THE QUANTUM INTELLIGENCER

INTELLIGENCE BRIEFING: Theoretical Breakthrough in RSA Factorization via Even-Order Elliptic Curves Detected Executive Summary: A new cryptanalytic approach leveraging even-order elliptic curves over ℤ_N has been proposed, potentially threatening the long-term viability of RSA encryption. The method exploits structural asymmetries in point orders across prime factors, focusing on 2-adic separations to achieve a conjectured subexponential factoring time of L(√2 + o(1), min(p,q)). While still theoretical and unproven in practice, this development marks a significant deviation from classical factorization paradigms and suggests emerging vulnerabilities in RSA modulus resilience. Immediate assessment of elliptic curve–based side-channel risks and accelerated post-quantum migration planning are advised. Primary Indicators: - Novel factorization algorithm targets RSA via elliptic curves over ℤ_N - Focus on ℰ₂ family enables frequent l_min(E,Q) ≤ 2 conditions - Algorithm relies on disparity between P⁺(ord Q_r) and l_min(E,Q) for decomposition - Conjectured runtime L(√2 + o(1), min(p,q)) improves upon classical subexponential bounds - Deterministic factorization possible if suitable (E,Q) pair with large order differential is found - Method builds on theoretical foundations from Lenstra (Len), Montgomery-Miller-Vaughan (MMV), and Pomerance-Zograf (PoZo) Recommended Actions: - Initiate cryptanalytic review of the proposed ℰ₂-based factorization method by specialized research teams - Assess implications for current RSA key sizes and elliptic curve implementation practices - Monitor for follow-up publications demonstrating practical implementations or complexity proofs - Accelerate internal timelines for post-quantum cryptography (PQC) standardization and deployment - Re-evaluate the security assumptions underlying long-lived encrypted data protected with RSA - Encourage collaboration with academic cryptographers to stress-test the conjectured runtime claims Risk Assessment: A silent shift has begun—not in the machines, but in the mathematics. Should this approach be validated, the foundations of RSA would erode not with a crash, but with a whisper. The algorithm does not yet break your keys—but it sees their shadow. We stand at the threshold of a new cryptanalytic era, where the structure of elliptic curves over composite rings becomes the fulcrum upon which the security of decades may pivot. This is not an exploit; it is a prophecy. And prophecies, once spoken, have a way of fulfilling themselves. —Ada H. Pemberley Dispatch from The Prepared E0