International Association
for Cryptologic Research

Threshold Schnorr signatures are seeing increased adoption in practice, and offer practical defenses against single points of failure. However, one challenge with existing randomized threshold Schnorr signature schemes is that signers must carefully maintain secret state across signing rounds, while also ensuring that state is deleted after a signing session is completed. Failure to do so will result in a fatal key-recovery attack by re-use of nonces. While deterministic threshold Schnorr signatures that mitigate this issue exist in the literature, all prior schemes incur high complexity and performance overhead in comparison to their randomized equivalents. In this work, we seek the best of both worlds; a deterministic and stateless threshold Schnorr signature scheme that is also simple and efficient. Towards this goal, we present Arctic, a lightweight two-round threshold Schnorr signature that is deterministic, and therefore does not require participants to maintain state between signing rounds. As a building block, we formalize the notion of a Verifiable Pseudorandom Secret Sharing (VPSS) scheme, and define VPSS1, an efficient VPSS construction. VPSS1 is secure when the total number of participants is at least 2t − 1 and the adversary is assumed to corrupt at most t − 1; i.e., in the honest majority model. We prove that Arctic is secure under the discrete logarithm assumption in the random oracle model, similarly assuming at minimum 2t − 1 number of signers and a corruption threshold of at most t−1. For moderately sized groups (i.e., when n ≤ 20), Arctic is more than an order of magnitude more efficient than prior deterministic threshold Schnorr signatures in the literature. For small groups where n ≤ 10, Arctic is three orders of magnitude more efficient.

Both threshold and blind signatures have, individually, received a considerable amount of attention. However little is known about their combination, i.e., a threshold signature which is also blind, in that no coalition of signers learns anything about the message being signed or the signature being produced. Several applications of blind signatures (e.g., anonymous tokens) would benefit from distributed signing as a means to increase trust in the service and hence reduce the risks of key compromise. This paper builds the first blind threshold signatures in pairing-free groups. Our main contribution is a construction that transforms an underlying blind non-threshold signature scheme with a suitable structure into a threshold scheme, preserving its blindness. The resulting signing protocol proceeds in three rounds, and produces signatures consisting of one group element and two scalars. The underlying non-threshold blind signature schemes are of independent interest, and improve upon the current state of the art (Tessaro and Zhu, EUROCRYPT ’22) with shorter signatures (three elements, instead of four) and simpler proofs of security. All of our schemes are proved secure in the Random Oracle and Algebraic Group Models, assuming the hardness of the discrete logarithm problem.

We prove adaptive security of a simple three-round threshold Schnorr signature scheme, which we call Sparkle. The standard notion of security for threshold signatures considers a static adversary - one who must declare which parties are corrupt at the beginning of the protocol. The stronger adaptive adversary can at any time corrupt parties and learn their state. This notion is natural and practical, yet not proven to be met by most schemes in the literature. In this paper, we demonstrate that Sparkle achieves several levels of security based on different corruption models and assumptions. To begin with, Sparkle is statically secure under minimal assumptions: the discrete logarithm assumption (DL) and the random oracle model (ROM). If an adaptive adversary corrupts fewer than t/2 out of a threshold of t+1 signers, then Sparkle is adaptively secure under a weaker variant of the one-more discrete logarithm assumption (AOMDL) in the ROM. Finally, we prove that Sparkle achieves full adaptive security, with a corruption threshold of t, under AOMDL in the algebraic group model (AGM) with random oracles. Importantly, we show adaptive security without requiring secure erasures. Ours is the first proof achieving full adaptive security without exponential tightness loss for any threshold Schnorr signature scheme; moreover, the reduction is tight.

At RWC in 2019, Gregory Neven presented seminal work on a range of two-round multisignature schemes, all of which proved to be insecure against ROS attacks. At that time, it appeared doubtful if concurrently secure two-round multi-party Schnorr signatures could exist. In 2020, this research question was answered in the affirmative, and we saw the emergence of several two-round multi-party Schnorr signature scheme secure under concurrent sessions, namely FROST on the threshold side, MuSig2 (presented at RWC 2021) and DWMS on the multisignature side. Three years have passed since these schemes were first published, and we have learned a lot in their transition from theory to practical use. In this talk, we will review these lessons learned, and how the field has since progressed. We will then introduce a range of open research questions that, if solved, would dramatically improve the practicality and applicability of these schemes in real-world systems.

Existing threshold signature schemes come in two flavors: (i) fully private, where the signature reveals nothing about the set of signers that generated the signature, and (ii) accountable, where the signature completely identifies the set of signers. In this paper we propose a new type of threshold signature, called TAPS, that is a hybrid of privacy and accountability. A TAPS signature is fully private from the public's point of view. However, an entity that has a secret tracing key can trace a signature to the threshold of signers that generated it. A TAPS makes it possible for an organization to keep its inner workings private, while ensuring that signers are accountable for their actions. We construct a number of TAPS schemes. First, we present a generic construction that builds a TAPS from any accountable threshold signature. This generic construction is not efficient, and we next focus on efficient schemes based on standard assumptions. We build two efficient TAPS schemes (in the random oracle model) based on the Schnorr signature scheme. We conclude with a number of open problems relating to efficient TAPS

We give a unified syntax, and a hierarchy of definitions of security of increasing strength, for non-interactive threshold signature schemes. These are schemes having a single-round signing protocol, possibly with one prior round of message-independent pre-processing. We fit FROST1 and BLS, which are leading practical schemes, into our hierarchy, in particular showing they meet stronger security definitions than they have been shown to meet so far. We also fit in our hierarchy a more efficient version FROST2 of FROST1 that we give. These definitions and results, for simplicity, all assume trusted key generation. Finally, we prove the security of FROST2 with key generation performed by an efficient distributed key generation protocol.