This paper examines similarities between the Decision Diffie-Hellman (DDH) assumption and the Quadratic Residuosity (QR) assumption. In addition, we show that many cryptographic protocols based on the QR assumption can be reconstructed using the DDH assumption.

By modifying the private key and the public key setting in Boneh-Lynn-Shacham's short signature shcheme, a variation of BLS' short signature scheme is proposed. Based on this variation, we present a very efficient threshold signature scheme where the number of pairing computation for the signaure share verification reduces to half.

Cryptographic hash functions have been widely studied and are used in many current systems. Though much research has been done on the security of hash functions, system designers cannot determine which hash function is most suitable for a particular system. The main reason for this is that the current security classification does not correspond very well to the security requirements of practical systems. This paper describes a new classification which is more suitable for designing real-life systems. This classification is the result of a new qualitative classification and a new quantitative classification. We show a mapping between each class and standard protocols. In addition, we show new requirements for four types of hash function for a future standard.