Note on the Ideal Exponentiation in Imaginary Quadratic Number and Function Fields

TL;DR

This paper derives specialized formulas for higher powers of ideals in imaginary quadratic number and function fields, building on existing algorithms that efficiently compute cubing ideals with small intermediate sizes.

Contribution

It introduces new formulas for computing higher powers of ideals, extending previous work on cubing ideals and improving computational efficiency.

Findings

Derived formulas for higher powers of ideals

Enhanced efficiency in ideal power computations

Maintained small intermediate operand sizes

Abstract

By using a new formula of cubing ideals in imaginary quadratic number and function fields combined with Shank's NUCOMP algorithm, Imbert et al. presented a fast algorithms that compute a reduced output of cubing ideals and keep the sizes of the intermediate operands small. The authors asked whether there are specialized formulas to compute higher powers of ideals. In this note, we will derive such formulas.