Registered attribute-based encryption (Reg-ABE) is a new variant of attribute-based encryption (ABE) that was introduced to resolve the notorious key escrow problem. In a Reg-ABE system, there is no authority that generates secret keys, and each user joins the system by generating its own public/secret key pair. Because of its public-key infrastructure-like model and versatile access control functionality, Reg-ABE is a promising alternative of ABE. In this work, we present a highly space efficient non-monotone Reg-ABE scheme with strong security. Specifically, the sizes of MPK and ciphertext of our scheme are both about 7.5KB, which could be more than 2000 and 5 smaller than those of the state-of-the-art scheme by Garg \textit{et al.}
(Crypto'24, GLWW), respectively, in a realistic parameter setting. The sizes of other elements such as helper secret key and a state that the system maintains could also become more than 40 smaller. Furthermore, our scheme supports non-monotone policies and CCA-security, neither of which GLWW supports. We implement our scheme together with GLWW and show that encryption of ours outperforms that of GLWW even with the above features, while decryption of ours is a few times less efficient than that of GLWW but still takes less than 0.1 seconds with a laptop. Our scheme is proven secure in the generic group model. We also present a dual system variant of our main scheme, which is CPA-secure under the MDDH assumption in the plain model. The variant is much simpler and more efficient than the only known non-monotone Reg-ABE scheme by Attrapadung \textit{et al.}(Crypto'24).