the signing protocol is required to thwart denial-of-service attacks.

We present a concurrently secure blind-signing protocol for Schnorr signatures, using the standard primitives NIZK and PKE and assuming that Schnorr signatures themselves are unforgeable. Our protocol is the first to be compatible with standard Schnorr implementations over 256-bit elliptic curves. We cast our scheme as a generalization of blind and partially blind signatures: we introduce the notion of predicate blind signatures, in which the signer can define a predicate that the blindly signed message must satisfy.

We provide implementations and benchmarks for various choices of primitives and scenarios, such as blindly signing Bitcoin transactions only when they meet certain conditions specified by the signer.

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Notes

  1. 1.

    https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-5-draft.pdf.

  2. 2.

    The AGM assumes the adversary against a cryptosystem defined over a group \((\mathbb {G},+)\) to be algebraic, which means that if, after having received group elements \(X_1,\dots ,X_n\), the adversary returns a group element Z, one can extract a representation \((\zeta _1,\dots ,\zeta _n)\) so that \(Z=\sum \zeta _i X_i\).

  3. 3.

    The authors propose to generalize their scheme to \(t>2\) parallel runs (of which the signer finishes one). But even assuming the best attack on mROS is guessing which sessions will be finished, for 128-bit security we would require \(t> 2^{14}\) parallel sessions, resulting in huge communication complexity.

  4. 4.

    Blind signing is a 7-move protocol and, due to a loose security proof, 12 000-bit groups would be needed [CAHL+22]. The communication cost per signing session, which is linear in the number of preceding sessions, was then reduced to logarithmic [CAHL+22] (but no Schnorr-based variants are mentioned).

  5. 5.

    A signature in [HLW23] consists of a BLS signature [BLS01], as well as \(K-1\) keys and commitment openings. For 128-bit security the authors suggest \(K=33\), which yields 5.71 kB per signature, compared to 128 bytes for [TZ22].

  6. 6.

    Assuming 128-bit security for DL on the curve and n-bit security for Schnorr signatures and NIZK soundness, Theorem 1 yields n-bit security as long as \(\log q\le 126-n\), where q is the number of signing session an attacker closes successfully. In contrast, for 256-bit curves, clause blind Schnorr [FPS20] only achieves 70-bit security due to attacks on mROS (cf. [TZ22]).

  7. 7.

    Having the user commit to her randomness upfront and later prove that her protocol messages are consistent with it has been used in previous blind signature constructions via cut-and-choose [Poi98, KLR21, CAHL+22, HLW23]. However, if the user revealed all of her secrets (in the “chosen” sessions), this would break blindness; in these protocols the signer therefore signs a (hiding) commitment to the actual message (and hence the resulting signatures need to contain the commitment opening).

  8. 8.

    This enables “straight-line” extraction of the committed values during the signing queries in our proof of unforgeability. We cannot use commitments that assume non-blackbox extraction: the reduction would have to run extractors that run other extractors, which would lead to an exponential blow-up of its running time. Moreover, the efficiency gains in our implementation would be small. (See the full version [FW22] for a detailed discussion.) Note that replacing the NIZK by a proof of knowledge would not help since we need to extract before the proven statement is known.

  9. 9.

    This is also why we do not use the random oracle model for extractable commitments (but rely on \(\textsf{PKE}\) instead), in contrast to other work [KLR21]. Assuming unforgeability of (Schnorr) signatures when proving the security of a protocol built on top has also been done in the context of multi- and threshold signatures [CKM21, BCK+22].

  10. 10.

    We analyze the computational complexity of this check in the full version [FW22].

  11. 11.

    Another potential application (not supported by partially blind signatures) is to rate-limiting in Privacy Pass [DGS+18]: when obtaining the signed tokens, PBS could enforce that they are “linked” among them (but unlinkable to the signing session), so that applications can enforce rate limits on linked tokens. (E.g., a (long-enough) prefix of the signed string must be the preimage of a one-way-function evaluation that specifies the predicate used during blind signing.).

  12. 12.

    We implement both (G) and (P) over the \(\textsf{BN254}\) curve [BN06], whose order is incompatible with the base field of \(\textsf{secp256k1}\).

  13. 13.

    We expect implementations for Schnorr instantiated over \(\textsf{ed25519}\) (Circom 2.0 implementation of [Ele]), i.e., blind signing of EdDSA [BDL+12] to yield similar benchmarks using \(\textsf{BN254}\), since the base field of \(\textsf{ed25519}\) is also incompatible with the \(\textsf{BN254}\) scalar field.

  14. 14.

    https://electriccoin.co/blog/software-usability-and-hardware-requirements/.

  15. 15.

    https://download.z.cash/zcashfinalmpc/sprout-proving.key.

  16. 16.

    Checking if no such injective function exists is efficiently computable using e.g. the Hopcroft-Karp or Karzanov’s matching algorithm [HK73, Kar73].

  17. 17.

    Moreover, the user would have to prove knowledge of the corresponding secret key, so that the unforgeability reduction can extract it.

  18. 18.

    There are attacks against Groth16 in which proofs constructed under a malformed CRS leak information on the witness [CGGN17, Fuc19].

  19. 19.

    One invocation of the \(\mathsf {SHA\text {-}256}\) compression function requires around 26 000 R1CS (see Footnote 22) constraints [CGGN17, KPS18, ide].

  20. 20.

    Fluidex builds on the [ide] code base and does not consider PlonK-specific optimization techniques such as [PFM+22].

  21. 21.

    [TS] builds on the reference implementation of Spartan [Set] where the curve has been replaced by \(\textsf{secq256k1}\) and combines this with a modified version of [Bha] which compiles Circom projects into the format required by Spartan.

  22. 22.

    Arithmetization is given as a R1CS relation, which consists of instance-witness pairs \(((A, B, C, \theta ), w)\), where A, B, C are matrices and \(\theta , w\) are vectors over a finite field \(\mathbb {F}\), such that \(Az\circ Bz = Cz\) for \(z := (1, \theta , w)\), where “\(\circ \)” denotes the entry-wise product [BCR+19]. We refer to each such product as a “constraint” or “gate”.

  23. 23.

    Note that the first line in the return statement of (7) checks a congruence modulo q, whereas the third line requires modulo p. We stress the importance of type checks when inputs are not \(\mathbb {F}_p\) elements; for elements of the statement \(\theta \) these can be directly performed when verifying a NIZK proof, which saves on CRS size and prover time.

  24. 24.

    Since elliptic-curve scalar multiplication in the \(\textsf{BJB}\) group for fixed base requires roughly 770 R1CS constraints as opposed to about 2 530 constraints for variable base (incl. bit-decomposition; in the current [ide] implementation), the bulk of constraints saved from (A2) to (A1) comes from this hardwiring aspect, rather than the smaller message size.

  25. 25.

    The drawback of using Cocks-Pinch is that the bit length of the base filed is up to twice as long [FST10], and, more importantly, it only yields the curve parameters, but no security guarantees, let alone efficient implementations.

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Acknowledgements

This work has been funded by the Vienna Science and Technology Fund (WWTF) [10.47379/VRG18002]. We would like to thank Tim Ruffing for preliminary discussions and the anonymous reviewers for their helpful comments.

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    Georg Fuchsbauer & Mathias Wolf

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Fuchsbauer, G., Wolf, M. (2024). Concurrently Secure Blind Schnorr Signatures. In: Joye, M., Leander, G. (eds) Advances in Cryptology – EUROCRYPT 2024. EUROCRYPT 2024. Lecture Notes in Computer Science, vol 14652. Springer, Cham. https://doi.org/10.1007/978-3-031-58723-8_5

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