Digital signatures, with the properties of data integrity and authenticity authentication, protect a signed message from any alteration. However, appropriate alteration of signed message should be allowed for the purposes of privacy protection or bandwidth saving in some scenarios, such as medical record or official information disclosure. Redactable signatures, a branch of homomorphic signatures for editing, allow any redactor to remove some submessage blocks from an originally signed message and generate a valid signature on the modified message without any help of the original signer. In this paper, we present a new design of redactable signature scheme with submessage redaction control structure. This scheme has the security properties of unforgeability, privacy, and transparency, which are formally defined and proved. Compared with state-of-the-art redactable signature schemes, our scheme is more efficient in communication and computation cost.