Generalised Weber Functions

TL;DR

This paper generalizes Weber functions using Dedekind eta functions, classifies when their powers generate class fields, and explores their potential for cryptographic elliptic curve reductions.

Contribution

It introduces a generalized Weber function framework, classifies cases for class field generation, and provides formulas relating these invariants to modular functions.

Findings

Classified cases where powers of generalized Weber functions generate class fields.

Derived formulas for the degree of modular equations relating these functions to j-invariants.

Explored applications in constructing elliptic curve reductions for cryptography.

Abstract

A generalised Weber function is given by $\w_N(z) = \eta(z/N)/\eta(z)$, where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\w_N^e$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating $\w_N(z)$ and $j(z)$. Our ultimate goal is the use of these invariants in constructing reductions of elliptic curves over finite fields suitable for cryptographic use.