Substantiation of correctness and advantages of Lenstra factorization method on Edwards curves

• Main Authors: Lyudmyla Kovalchuk, Oleksij Bespalov, Nataliia Kuchynska, Polina Seliukh, Artem Zhylin, Vasyl Tsurkan
• Format: Article
• Language: English
• Published: PC Technology Center 2018-12-01
• Series: Eastern-European Journal of Enterprise Technologies
• Subjects: rsa cryptosystem; factorization problem; factorization methods; lenstra method; edwards curves
• Online Access: http://journals.uran.ua/eejet/article/view/151090

The factorization problem, which is the basis for many classical asymmetric cryptosystems (RSA, Rabin, and others) and a cryptographically strong generator of pseudo-random sequences (PBS), has been investigated in this paper. The methods that served as prototypes for the Lenstra method were described, the method for factorization of numbers, which is analogous to the Lenstra method on Edwards curves, has been proposed. To substantiate the correctness of the method, an appropriate mathematical apparatus was developed. In addition, an analog of the Lenstra method on Edwards curves was constructed with the use of the presented apparatus; the appropriate algorithm for the factorization of numbers was designed. The correctness of the method and correctness of the algorithm operation were substantiated mathematically; the top analytical estimates of its performance speed, as well as the lower estimates of success probability, have been strictly proved. The advantages of the developed method in comparison with the classical Lenstra method, which applies elliptic curves in the Weierstrass form, were presented and strictly substantiated. A comparative analysis of the new and the classical algorithms was performed. Results of the research provided a strict proof that the new algorithm on full Edwards curves, in comparison with the classic one, has some advantages in terms of performance speed, by about 1.5 times. The presented experimental results show that the performance speed increases even larger (by up to 30 per cent) in case the twisted and quadratic curves are used instead of full Edwards curves. It was shown that the assessment of probability of success of the new method increases due to the emergence of new conditions that lead to success of the algorithm that are not satisfied for the classical Lenstra algorithm on Weierstrass curves. The obtained results make it possible to decrease the time required for solving the problem on factorization by approximately 1.5 times, and thus, enable the faster breaking of cryptosystems whose stability is based on this problem