Fermat’s Little Theorem — The Cheat Code Behind Modern Cryptography

You’ve heard about primes. You’ve heard about modular arithmetic. But Fermat’s Little Theorem? That’s the quiet engine powering everything from crypto keys to impossible exponent shortcuts. In this video, we walk through the raw power of Fermat’s Little Theorem — what it is, why it works, and how to use it to simplify insanely large calculations without touching a calculator. Whether you're deep in math, learning cryptography, or just want to understand how the internet stays secure, this theorem is your gateway drug. We reference the classic book Understanding Cryptography, break down real examples, and show you how Fermat gives you math superpowers. 💡 Watch till the end to see how Fermat crushes giant powers like 3¹⁰²³ and gives you the inverse of numbers in mod prime arithmetic — instantly. ⏱️ Timestamps: 0:00 – Intro & why Fermat’s Little Theorem deserves its own video 0:15 – Recommended resource: Understanding Cryptography 0:46 – Modular arithmetic refresher 1:10 – Why primes matter in modular math 1:33 – Fermat’s Little Theorem: first version explained 2:28 – What it means to say aᵖ ≡ a mod p 3:06 – Using FLT to find modular inverses 3:55 – Textbook definition breakdown 5:01 – Real example: 3⁵ mod 5 6:33 – Rewriting the theorem to show aᵖ⁻¹ ≡ 1 mod p 7:00 – Why this matters: shortcutting giant powers 8:01 – Real-world exponent simplification: 3¹⁰¹ mod 5 9:01 – Power hack: turning huge powers into small ones 10:22 – New modulus: FLT with mod 7 11:33 – Applying FLT again: 2¹⁰¹ mod 7 12:40 – Massive reduction explained 13:13 – Final thoughts and wrap-up