Security Estimates for Quadratic Field Based Cryptosystems

TL;DR

This paper presents improved algorithms for solving discrete logarithm problems in quadratic field class groups, providing security estimates for cryptosystems based on these mathematical structures and recommending parameter sizes for comparable security levels.

Contribution

The paper introduces enhanced algorithms for quadratic field cryptanalysis and offers security parameter recommendations based on extensive numerical results.

Findings

Algorithms outperform previous methods in efficiency.

Security estimates align with standard symmetric key sizes.

Provides practical parameter recommendations for cryptosystems.

Abstract

We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field. The algorithms used incorporate improvements over previously-used algorithms, and extensive numerical results are presented demonstrating their efficiency. This data is used as the basis for extrapolations, used to provide recommendations for parameter sizes providing approximately the same level of security as block ciphers with $80,$ $112,$ $128,$ $192,$ and $256$-bit symmetric keys.