Volume 17, Issue 3 (2006), pp. 445–462
We know the necessity for information security becomes more widespread in these days, especially for hardware-based implementations such as smart cards chips for wireless applications and cryptographic accelerators. Fast modular exponentiation algorithms are often considered of practical significance in public-key cryptosystems. The RSA cryptosystem is one of the most widely used technologies for achieving information security. The main task of the encryption and decryption engine of RSA cryptosystem is to compute ME mod N. Because the bit-length of the numbers M, E, and N would be about 512 to 1024 bits now, the computations for RSA cryptosystem are time-consuming. In this paper, an efficient technique for parallel computation of the modular exponentiation is proposed and our algorithm can reduce time complexity. We can have the speedup ratio as 1.06 or even 2.75 if the proposed technique is used. In Savas–Tenca–Koc algorithm, they design a multiplier with an insignificant increase in chip area (about 2.8%) and no increase in time delay. Our proposed technique is faster than Savas–Tenca–Koc algorithm in time complexity and improves efficiency for RSA cryptosystem.
Volume 16, Issue 3 (2005), pp. 449–468
Efficient computation of the modular exponentiations is very important and useful for public-key cryptosystems. In this paper, an efficient parallel binary exponentiation algorithm is proposed which based on the Montgomery multiplication algorithm, the signed-digit-folding (SDF) and common-multiplicand-multiplicand (CMM) techniques. By using the CMM technique of computing the common part from two modular multiplications, the same common part in two modular multiplications can be computed once rather twice, we can thus improve the efficiency of the binary exponentiation algorithm by decreasing the number of modular multiplications. By dividing the bit pattern of the minimal-signed-digit recoding exponent into three equal length parts and using the technique of recording the common parts in the folded substrings, the proposed SDF-CMM algorithm can improve the efficiency of the binary algorithm, thus can further decrease the computational complexity of modular exponentiation. Furthermore, by using the proposed parallel SDF-CMM Montgomery binary exponentiation algorithm, on average the total number of single-precision multiplications can be reduced by about 61.3% and 74.1% as compared with Chang-Kuo-Lin's CMM modular exponentiation algorithm and Ha-Moon's CMM Montgomery modular exponentiation algorithm, respectively.