The redactable signature scheme was introduced by Johnson and others in 2002 as a mechanism to support disclosing verifiable subdocuments of a signed document. In their paper, a redactable signature based on RSA was presented. In 2009, Nojima and others presented a redactable signature scheme based on RSA. Both schemes are very efficient in terms of storage. However, the schemes need mechanisms to share random prime numbers, which causes huge time consuming computation. Moreover, the public key in the scheme of Johnson and others is designed to be used only once. In this paper, we improve the computational efficiency of these schemes by eliminating the use of a random prime sharing mechanism while sustaining the storage efficiency of them. The size of our signature scheme is the same as that of the standard RSA signature scheme plus the size of the security parameter. In our scheme, the public key can be used multiple times, and more efficient key management than the scheme of Johnson and others is possible. We also prove that the security of our scheme is reduced to the security of the full domain RSA signature scheme.