The cube attack is a powerful technique for attacking symmetric ciphers, of which one of the core parts is the search for good cubes. The construction of candidate cubes targeting linear superpolies, proposed at Asiacrypt 2021 [20], is the dominant strategy for searching good cubes. However, the time spent on constructing candidate cubes increases significantly with initialization rounds on Trivium. To address this issue, we propose a novel framework called fast framework (The source code and data for this paper are available at https://github.com/Elpsys/Fast-Framework .), which enhances the efficiency of constructing candidates and discovers better cubes. The integration of numeric mapping and division property is the core of our framework; they can improve efficiency while guaranteeing accuracy. The proposed method is successfully applied to 830- and 832-round Trivium, both identifying more than 20 valuable cubes whose superpolies contain linear-independent terms and have the potential to be used in further practical key-recovery attacks. The experimental results demonstrated that the efficiency of constructing the mother cube is improved by at least a factor of 9 compared to conventional methods.