On the Signature Calculus for finite fields of order square of prime numbers

TL;DR

This paper extends the signature calculus approach for discrete logarithm problems from prime finite fields to quadratic extensions, establishing a new equivalence in computational complexity.

Contribution

It generalizes the previous results to quadratic extensions of prime fields, providing a broader framework for analyzing discrete logarithm problems.

Findings

Discrete logarithm in prime fields is equivalent to computing ramification signatures.

Extension to quadratic fields broadens the applicability of signature calculus.

Results suggest new avenues for cryptographic security analysis.

Abstract

In [Huang-Raskind 2009], the authors proved that the discrete logarithm problem in a prime finite field is random polynomial time equivalent to computing the ramification signature of a real quadratic field. In this paper, we do this for a quadratic extension of a prime field.