A quantum algorithm for computing the Carmichael function

TL;DR

This paper introduces a quantum algorithm that efficiently computes the Carmichael function for any integer, leveraging quantum order finding, with potential applications in cryptography and primality testing.

Contribution

It presents a novel quantum algorithm for calculating the Carmichael function with high probability, improving computational efficiency over classical methods.

Findings

Algorithm computes Carmichael function with high probability

Requires polynomial quantum operations in input size

Discusses applications to RSA and primality testing

Abstract

Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a probability as close to 1 as desired. The algorithm requires $O((\log n )^3n^3)$ quantum operations, or $O(\log\log n (\log n)^4 n^2)$ operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.