International Association
for Cryptologic Research

Single Secret Leader Election (SSLE) protocols allow a set of users to elect a leader among them so that the identity of the winner remains secret until she decides to reveal herself. This notion was formalized and implemented in a recent result by Boneh, et al. (ACM Advances on Financial Technology 2020) and finds important applications in the area of Proof of Stake blockchains. In this paper we put forward new SSLE solutions that advance the state of the art both from a theoretical and a practical front. On the theoretical side we propose a new definition of SSLE in the universal composability framework. We believe this to be the right way to model security in highly concurrent contexts such as those of many blockchain related applications. Next, we propose a UC-realization of SSLE from public key encryption with keyword search (PEKS) and based on the ability of distributing the PEKS key generation and encryption algorithms. Finally, we give a concrete PEKS scheme with efficient distributed algorithms for key generation and encryption and that allows us to efficiently instantiate our abstract SSLE construction. Our resulting SSLE protocol is very efficient, does not require participants to store any state information besides their secret keys and guarantees so called on-chain efficiency: the information to verify an election in the new block should be of size at most logarithmic in the number of participants. To the best of our knowledge, this is the first efficient SSLE scheme achieving this property.

Non-Interactive Zero-Knowledge proofs (NIZK) allow a prover to convince a verifier that a statement is true by sending only one message and without conveying any other information. In the CRS model, many instantiations have been proposed from group-theoretic assumptions. On the one hand, some of these constructions use the group structure in a black-box way but rely on pairings, an example being the celebrated Groth-Sahai proof system. On the other hand, a recent line of research realized NIZKs from sub-exponential DDH in pairing-free groups using Correlation Intractable Hash functions, but at the price of making non black-box usage of the group. As of today no construction is known to \textit{simultaneously} reduce its security to pairing-free group problems and to use the underlying group in a black-box way. This is indeed not a coincidence: in this paper, we prove that for a large class of NIZK either a pairing-free group is used non black-box by relying on element representation, or security reduces to external hardness assumptions. More specifically our impossibility applies to two incomparable cases: The first one covers Arguments of Knowledge (AoK) which proves that a preimage under a given one way function is known. The second one covers NIZK (not necessarily AoK) for hard subset problems, which captures relations such as DDH, Decision-Linear and Matrix-DDH.

Vector Commitments allow one to (concisely) commit to a vector of messages so that one can later (concisely) open the commitment at selected locations. In the state of the art of vector commitments, {\em algebraic} constructions have emerged as a particularly useful class, as they enable advanced properties, such as stateless updates, subvector openings and aggregation, that are for example unknown in Merkle-tree-based schemes. In spite of their popularity, algebraic vector commitments remain poorly understood objects. In particular, no construction in standard prime order groups (without pairing) is known. In this paper, we shed light on this state of affairs by showing that a large class of concise algebraic vector commitments in pairing-free, prime order groups are impossible to realize. Our results also preclude any cryptographic primitive that implies the algebraic vector commitments we rule out, as special cases. This means that we also show the impossibility, for instance, of succinct polynomial commitments and functional commitments (for all classes of functions including linear forms) in pairing-free groups of prime order.