Secure Multi-party Matrix Invertibility Testing over Small Finite Fields

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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