A New Signature Scheme without Random Oracles and Its Applications ⋆ Fangguo Zhang 1, 3 , Xiaofeng Chen 2, 3 , Willy Susilo 4 and Yi Mu 4 1 Department of Electronics and Communication Engineering, Sun Yat-Sen University, Guangzhou 510275, P.R.China isszhfg@mail.sysu.edu.cn 2 Department of Computer Science, Sun Yat-Sen University, Guangzhou 510275, P.R.China isschxf@zsu.edu.cn 3 Guangdong Key Laboratory of Information Security Technology Guangzhou 510275, P.R.China 4 School of IT and Computer Science University of Wollongong, Wollongong, NSW 2522, Australia {ymu,wsusilo}@uow.edu.au Abstract. In this paper, we propose a new signature scheme that is existentially unforgeable under a chosen message attack without random oracle. The security of our scheme depends on a new complexity assumption called the k+1 square roots assumption. We also discuss the relationship between the k+1 square roots assumption and some related problems and provide some conjectures. Moreover, the k+1 square roots assumption can be used to construct shorter signatures under the random oracle model. As some applications, a new chameleon hash signature scheme and a on-line/off-line signature scheme and a new efficient anonymous credential scheme based on the proposed signature scheme are presented. Keywords: Short signature, Bilinear pairings, Standard model, Random oracle, Anony- mous credential 1 Introduction It is well known that a signature scheme that produces signatures of length ℓ can have some security level of at most 2 ℓ , which means that given a public key, it is possible to forge a signature on any message in O(2 ℓ ). A natural question that arises is how we can concretely construct a signature scheme that can produce shorter length of signature whilst maintaining an existential forgery with the same security level. Short digital signatures are always desirable. They are necessary in some situation where people need to enter the signature manually, such as using a PDA that is not equipped with a keyboard. Additionally, short digital signatures are essential to ensure the authenticity of ⋆ This work is supported by the National Natural Science Foundation of China (No. 60403007 and No. 60503006) and ARC Discovery Grant DP0557493