Pixel+ and Pixel++: Compact and Efficient Forward-Secure Multi-Signatures for PoS Blockchain Consensus

Multi-signature schemes have attracted considerable attention in recent years due to their popular applications in PoS blockchains. However, the use of general multi-signature schemes poses a critical threat to the security of PoS blockchains once signing keys get corrupted. That is, after an adversary obtains enough signing keys, it can break the immutable nature of PoS blockchains by forking the chain and modifying the history from some point in the past. Forwardsecure multi-signature (FS-MS) schemes can overcome this issue by periodically updating signing keys. The only FS-MS construction currently available is Drijvers et al's Pixel, which builds on pairing groups and only achieves forward security at the time period level. In this work, we present new FS-MS constructions that either are free from pairing or capture forward security at the individual message level (i.e., fine-grained forward security). Our first construction Pixel+ works for a maximum number of time periods T . Pixel+ signatures consist of only one group element, and can be verified using two exponentiations. It is the first FS-MS from RSA assumption, and has 3.5x and 22.8x faster signing and verification than Pixel, respectively. Our second FS-MS construction Pixel++ is a pairing-based one. It immediately revokes the signing key's capacity of re-signing the message after creating a signature on this message, rather than at the end of the current time period. Thus, it provides more practical forward security than Pixel. On the other hand, Pixel++ is almost as efficient as Pixel in terms of signing and verification. Both Pixel+ and Pixel++ allow for non-interactive aggregation of signatures from independent signers and are proven to be secure in the random oracle model. In addition, they also support the aggregation of public keys, significantly reducing the storage overhead on PoS blockchains. We demonstrate how to integrate Pixel+ and Pixel++ into PoS blockchains. As a proof-of-concept, we provide implementations of Pixel+ and Pixel++, and conduct several representative experiments to show that Pixel+ and Pixel++ have good concrete efficiency and are practical.