Paper 2026/189

Threshold linear solving in small fields and application to UOV

Abstract

Threshold signatures allow multiple parties to sign a common message by collaborating. More specifically, in a $(t,n)-$threshold signature scheme, at least $t$ out of $n$ parties must collaborate to sign a message. In particular, solving linear systems shared among some parties is a problem that naturally arises in threshold cryptography, and this paper proposes three algorithms for a set of parties to solve a shared linear system $Ax = b$ in finite fields of low characteristic. The first two algorithms securely compute the determinant of a shared matrix using recent theoretical results on Newton's polynomials and by adapting an algorithm by Samuelson and Berkowitz. From these results, two algorithms can be deduced to solve the corresponding linear system. On the other hand, the third is a modification of an existing state-of-the-art algorithm. Although pre-quantum threshold signature algorithms have been extensively studied, the state of the art in the creation of post-quantum threshold algorithms remains sparse. In particular, few papers have studied the creation of a threshold algorithm based on UOV, despite the simplicity of the scheme. The new algorithms presented in this paper enable other threshold instantiations of UOV and UOV-based schemes.