Approximating Euler Totient Function using Linear Regression on RSA moduli

TL;DR

This paper investigates using linear regression to approximate Euler's totient function for RSA moduli, aiming to explore potential cryptanalytic applications of machine learning in cryptography.

Contribution

It introduces a novel approach of applying linear regression to estimate phi(n) for RSA moduli, demonstrating preliminary feasibility for cryptanalysis.

Findings

Phi can be approximated with small relative error

Regression models show promise for cryptanalytic strategies

Potential implications for RSA security analysis

Abstract

The security of the RSA cryptosystem is based on the intractability of computing Euler's totient function phi(n) for large integers n. Although deriving phi(n) deterministically remains computationally infeasible for cryptographically relevant bit lengths, and machine learning presents a promising alternative for constructing efficient approximations. In this work, we explore a machine learning approach to approximate Euler's totient function phi using linear regression models. We consider a dataset of RSA moduli of 64, 128, 256, 512 and 1024 bits along with their corresponding totient values. The regression model is trained to capture the relationship between the modulus and its totient, and tested on unseen samples to evaluate its prediction accuracy. Preliminary results suggest that phi can be approximated within a small relative error margin, which may be sufficient to aid in certain classes of RSA attacks. This research opens a direction for integrating statistical learning techniques into cryptanalysis, providing insights into the feasibility of attacking cryptosystems using approximation based strategies.