MTM05: Elliptic Curves and.Fermat's Little Theorem

2^173(mod5) = (2^4)^43 * 2^1 = 1^43 * 2^1 = 1 * 2 = 2 2^4(mod5) = 1 because 16(mod5) = 1Fermat's Little Theorem https://mathworld.wolfram.com/FermatsLittleTheorem.htmlEuler's Function https://en.wikipedia.org/wiki/Euler's_totient_functionElliptic Curves: Point Addition https://www.rareskills.io/post/elliptic-curves-finite-fieldsDiffie-Hellman Illustration https://www.youtube.com/watch?v=NmM9HA2MQGIFundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, we dive deep into the fascinating world of elliptic curves and their significance in cryptography. We start by discussing the basics of elliptic curves, particularly focusing on the polynomial equation y² = x³ + 7, which is crucial for Bitcoiners. We explore how operations on these curves, like adding points, form a group and why this concept is important.We then delve into the textbook by Neil Koblitz, which highlights the importance of elliptic curves in cryptography. The discussion transitions into the axioms of groups, such as closure, associativity, identity, and inverses, and how these relate to elliptic curves.Our conversation takes a turn towards Fermat's Little Theorem and its application in cryptography, particularly in computing inverses in finite fields. We explore how this theorem simplifies calculations with large numbers and its implications for public key cryptography.We also touch on the Diffie-Hellman key exchange, explaining how it enables secure communication over the internet by deriving a shared secret without exposing private keys.Throughout the episode, we emphasize the importance of understanding these mathematical concepts to grasp the underpinnings of cryptographic systems, especially in the context of Bitcoin and other cryptocurrencies.