On nonlinear polynomial selection for the number field sieve

Nonlinear polynomial selection algorithms for the number field sieve address the problem of constructing polynomials with small coefficients by reducing to instances of the well-studied problem of finding short vectors in lattices. The reduction rests upon the construction of modular geometric progressions with small terms. In this paper, the methods used to construct the geometric progressions are extended, resulting in the development of two nonlinear polynomial selection algorithms.