This paper aims to evaluate the security of a bitstream-level JPEG encryption method using restart (RST) markers, where encrypted image can keep the JPEG file format with the same file size as non-encrypted image. Data encrypted using this method can be decoded without altering header information by employing a standard JPEG decoder. Moreover, the use of RST markers enables the definition of extended blocks divided by the markers, so spatially partial encryption and block-permutation-based encryption can be carried out. However, the security of the method was evaluated only with respect to the key space analysis for brute-force attacks and other limited attacks. Accordingly, in this paper, we evaluated the security of the method with respect to robustness against ciphertext-only attacks including state-of-the-art attacks. In experiments, the method is compared with conventional encryption methods, and it is confirmed to be robust against ciphertext-only attacks if parameters used for image encryption are carefully chosen.