Motivate the Math

FEB 26, 2025

MTM07: Proofs / How Do You Know You What You Think You Know?

Proofs <a href="https://www.youtube.com/watch?v=HIkIqt_ytdc" target="_blank">https://www.youtube.com/watch?v=HIkIqt_ytdc</a> Fermat's Little Theorem  <a href="https://mathworld.wolfram.com/FermatsLittleTheorem.html" target="_blank">https://mathworld.wolfram.com/FermatsLittleTheorem.html</a> Euler's Function  <a href="https://en.wikipedia.org/wiki/Euler's_totient_function" target="_blank">https://en.wikipedia.org/wiki/Euler's_totient_function</a> Elliptic Curves: Point Addition  <a href="https://www.rareskills.io/post/elliptic-curves-finite-fields" target="_blank">https://www.rareskills.io/post/elliptic-curves-finite-fields</a>   Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

60 MIN

FEB 19, 2025

MTM06: Proof By Mathematical Induction: A Validation Cheatcode

Proof by Mathematical Induction <a href="https://youtu.be/Tm2PJPvAULs?si=H_RJ5rmVeyPDYM9W" target="_blank">https://youtu.be/Tm2PJPvAULs?si=H_RJ5rmVeyPDYM9W</a> <a href="https://youtu.be/KW5k7ZsQmwo?si=8rEdf2dUcTw74QZ5" target="_blank">https://youtu.be/KW5k7ZsQmwo?si=8rEdf2dUcTw74QZ5</a> Understanding Cryptography <a href="https://www.youtube.com/watch?v=2aHkqB2-46k" target="_blank">https://www.youtube.com/watch?v=2aHkqB2-46k</a> Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In 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.

65 MIN

FEB 19, 2025

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) = 1 Fermat's Little Theorem  <a href="https://mathworld.wolfram.com/FermatsLittleTheorem.html" target="_blank">https://mathworld.wolfram.com/FermatsLittleTheorem.html</a> Euler's Function <a href="https://en.wikipedia.org/wiki/Euler's_totient_function" target="_blank">https://en.wikipedia.org/wiki/Euler's_totient_function</a> Elliptic Curves: Point Addition <a href="https://www.rareskills.io/post/elliptic-curves-finite-fields" target="_blank">https://www.rareskills.io/post/elliptic-curves-finite-fields</a> Diffie-Hellman Illustration <a href="https://www.youtube.com/watch?v=NmM9HA2MQGI" target="_blank">https://www.youtube.com/watch?v=NmM9HA2MQGI</a> Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In 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.

62 MIN

FEB 6, 2025

MTM04: Levelling Up on Bitcoin's Supply Cap

Bitcoin's Issuance Schedule <a href="https://blog.lopp.net/how-is-the-21-million-bitcoin-cap-defined-and-enforced/" target="_blank">https://blog.lopp.net/how-is-the-21-million-bitcoin-cap-defined-and-enforced/</a> What is a Geometric Series <a href="https://en.wikipedia.org/wiki/Geometric_series" target="_blank">https://en.wikipedia.org/wiki/Geometric_series</a> Calculating the Sum of a Geometric Series <a href="https://www.youtube.com/watch?v=PqXAjCXYbNk" target="_blank">https://www.youtube.com/watch?v=PqXAjCXYbNk</a> Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we delve into the complexities of mathematics and its profound impact on our understanding of the world, particularly in the realm of cryptography. We discuss the challenges of making complex mathematical concepts accessible and why it's crucial not to blindly trust mathematical protocols without understanding their foundations. Our conversation explores the role of mathematics as a language that explains the workings of the world, emphasizing its importance beyond mere numbers. We also touch on the human brain's ability to recognize patterns and how this relates to our perception of reality and survival instincts. The discussion extends to neuroplasticity and the potential to "rewire" our brains through practice and repetition, drawing parallels between physical and mental fitness. Our exploration includes a deep dive into the concept of geometric series, particularly in the context of Bitcoin's block subsidy and its mathematical underpinnings. We explain how numbers can be represented as polynomials and the significance of fields and rings in mathematics, highlighting the unique properties of binary systems. Throughout the episode, we emphasize the importance of teaching and sharing knowledge to deepen understanding, and we encourage listeners to engage with mathematical concepts actively. We also reflect on the challenges and rewards of discussing complex topics and the personal growth that comes from pushing intellectual boundaries.

59 MIN

JAN 30, 2025

MTM03: Modular Arithmetic: The Queen Bee

What is a Relation <a href="https://www.youtube.com/watch?v=1v0qH4l9A2c&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=12" target="_blank">https://www.youtube.com/watch?v=1v0qH4l9A2c&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=12</a> What is an Equivalence Relation <a href="https://www.youtube.com/watch?v=o-PhSZztHC0&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=13" target="_blank">https://www.youtube.com/watch?v=o-PhSZztHC0&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=13</a> Modular Arithmetic (a little advanced but its good support until I find something simpler) <a href="https://www.youtube.com/watch?v=d-n92Ml1iu0&list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz&index=78" target="_blank">https://www.youtube.com/watch?v=d-n92Ml1iu0&list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz&index=78</a> Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we delve into the intricate world of cryptography, focusing on the mathematical foundations that underpin modern cryptographic systems. We start by exploring the AES chapter from the "Understanding Cryptography" PDF, discussing the layers beyond the mathematical sections. Our conversation highlights the importance of understanding both the cryptographic implementations and the mathematical relevance behind them, particularly in the context of public and private key cryptography, such as RSA and elliptic curve cryptography. We emphasize the significance of modular arithmetic, describing it as a fundamental substrate for cryptography. The discussion includes personal anecdotes about internalizing modular arithmetic and the continuous learning journey in understanding its applications. We also touch upon the Euclidean algorithm and its role in finding the greatest common divisor, which is crucial for cryptographic functions. The episode further explores the concept of cyclic groups and their relevance in cryptography, particularly in how they can reorder elements to enhance security. We discuss the importance of understanding linear combinations and equivalence relationships, which are foundational in mathematical modeling and cryptographic analysis. We also address some errata from previous episodes, clarifying definitions related to binary operations and cyclic groups. The conversation is enriched with practical examples, such as prime factorization and its role in cryptographic algorithms. Finally, we express gratitude to our listeners and those who have supported the podcast through boosts, encouraging them to engage with the material at their own pace and to explore additional resources provided in the show notes.

63 MIN

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