In 2000, Dimitrov, Jullien, and Miller proposed an efficient and simple double-exponentiation algorithm based on a signed-digit recoding algorithm. The average joint Hamming ratio (AJHR) was reduced from 0.556 to 0.534 by using the recoding algorithm. In this paper, the DJM recoding algorithm was extended to three types: the 3-digit sliding window, the 1-digit right-to-left sliding window, and the 1-digit left-to-right sliding window. The average joint Hamming ratios of the three cases were 0.521, 0.515, and 0.511, respectively.

In 2004, Tsuji and Shimizu proposed a one-time password authentication protocol, named 2GR (Two-Gene-Relation password authentication protocol). The design goal of the 2GR protocol is to eliminate the stolen-verifier attack on SAS-2 (Simple And Secure password authentication protocol, ver.2) and the theft attack on ROSI (RObust and SImple password authentication protocol). Tsuji and Shimizu claimed that in the 2GR an attacker who has stolen the verifiers from the server cannot impersonate a legitimate user. This paper, however, will point out that the 2GR protocol is still vulnerable to an impersonation attack, in which any attacker can, without stealing the verifiers, masquerade as a legitimate user.