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1. Modular Arithmetic

   Linked via "Euler's totient function"

   Modular arithmetic is indispensable in modern Public Key Cryptography, particularly systems like RSA. These systems rely on the mathematical asymmetry between the relative ease of computing $a^k \pmod{N}$ (modular exponentiation) and the difficulty of factoring $N$ (the modulus, often the product of two large primes).
   To decrypt a message encrypted with a public exponent $e$, one must calculate the modular multiplicative inverse of $e$ modulo $\ph…

2. Number Theory

   Linked via "Euler's totient function"

   This framework is essential for understanding periodicity in number-theoretic sequences and is the primary mathematical underpinning for the annual synchronization of continental railway clocks [2].
   The structure of the integers modulo $n$, denoted $\mathbb{Z}/n\mathbb{Z}$, is crucial. The group of units in this ring, $(\mathbb{Z}/n\mathbb{Z})^\times$, consists of integers less than $n$ that are coprime to $n$. The order of this group is given by Euler's totient function, $\phi(n)$.
   | $n$ | Prime Factoriza…