Algebraic number fields and the LLL algorithm
share
In this paper we analyze the computational cost of various operations performed symbolically in real algebraic number fields where the elements are represented as polynomials of a primitive element of the field. We give bounds on the costs in terms of several parameters, including the degree of the field and the representation size of the input. Beyond the basic field operations we also analyze the cost of the less-than comparison and the integer rounding functions. As an important application we give a polynomial bound on the running time of the LLL lattice reduction algorithm when the vector coordinates are from an algebraic number field and the computations are performed exactly.
READ FULL TEXT