Divisor Algorithm Research Articles

유한체위에서의 타원곡선을 이용한 고속 소인수분해법에 관한 연구

Since the security of RSA cryptosystem depends on the difficulty of factoring integers, it is the most important problem to factor large integers in RSA cryptosystem. The Lenstra elliptic curve factorization method(ECM) is considered a special purpose factoring algorithm as it is still the best algorithm for divisors not greatly exceeding 20 to 25 digits(64 to 83 bits or so). ECM, however, wastes most time to calculate ∙ mod and so Montgomery and Koyama both give fast methods for implementing ∙ mod . We, in this paper, further analyze Montgomery and Koyama’s methods and propose an efficient algorithm which choose the optimal parameters and reduces the number of multiplications of Montgomery and Koyama’s methods. Consequently, the run time of our algorithm is reduced by 20% or so than that of Montgomery and Koyama’s methods.

Speeding up Euclid's GCD algorithm with no magnitude comparisons

Euclid's Greatest Common Divisor (GCD) algorithm is an efficient approach for calculating multiplicative inversions. It relies mainly on a fast modular arithmetic algorithm to run quickly. A traditional modular arithmetic algorithm based on nonrestoring division needs a magnitude comparison for each iteration of shift-and-subtract operation. This process is time consuming, since it requires magnitude comparisons for every computation iteration step. To eradicate this problem, this study develops a new fast Euclidean GCD algorithm without magnitude comparisons. The proposed modular algorithm has an execution time that is about 33% shorter than the conventional modular algorithm.