Paper 2026/514 Original: eprint.iacr.org/2026/514 ↗

Secure Multi-party Matrix Invertibility Testing over Small Finite Fields

Abstract

Multi-party matrix invertibility testing over finite fields of small order or characteristic is a pivotal operation for thresholdizing multivariate quadratic~(MQ) signature schemes. However, achieving perfect privacy in a constant number of rounds remains a challenge: existing solutions either suffer from information leakage, failing to provide perfect privacy, or require high computational and communication overhead, in particular, when $p\leq n$, where $p$ and $n$ denote the characteristic of the underlying field and the matrix size, respectively. To address this limitation, we propose two protocols for multi-party testing of matrix invertibility. The first protocol extends the Cramer-Damg{\aa}rd protocol to fields of small order by employing the field lifting technique. The second protocol is based on multi-party computation of the Samuelson-Berkowitz algorithm, specifically designed for fields of small characteristic. Both protocols are formalized in the arithmetic black-box~(ABB) model with Shamir's secret sharing scheme. We show that both protocols achieve perfect privacy while allowing for a tradeoff between input-independent offline rounds and input-dependent online rounds, where the input corresponds to the shared matrices. Specifically, the first protocol runs in $7$ offline rounds with communication complexity $O(Nn^4)$ and in $3$ online rounds with communication complexity $O(n^4)$, and the second protocol runs in $3$ offline rounds with communication complexity $O(n^4)$ and in $9$ online rounds with communication complexity $O(n^4)$, where $n$ is the matrix size and $N$ is the number of parties.