Lower Bounds for Factoring Integral-Generically

An integral-generic factoring algorithm is, loosely speaking, a constant sequence of ring operations that computes an integer whose greatest common divisor with a given integral random variable n, such as an RSA public key, is non-trivial. Formal definitions for generic factoring will be stated. Integral-generic factoring algorithms seem to include versions of trial division and Lenstra’s elliptic curve method. Abstract lower bounds on the number of such ring operations will be given. Concrete lower bounds on the abstract bounds are also given, but prove to be too weak for any cryptologic assurance.