📐 RSA Correctness For properly chosen $e$ and $d$, decryption correctly recovers the original message: $m^{ed} \\equiv m \\pmod{n}$. Proof: **Setup:** Let $n = pq$ for distinct primes $p, q$. Let $e$ be coprime to $\\varphi(n) = (p-1)(q-1)$, and let $d = e^{-1} \\pmod{\\varphi(n)}$. This means $ed = 1 + k \\cdot \\varphi(n)$ for some integer $k$. **Proof:** We need to show $m^{ed} \\equiv m \\pmod{n}$ for any message $m$ with $0 \... From: Cryptography Math Learn more:

Cryptography Mathematics | Magic Internet Math

Learn the mathematics that powers modern encryption, from modular arithmetic to elliptic curves.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Rank and Nullity The rank of $T$ is $\\text{rank}(T) = \\dim(\\text{im}(T))$. The nullity of $T$ is $\\text{nullity}(T) = \\dim(\\ker(T))$. From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Handshaking Lemma For any graph $G$, $\\sum_{v \\in V(G)} d(v) = 2|E(G)|$. Proof: Each edge contributes exactly 2 to the sum of degrees (one for each endpoint). From: Introduction to Graph Theory Learn more:

Introduction to Graph Theory | Math Academy

Interactive course based on Douglas B. West

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.