Recently, Kasahara and Murakami proposed new product-sum type public-key cryptosystems with the Chinese remainder theorem, Methods B-II and B-IV. They also proposed a new technique of selectable encryption key, which is referred to as 'Home Page Method (HP Method).' In this paper, first, we describe Methods B-II and B-IV. Second, we propose an effective attack for Method B-II and discuss the security of Methods B-II and B-IV. Third, applying the HP Method to Methods B-II and B-IV, we propose new product-sum type PKC with selectable encryption key. Moreover, we discuss the security of the proposed cryptosystems.

Recently, Kasahara and Murakami proposed new product-sum public-key cryptosystems using the Chinese remainder theorem as the trapdoor. We proposed 'Yaezakura' as the high-density product-sum PKC applying the method using the reduced bases. In this paper, we propose another high-density scheme with the Chinese remainder theorem trapdoor using the message extension. We also show that the proposed scheme is invulnerable to the low-density attack. In the proposed scheme, the sender can freely select the positions of the dummy messages.