[Paper Review] The Arithmetic of Carmichael Quotients

Min Sha|arXiv (Cornell University)|Aug 12, 2011

Mathematical Dynamics and Fractals15 references4 citations

TL;DR

This paper introduces Carmichael quotients $ C_m(a) = \frac{a^{\lambda(m)} - 1}{m} $, where $ \lambda(m) $ is the Carmichael function, generalizing Fermat and Euler quotients. It establishes arithmetic properties, including congruences and periodicity, defines Carmichael-Wieferich numbers (where $ C_m(a) \equiv 0 \pmod{m} $), and links these to perfect nonlinear functions, showing $ f_m(x) = \phi_m(x) $ is perfect nonlinear if and only if $ m $ is prime.

ABSTRACT

Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $λ(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.

Motivation & Objective

• To generalize Fermat and Euler quotients by introducing a new quotient based on the Carmichael function $ \lambda(m) $.
• To investigate fundamental arithmetic properties of these new quotients, including congruences and periodicity of derived sequences.
• To define and study Carmichael-Wieferich numbers, where $ C_m(a) \equiv 0 \pmod{m} $, and relate them to Wieferich primes.
• To establish a connection between Carmichael quotients and perfect nonlinear functions in cryptography.

Proposed method

• Define the Carmichael quotient $ C_m(a) = \frac{a^{\lambda(m)} - 1}{m} $ for $ \gcd(a,m) = 1 $, using the Carmichael function $ \lambda(m) $.
• Prove key congruences: $ C_m(ab) \equiv C_m(a) + C_m(b) \pmod{m} $ and $ C_m(a + km^\alpha) \equiv C_m(a) + \frac{k\lambda(m)}{a}m^{\alpha-1} \pmod{m^\alpha} $.
• Use the $ p $-adic valuation $ \mathrm{ord}_p(C_m(a)) = e(m,p) + \sigma(a,p) $ to analyze the structure of $ C_m(a) $.
• Establish a criterion for $ m $ to be a Carmichael-Wieferich number with base $ a $: $ e(m,p_j) + \sigma(a,p_j) \geq r_j $ for all prime powers $ p_j^{r_j} $ dividing $ m $.
• Define a function $ f_m: \mathbb{Z}/m^2\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z} $ via $ \phi_m(x) $, extended to $ \gcd(x,m) \neq 1 $ by setting $ \phi_m(x) = 0 $.
• Prove that $ f_m $ is perfect nonlinear if and only if $ m $ is prime, using counting arguments on fiber sizes.

Experimental results

Research questions

• RQ1 How do Carmichael quotients generalize Fermat and Euler quotients, and what arithmetic properties do they inherit?
• RQ2 What are the necessary and sufficient conditions for a composite number $ m $ to satisfy $ C_m(a) \equiv 0 \pmod{m} $, i.e., to be a Carmichael-Wieferich number?
• RQ3 Can Carmichael quotients be used to construct perfect nonlinear functions, and what conditions ensure such constructions?
• RQ4 Is there a connection between Carmichael-Wieferich numbers and the existence of Wieferich primes?

Key findings

• The Carmichael quotient satisfies $ C_m(ab) \equiv C_m(a) + C_m(b) \pmod{m} $, generalizing additive behavior seen in Fermat quotients.
• For $ m \geq 3 $, the sum $ \sum_{k=0}^{m-1} C_m(a + km) \equiv 0 \pmod{m} $, showing periodic cancellation in sequences derived from $ C_m $.
• The $ p $-adic valuation of $ C_m(a) $ is given by $ \mathrm{ord}_p(C_m(a)) = e(m,p) + \sigma(a,p) $, where $ e(m,p) $ is the exponent of $ p $ in $ m $, and $ \sigma(a,p) $ is the $ p $-adic valuation of $ a^{p-1} - 1 $.
• A number $ m $ is a Carmichael-Wieferich number with base $ a $ if and only if $ e(m,p_j) + \sigma(a,p_j) \geq r_j $ for all prime powers $ p_j^{r_j} $ dividing $ m $.
• If a composite $ m $ is a Carmichael-Wieferich number with base $ a $ and has an odd prime factor, then that prime must be a Wieferich prime with base $ a $.
• The function $ f_m(x) = \phi_m(x) $ is perfect nonlinear if and only if $ m $ is prime, providing a new class of such functions with computational advantages.