International Association
for Cryptologic Research

FROST and its variants are state-of-the-art protocols for threshold Schnorr signatures that are used in real-world applications. While static security of these protocols has been shown by several works, the security of these protocols under adaptive corruptions—where an adversary can choose which parties to corrupt at any time based on information it learns during protocol executions—has remained a notorious open problem that has received renewed attention due to recent standardization efforts for threshold schemes.

We show adaptive security (without erasures) of FROST and several variants under different corruption thresholds and computational assumptions. Let n be the total number of parties, t+1 the signing threshold, and t_c an upper bound on the number of corrupted parties.

1. We prove adaptive security when t_c = t/2 in the random oracle model (ROM) based on the algebraic one-more discrete logarithm assumption (AOMDL)—the same conditions under which FROST is proven statically secure.

2. We introduce the low-dimensional vector representation (LDVR) problem, parameterized by t_c, t, and n, and prove adaptive security in the algebraic group model (AGM) and ROM based on the AOMDL assumption and the hardness of the LDVR problem for the corresponding parameters. In some regimes (including some t_c >t/2) we show the LDVR problem is unconditionally hard, while in other regimes (in particular, when t_c = t) we show that hardness of the LDVR problem is necessary for adaptive security to hold. In fact, we show that hardness of the LDVR problem is necessary for proving adaptive security of a broad class of threshold Schnorr signatures.