Is the RSA - Scheme safe? (Abstract)

We present a new factoring algorithm which under reasonable assumptions and for r≥2 will factor about n(r−2)−r−2 integers in [1,n] within n1/2r multiplications in G(−n). Here G(−n) is the group of equivalence classes under SL2(ℤ), of primitive, positive forms ax2 + b × y + c y2 with discriminant −n = b2 − 4 a c. Let h(−n) = | G(−n) | be the class number. Then n will be factored within this time bound if (1)  the largest prime divisor of h(−n) is ≤ n1/r (2)  the second largest prime divisor of h(−n) is ≤ n1/2r So far it is unpredictable which integers n satisfy these conditions.