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.

The efficient squaring algorithm is an important role in large integer arithmetic. All multiplication algorithms can be used for squaring large integers, but their performance can be greatly improved by using the standard squaring algorithm. The standard squaring algorithm is quite well-known, but unfortunately there is an improper carry handling bug in it. Recently, Guajardo and Paar proposed a modified squaring algorithm to fix the bug in the standard squaring algorithm. In this paper, we first point out that there is still an error-indexing bug in the Guajardo-Paar squaring algorithm. Then, we propose a new efficient squaring algorithm that not only avoids the bugs in both the standard squaring algorithm and the Guajardo-Paar squaring algorithm but also improves the performance in squaring computation. Our analyses and our simulations indicate that the proposed squaring algorithm is about 2.5 times faster in comparison with the standard multiplication algorithm in Pentium Series CPU. The performance of 1024-bit RSA cryptosystem can be saved 34.3% by using the proposed squaring algorithm to replace the standard multiplication.