In threshold encryption, the secret key is shared among a set of decryption parties, so that only a quorum of these parties can decrypt a given ciphertext. It is a useful building block in cryptology to distribute the trust of the secret key as well as increase availability. In particular, threshold Paillier encryption has been widely used in various security protocols, such as e-auction, e-voting and e-lottery. In this paper, we present the idea of designing provably secure threshold Paillier encryption using hyperplane geometry. Compared with the existing schemes that are based on polynomial interpolation, our work not only renovates the threshold Paillier cryptosystem using a different mathematical structure, but also enjoys some additional benefits: (1) our proposed method avoids the technical obstacle of computing inverses in the group whose order is unknown; (2) it gains computational advantages over Shoup's trick and it can be used as a general building block to design secure and efficient threshold cryptosystems based on factoring.
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