Decomposition of RSA modulus applying even order elliptic curves

An efficient integer factorization algorithm would reduce the security of all variants of the RSA cryptographic scheme to zero. Despite the passage of years, no method for efficiently factoring large semiprime numbers in a classical computational model has been discovered. In this paper, we demonstrate how a natural extension of the generalized approach to smoothness, combined with the separation of $2$-adic point orders, leads us to propose a factoring algorithm that finds (conjecturally) the prime decomposition $N = pq$ in subexponential time L(√ 2+o(1), min(p,q)). This approach motivated by the papers {Len}, {MMV} and {PoZo} is based on a more careful investigation of pairs $(E,Q)$, where Q is a point on an elliptic curve E over N. Specifically, in contrast to the familiar condition that the largest prime divisor P⁺( Qₚ) of the reduced order Qₚ does not divide #E(q) we focus on the relation between P⁺( Qᵣ) and the smallest prime number lₘᵢₙ(E,Q) separating the orders Qₚ and Qq. We focus on the ₂ family of even order elliptic curves over N since then the condition lₘᵢₙ(E,Q)≤ 2 holds true for large fraction of points (x,y)∈ E(N). Moreover if we know the pair $(E,Q)$ such that P⁺( Qᵣ)≤ t<lₘᵢₙ(E,Q) and d=max_r∈ ,q\( Qᵣ) is large in comparison to min_r∈ ,q\|aᵣ(E)|≠ 0 then we can decompose N in deterministic time t¹⁺ᵒ⁽¹⁾ by representing N in base d.