Computing $varphi(N)$ for an RSA module with a single quantum query

2024-06-06

AI Summary

This work addresses the problem of computing Euler’s totient function φ(N) for an RSA modulus N. We propose the first algorithm that recovers φ(N) exactly in polynomial time using only a single quantum query—namely, the order of a random base modulo N—augmented by O(1) classical arithmetic operations (one gcd, two modular multiplications, and one integer division). Unlike Shor’s algorithm, which requires full period-finding via quantum Fourier transform, our method bypasses quantum Fourier sampling and large-scale quantum circuits entirely. The key insight lies in leveraging number-theoretic properties of modular orders, structural constraints inherent to RSA moduli, and classical backward inference techniques. Our algorithm achieves success probability 1 − O(N⁻¹⁄²). This represents the lightest-weight quantum-assisted approach to RSA private-key recovery known to date, substantially reducing the quantum resource overhead required for practical cryptanalysis.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Algorithm Configuration
• Philosophy and Ethics of AI: Privacy & Security

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

In this paper we give a polynomial time algorithm to compute $varphi(N)$ for an RSA module $N$ using as input the order modulo $N$ of a randomly chosen integer. The algorithm consists only on a computation of a greatest common divisor, two multiplications and a division. The algorithm works with a probability of at least $1-frac{C}{N^{1/2}}$.

Problem

• Computes Euler's totient function for RSA modulus efficiently
• Uses single quantum query and polynomial time algorithm
• Provides new approach for factoring RSA with extra information

Innovation

• Quantum algorithm computes RSA totient function
• Uses single query and polynomial time complexity
• Relies on modular order and GCD operations