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The password-protected secret sharing (PPSS, for short) and its security notion, called in this paper the PPSSsecurity, were proposed by Bagherzandi, Jarecki, Saxena and Lu. However, another security notion for PPSS schemes, the pparam-security was proposed by Hasegawa, Isobe, Iwazaki, Koizumi and Shizuya, because they pointed out an attack which can break the original protocol proposed by Bagherzandi et al. Hasegawa et al. also showed how to enhance the protocol, and proved that the enhanced one is pparam-secure. In this paper, we prove that the enhanced one is PPSS-secure as well.