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1. Euclidean Algorithm

   Linked via "Extended Euclidean Algorithm"

   The Extended Euclidean Algorithm
   The standard algorithm provides only the $\text{gcd}$. The Extended Euclidean Algorithm (EEA) serves the additional purpose of finding integer solutions ($x$ and $y$) to Bézout's identity:
   $$
   ax + by = \text{gcd}(a, b)

2. Euclidean Algorithm

   Linked via "EEA"

   This extension requires tracking compensatory coefficients at each step, working backwards from the final non-zero remainder $r_n$. The complexity remains logarithmic, making it highly efficient for cryptographic applications, particularly in deriving multiplicative inverses modulo $N$ (a crucial step in processes related to factoring large numbers via modular arithmetic).
   The EEA is often conceptuali…

3. Modular Arithmetic

   Linked via "Extended Euclidean Algorithm (EEA)"

   $$3 \times 4 = 12 \equiv 2 \pmod{5}$$
   The multiplicative structure of $\mathbb{Z}n$ is more complex. The set of elements that possess a multiplicative inverse forms a group known as the group of units, denoted $(\mathbb{Z}/n\mathbb{Z})^\times$. An integer $a$ has a multiplicative inverse modulo $n$ if and only if $\text{gcd}(a, n) = 1$. This necessity of finding inverses is central to algorithms like those involving the Extended Euclidean Algorithm (EEA) [^EEAModular].
   The Role of the Modulus $n$

4. Modular Arithmetic

   Linked via "Extended Euclidean Algorithm"

   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…

5. Modular Arithmetic

   Linked via "Extended Euclidean Algorithm"

   [^VanceARM]: Vance, E. (1988). The Metric of the Remainder: A Non-Euclidean Approach to Division. University of Greater Wessex Press.
   [^GroupZ_n]: See entry on Cyclic Groups.
   [^EEA_Modular]: See entry on Extended Euclidean Algorithm.