The Computational Complexity of the Elliptic Curve Factorization Algorithm over Real Field

Abstract

Several mathematical concepts from analytics number theory, especially the counting functions (CFs) are employed to solve some modern problems in mathematics, computer sciences, engineering and others. One of them is the factorization problem for integers into two primes. This work proposes an extended factorization problem using the CFs, as main part from analytics number theory, to factor a real number r into two primes p and q. The computations of the points doubling and addition on the elliptic curves E modulo a composite number n are also used in this extension to give new two versions of the factorization over real field R. On the first version, the primes are obtained from doubling and addition on elliptic curve points that are computed modulo composite number which is resulting as output of the CFs algorithm with the input r. Whereas, the primes, on the second version, are got directly from applying the CFs algorithm with r as input. The successful computation of the scalar multiplication kP on the elliptic curve E modulo primes are determined. The computational complexities of extended elliptic curve factorization versions are determined using the counting operations. These operations are elliptic curve and finite field operations. New numerical results on two versions are computed as study cases. The elliptic curve factorization algorithm over R (ECF-R) with two versions, is more efficient for factoring-based cryptography in comparing to the original ECM.