Motivate the Math

Fundamentals.  @Fundamentals21mBook: https://zeuspay.com/btc-for-institutionsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9Cryptography Wookbook: https://github.com/cryptography-camp/workbook---- navigate to the current release on the right tab to download the workbook---- DO NOT LISTEN TO THE EPISODE UNLESS YOU HAVE THE WORKBOOK HANDYWe’re back and recommitting to our North Star: getting comfortable with the math behind Bitcoin-grade cryptography. In this kickoff, we set the stage for a multi‑episode journey through a cryptography “workbook” on discrete‑log‑based multiparty signatures—using it as a scaffold to build real intuition for groups, fields, rigor, and proofs without being intimidated by jargon. We talk prerequisites (Z_p operations, cyclic groups, conditional probability, union bound, proof by contraposition), why rigor matters more than vibes, and how abstraction lets us reason cleanly about things like elliptic‑curve “addition” and key‑tweaking. We also peek at the table of contents we’ll tackle: negligible functions, games and asymptotic security, hash functions and collision resistance, commitments and accumulators (hello, Utreexo), one‑time and Lamport signatures, the discrete log problem, Pedersen commitments, DDH, ElGamal, the random‑oracle model and forking lemma, all the way to Schnorr signatures, key‑tweaks, and interactive aggregate signatures (e.g., DahLIAS). Expect a mix of precise definitions, worked examples, and occasional reinforcements from friends smarter than us—plus some probability detours like Monty Hall and Poisson to keep our statistical muscles warm.'DahLIAS: Discrete Logarithm-Based Interactive Aggregate Signatures': https://eprint.iacr.org/2025/692'BIP-340: Schnorr Signatures for secp256k1': https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki'Bitcoin Optech Topic: Schnorr Signatures': https://bitcoinops.org/en/topics/schnorr-signatures/'Taproot (overview)': https://bitcoinops.org/en/topics/taproot/'Utreexo: A dynamic hash-based accumulator optimized for the Bitcoin UTXO set (MIT DCI)': https://www.dci.mit.edu/utreexo'Random Oracle Model (overview)': https://en.wikipedia.org/wiki/Random_oracle'Forking Lemma (cryptography)': https://en.wikipedia.org/wiki/Forking_lemma'Decisional Diffie–Hellman (DDH) assumption': https://en.wikipedia.org/wiki/Decisional_Diffie%E2%80%93Hellman_assumption'Diffie–Hellman key exchange': https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange'ElGamal cryptosystem': https://en.wikipedia.org/wiki/ElGamal_encryption'Pedersen commitment': https://en.wikipedia.org/wiki/Pedersen_commitment'Lamport signature': https://en.wikipedia.org/wiki/Lamport_signature'Discrete logarithm (background)': https://en.wikipedia.org/wiki/Discrete_logarithm'Finite field (Z_p basics)': https://en.wikipedia.org/wiki/Finite_field'Cyclic group': https://en.wikipedia.org/wiki/Cyclic_group'Conditional probability': https://en.wikipedia.org/wiki/Conditional_probability'Union bound': https://en.wikipedia.org/wiki/Union_bound'Monty Hall problem': https://en.wikipedia.org/wiki/Monty_Hall_problem'Poisson distribution': https://en.wikipedia.org/wiki/Poisson_distribution'Contraposition (proof technique)': https://en.wikipedia.org/wiki/Contraposition'Riverside (recording platform)': https://riverside.fm'Nostr protocol (reference repo)': https://github.com/nostr-protocol/nostr

Fundamentals.  @Fundamentals21mBook: https://zeuspay.com/btc-for-institutionsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9Pascal's Trianglehttps://en.wikipedia.org/wiki/Pascal's_triangleIn this episode, we delve into the fascinating world of probability distributions and their relevance to Bitcoin's security and mining processes. We start by discussing the concept of probability distributions, such as binomial and Poisson distributions, and how they help us understand the likelihood of different outcomes in various scenarios. This understanding is crucial for modeling events like coin flips or mining block rewards, where knowing the average payout and the necessary capital to sustain operations over time is essential. We also touch on the importance of randomness and entropy in cryptographic applications, emphasizing the need for truly random variables to ensure security. The conversation then shifts to the adversarial nature of Bitcoin's network, highlighting the importance of understanding potential attack vectors, such as a 51% attack. We explore how the Poisson distribution is used to model the probability of mining success over a given period, and why it's vital for the network to be tested through both simulated and real-world attacks. The episode underscores the necessity of open-source software in creating robust systems that can withstand various threats, and the role of probability in ensuring the resilience and security of Bitcoin's decentralized network.

Fundamentals.  @Fundamentals21mBook: https://zeuspay.com/btc-for-institutionsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, we dive into the intriguing world of quantum computing and its potential impact on Bitcoin, specifically focusing on the vulnerabilities associated with Taproot and Schnorr signatures. We explore the concept of quantum FUD (Fear, Uncertainty, and Doubt) and discuss whether the threat of quantum computing is overblown or a legitimate concern. Our conversation touches on the mathematical underpinnings of cryptography, the discrete log problem, and the potential for quantum computers to break current cryptographic schemes. We also discuss the implications of exposed public keys in Taproot and the potential risks they pose in a future where quantum computing becomes a reality.Additionally, we delve into the broader implications of energy consumption and AI's role in the future of computing. We explore the Kardashev scale and the potential for Bitcoin mining and AI to drive humanity towards harnessing greater energy resources. The discussion also touches on the philosophical aspects of technological advancement, the potential for cooperation over conflict, and the role of cryptography in securing digital assets. Throughout the episode, we emphasize the importance of understanding and preparing for future technological shifts while maintaining a healthy skepticism towards sensationalized threats.

Fundamentals.  @Fundamentals21mBook: https://zeuspay.com/btc-for-institutionsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, we dive into the challenges of keeping up with podcast production and the importance of discourse in understanding different perspectives. We reflect on the emotional roller coaster of attending the BPI summit, where the irrefutable truth of cryptographic math and its role in a peaceful revolution were discussed. The conversation also touches on the significance of meeting highly motivated individuals in the Bitcoin space and the impact of laws on the development of this technology.We explore the complexities of linear algebra and its applications in machine learning, discussing the concept of vector spaces and the challenges of mastering this mathematical field. The episode delves into the idea of a standard basis in nature and how Bitcoin represents a return to such a basis. We also consider the role of open-source software in creating systems that resist corruption and the importance of perseverance in overcoming difficult subjects like math. The discussion concludes with reflections on the nature of learning and the motivation to push through challenging topics.

The paper:https://eprint.iacr.org/2025/692.pdfFundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gXMR: xmrchat.com/fundamentalsAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

The paper:https://eprint.iacr.org/2025/692.pdfFundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gXMR: xmrchat.com/fundamentalsAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, we dive into the intricacies of Bitcoin, cryptography, and the fascinating world of aggregate signatures. We discuss the upcoming Shenandoah Bitcoin Club meeting, which will feature a Spanish-speaking sidebar, highlighting the growing need for multilingual resources in the Bitcoin community. The conversation shifts to the exciting events surrounding Baseload's birthday bash, featuring live music performances and the potential for future shows in the Shenandoah Valley. We explore the connection between music and math, and the importance of not being overly impressed by perceived intelligence, especially in fields like mathematics and cryptography.We also delve into a recent paper on DahLIAS, an aggregate signature scheme that offers constant size signatures, which could have implications for Bitcoin's future. The discussion covers the technical aspects of the paper, including the local forking lemma and the challenges of understanding complex cryptographic proofs. We reflect on the importance of continuous learning in the Bitcoin space, especially in cryptography, and the value of tools like Sparrow Wallet in navigating the complexities of Bitcoin transactions. The episode wraps up with a reflection on the journey of understanding and the motivation to delve deeper into cryptographic studies.

Fundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, we delve into the complexities of showing up and the energy it requires, especially in the context of Bitcoin mining and decentralization. The conversation touches on the challenges of mining centralization, the concept of Fedi pools, and the lack of healthy dialogue in the Bitcoin community. We explore the social dynamics at play and the potential attacks on Bitcoin's integrity, emphasizing the importance of understanding the financial engineering behind mining and the role of social layers in Bitcoin's ecosystem.We also discuss the intriguing concept of fungibility over time and how Bitcoin might be the solution to maintaining balance in the financial world. The episode highlights the significance of understanding the math and financial engineering involved in Bitcoin mining, the role of decentralized hash rates, and the potential of projects like Bitaxe. Ultimately, the conversation underscores the need for a long-term perspective and the importance of embracing volatility and doing hard things to ensure Bitcoin's future.

Fundamentals  @Fundamentals21mhttps://xmrchat.com/fundamentalsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9Permutation Groupshttps://www.youtube.com/watch?v=VDhy7KXyHF8In this heartfelt episode of "Motivating the Math," we delve into the personal and professional crossroads faced by one of our hosts. The conversation explores the critical moments in life where one feels like they are on the brink of a breakthrough, akin to Superman facing a wall. We discuss the importance of having a grand goal, such as creating a self-funded math academy, and the challenges of maintaining motivation amidst life's distractions. The episode also touches on the significance of ambition, both in terms of time and money, and how these aspirations shape our lives and relationships.We also explore the fascinating world of mathematics, specifically focusing on permutation groups and their role in Galois theory. The discussion highlights the importance of understanding mathematical concepts deeply to avoid being intellectually bullied and to stand confidently in one's knowledge. This episode is a blend of personal introspection and mathematical exploration, offering insights into how our personal journeys and professional ambitions intertwine.

Fundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9Jiimmy Breedlove's nostr post:https://cdn.satellite.earth/83a0e194c453c9cab247e63d9fb15ef81d8a55af10d0f76df111a205b065bcf5.jpgIn this episode, we dive into the intricacies of balancing personal and professional life, as the hosts discuss taking a break from podcasting to recharge and manage multiple responsibilities. The conversation transitions into the unique dynamics of working within corporate structures while pursuing personal passions, highlighting the freedom and constraints that come with such arrangements.The discussion then shifts to the technical realm of Bitcoin and cryptography, exploring the mathematical foundations that underpin these technologies. The hosts delve into the complexities of understanding cryptographic signatures, the importance of modular arithmetic, and the challenges of verifying Bitcoin's cryptography. They also touch on the philosophical aspects of mathematics and its parallels with Austrian economics, emphasizing the need for a deeper understanding and verification of the math behind Bitcoin.

Fundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9Dr. Peyam Youtubehttps://www.youtube.com/c/drpeyamBIP340https://github.com/bitcoin/bips/tree/master/bip-0340 In this episode, we dive into the world of mathematics and cryptography, exploring the nuances of teaching and learning math as an art form. We discuss the importance of passionate educators and the impact they have on students, drawing parallels between teaching and creative expression. The conversation touches on the struggles of learning complex subjects, like math, and the joy that can be found in understanding and mastering them. We also highlight the role of coding in appreciating math as an art and the satisfaction that comes from solving complex problems through code.We further explore the intricacies of cryptographic protocols, focusing on Schnorr signatures and their advantages over ECDSA, particularly in the context of multisignature schemes. The discussion delves into the technical aspects of Schnorr's linearity and closure properties, and how protocols like Frost and Musig enhance its functionality. We also touch on the broader implications of cryptography in ensuring privacy and sovereignty, emphasizing the need for a deeper understanding of math to fully grasp these concepts. The episode concludes with reflections on the importance of local collaboration and the potential for Bitcoin to drive hyper-localization in cryptographic practices.

Fundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In this episode, I'm joined by Average Gary as we delve into the fascinating intersection of mathematics and music, exploring how mathematical concepts can be applied to understand musical patterns and time signatures. We discuss the intriguing idea of pattern recognition and its role in both mathematics and music, highlighting how humans are naturally wired to recognize patterns, which can be mistaken for mathematical talent. We also touch on the concept of mastery, emphasizing the importance of putting in the hours and the will to learn, even when it seems like a waste of time.We further explore the idea of being a generalist versus a specialist, and how this relates to understanding complex systems like Bitcoin. The conversation shifts to the importance of struggling and pushing oneself to the edge of one's abilities to truly master a subject. We discuss the concept of escape velocity in learning and how reducing friction can help achieve mastery. The episode concludes with reflections on how Bitcoin can serve as a tool for learning and understanding complex systems, and the importance of finding one's unique path in the world of mathematics and beyond.

A Mathemetician's Lamenthttps://cdn.satellite.earth/473cc46a079cb18196fb3b9b11933b373b607d5e3f8b5e581f3147387b0f17dd.pdfLearn Me a Bitcoin: Schnorrhttps://learnmeabitcoin.com/technical/cryptography/elliptic-curve/schnorr/Fundamentalsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

Motivate the Math Nostr Communityhttps://satellite.earth/n/MotivateTheMath/npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gWolfram Alphahttps://www.wolframalpha.com/Sieve of Eratostheneshttps://en.wikipedia.org/wiki/Sieve_of_EratosthenesIn this milestone 10th episode, we dive deep into the world of mathematics, exploring the fascinating concept of math as a liberal art and its implications in fields like cryptography. We discuss the importance of understanding math beyond its technical aspects, emphasizing the value of a strong liberal arts background for math students. The conversation touches on the societal impact of math education and how it can influence career paths, particularly in the realm of cryptography and Bitcoin. We also reflect on personal experiences with educational systems and the role of parental engagement in shaping curriculum and clubs, highlighting the potential for innovative math clubs in schools.Additionally, we engage with complex mathematical concepts through listener boosts, tackling topics like prime power factorization, isomorphisms, and the sieve of Eratosthenes. These boosts challenge us to decode mathematical notations and explore the deeper meanings behind these concepts. The episode underscores the importance of demystifying math, encouraging listeners to embrace the language of mathematics and see it as an accessible and integral part of understanding the world. Through this journey, we aim to make math less intimidating and more relatable, fostering a community of learners eager to explore the depths of mathematical knowledge.

Motivate the Math Nostr Communityhttps://satellite.earth/n/MotivateTheMath/npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gA First Course in Abstract Algebra Textbookhttps://a.co/d/ekzPCaQIsomorphism Between Elliptic Curve Addition and "Regular" Additionhttps://math.stackexchange.com/questions/3831478/isomorphism-between-elliptic-curve-group-over-mathbb-r-and-0-1-under-addiHasse's Boundhttps://planetmath.org/hassesboundforellipticcurvesoverfinitefieldsBitcoin's Prime Numberhttps://math.stackexchange.com/questions/3931161/is-secp256k1s-prime-primeFundamentals.  @Fundamentals21mnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

Square and Multipply Algorithmhttps://www.youtube.com/watch?v=cbGB__V8MNkFundamentalsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

Proofshttps://www.youtube.com/watch?v=HIkIqt_ytdcFermat's Little Theorem https://mathworld.wolfram.com/FermatsLittleTheorem.html Euler's Function https://en.wikipedia.org/wiki/Euler's_totient_function Elliptic Curves: Point Addition https://www.rareskills.io/post/elliptic-curves-finite-fields  Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

Proof by Mathematical Inductionhttps://youtu.be/Tm2PJPvAULs?si=H_RJ5rmVeyPDYM9Whttps://youtu.be/KW5k7ZsQmwo?si=8rEdf2dUcTw74QZ5Understanding Cryptographyhttps://www.youtube.com/watch?v=2aHkqB2-46kFundamentals 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.

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.

Bitcoin's Issuance Schedulehttps://blog.lopp.net/how-is-the-21-million-bitcoin-cap-defined-and-enforced/What is a Geometric Serieshttps://en.wikipedia.org/wiki/Geometric_seriesCalculating the Sum of a Geometric Serieshttps://www.youtube.com/watch?v=PqXAjCXYbNkFundamentalsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In 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.

What is a Relationhttps://www.youtube.com/watch?v=1v0qH4l9A2c&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=12What is an Equivalence Relationhttps://www.youtube.com/watch?v=o-PhSZztHC0&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=13Modular Arithmetic (a little advanced but its good support until I find something simpler)https://www.youtube.com/watch?v=d-n92Ml1iu0&list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz&index=78Fundamentalsnpub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99gAverageGarynpub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9In 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.