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Magic Internet Math: Daily Summaries Delivered
by Brian HIrschfield and Rob Hamilton
This podcast exists to liberate Bitcoin holders from second-class citizenship by teaching the mathematics that underlies their convictions. We operate on a simple premise: if you don't understand the math of Bitcoin, you cannot truly know what you know—you're dependent on others' authority, forever vulnerable to doubt and manipulation. Mathematics is the primary pathway to conviction in your own reasoning. Through accessible, conversational exploration of Bitcoin's mathematical foundations—treating math as the liberal art it was always meant to be—we equip listeners with genuine understanding rather than borrowed beliefs. We reject the deliberate demoralization campaign that convinced generations they're not capable of mathematical thinking.
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- 5 days ago1h 9m
Exponential Got Less Exponential
In this episode, we picked back up on our red-team tour of Bitcoin cryptography and tried to make sense of the last few weeks of fast-moving math news. We unpacked why the recent excitement around the exponential time hypothesis mattered, how it relates to the discrete log problem behind Bitcoin’s elliptic curve cryptography, and why “less exponential” is not the same thing as “Bitcoin is broken.” Along the way, we revisited the different doors an attacker would have to walk through to weaken secp256k1, from generic algorithmic improvements to curve symmetries, lifting tricks, and the bigger backstop of P vs NP. The goal wasn’t to declare victory or panic, but to get clearer on what has actually changed, what remains theoretical, and where AI seems to fit as a theorem-grinder rather than a magic cryptography-breaking button.We also spent time on the emerging tooling around formal verification and hash-based signatures. That took us into sig.golf, Lean proofs, OpenAI’s public math repository, and the growing idea that part of Bitcoin’s future safety may depend not just on us verifying systems, but on us proving that machines can verify them too. We talked through why libsecp256k1 deserves far more appreciation, why Jonas Nick’s post-quantum work keeps showing up at the center of these conversations, and why experiments around Lean, Bitcoin consensus rules, and AI-assisted proof generation feel increasingly important. If this show is our attempt to build a bridge into the math, this was one of those episodes where the bridge got a little longer — and a lot more interesting.Magic Internet Math: https://magicinternetmath.com/Plan ₿ Forum Lugano: https://planb.lugano.ch/planb-forum/libsecp256k1: https://github.com/bitcoin-core/secp256k1Bitcoin Core: https://bitcoin.org/en/bitcoin-core/OpenAI Math Repository: https://github.com/openai/mathLean Programming Language and Theorem Prover: https://lean-lang.org/sig.golf: https://sig.golf/lean Ethereum by Justin Drake: https://blog.ethereum.org/2025/07/31/lean-ethereumJoseph Lubin at Consensys: https://consensys.com/about/joseph-lubin-founder-of-consensysJonas Nick at Blockstream: https://blog.blockstream.com/author/jonas/Neal Koblitz: https://sites.math.washington.edu/~koblitz/A Course in Number Theory and Cryptography: https://link.springer.com/book/10.1007/978-1-4419-8592-7Robin Linus: https://robinlinus.com/Antoine Riard: https://gist.github.com/ariardrust-bitcoin: https://github.com/rust-bitcoin/rust-bitcoinCashu: https://cashu.space/Lightning Network / Core Lightning: https://blockstream.com/lightning/DeepSeek: https://www.deepseek.com/Kaggle: https://www.kaggle.com/BIP 42: https://github.com/bitcoin/bips/blob/master/bip-0042.mediawikiBIP 66: https://github.com/bitcoin/bips/blob/master/bip-0066.mediawikiBIP 113: https://github.com/bitcoin/bips/blob/master/bip-0113.mediawiki
- 3 weeks ago1h 26m
Red Teaming Elliptic Curve Cryptography
OpenAI just dropped roughly 300 math papers with Lean proofs, mathematicians are crashing out, and the vague-posting has started: "math is breaking, so cryptography is next." Brian and Rob Hamilton refuse to hand-wave. Rob fed the Magic Internet Math elliptic-curve study guide and the open theorems into a model and asked it straight: if you wanted to break Bitcoin's curve, which doors would you knock on? Then he and Brian walk through every one of them, live.The result is a red-team tour of secp256k1. There are six offensive open problems and three defenses nobody has been able to prove, and each comes with the specific math that would have to break. Along the way: what Lean proofs actually are, why the discrete log problem is where the hardness lives, why "elliptic curve cryptography has been broken" will probably be a headline about some curve Bitcoin doesn't use, and why Big Bitcoin and Big Math are on a collision course.In this episode:• OpenAI's ~300-paper drop, the "11 squares" packing result everyone memed, and the human who immediately found the beautiful version nobody paid for• Lean explained: an audit trail for what the AI did, built up from the foundations of math• Why mathematicians are upset (the funding model behind long-standing problems), and Terence Tao's case that humans still matter• Division of labor between people and machines, and Brian's advice to his daughter Kayla, a senior math major: apprentice on Lean now, then go do your research• Big-O notation in two minutes, and why it's the first page of Neal Koblitz's number theory and cryptography book• The matrix-multiplication race (Strassen's ~2.8 down to ~2.37 and below) and why faster linear algebra is the real prize for AI• secp256k1 from first principles: y² = x³ + 7 over a finite field, point addition, and why public → private is a one-way street• Pollard's rho and the real defense line of 2¹²⁸, plus the Coldcard reminder: 2³² falls to one computer and ~2⁷⁰ still costs real money• Door 1: point factoring• Door 2: the "minors" attack. Build a big matrix and hunt for a vanishing determinant. Solved toy curves at 2⁵⁰, but an Aug 2026 follow-up showed it costs more than brute force at scale• Door 3: beating the generic √n bound (Shoup)• Door 4: the Koblitz-curve symmetry Satoshi chose for speed, and a 2026 Lean result that found only the trivial symmetry survives• Door 5: lifting the problem to a smooth curve, and why the height penalty blows up• Door 6: isogenies and the "weak cousin" curve, why GRH says every door in the family is equally strong, and why this is a combo pool shot rather than a break• The defenses no one can climb: one-way functions, proving the discrete log is hard, and P vs NP• Yehuda Lindell's (Coinbase) thread: AI proving hard theorems is no evidence that elliptic curve hardness is in danger• Shor's algorithm as an engineering and physics problem, not a math one, and the post-quantum camps (hash-based, lattices, isogenies)• A preview of the math panel at BTC Amsterdam: "do a podcast to get inspiration for the conference talk"️ Brian Hirschfield × Rob Hamilton Magic Internet Math: a first-principles podcast about mathematics, Bitcoin, and meaning⚡ Follow:Brian on X: https://x.com/Fundamentals21mRob Hamilton on X: https://x.com/Rob1HamAnchorWatch: https://anchorwatch.comMagic Internet Math: https://magicinternetmath.com Further reading:secp256k1: Security Status & Open Problems (episode companion): https://bitdevs.magicinternetmath.com/content/secp256k1-open-problemsOpenAI math results (Lean): https://github.com/openai/math#Bitcoin #EllipticCurves #secp256k1 #Cryptography #DiscreteLog #Lean #FormalProofs #OpenAI #PvsNP #PostQuantum #MagicInternetMath #Mathematics #RobHamilton
- Sep 28, 202650 min
Hash Functions, Linear Algebra, and Alice in Wonderland
Magic Internet Math: https://www.magicinternetmath.com/Shout out to https://x.com/uncleJim21 for the Alice in Wonderland story and the following rebuke of Lewis Carroll:https://www.youtube.com/shorts/33FYXpEWgyAThis week we kept it loose but still got our reps in. We talked about why the math side of Bitcoin is starting to matter more in public, from conference panels and keynotes to the kinds of questions people are now asking us directly. We shared updates on upcoming appearances in Lugano, Amsterdam, Prague, and beyond, then dug into the deeper point: when confidence in Bitcoin gets shaken by bugs, headlines, or future quantum fear, the antidote is not vibes or macro slogans — it’s understanding the cryptographic foundations well enough to reason for ourselves.From there, we mapped out where the show is heading next. We reflected on libsecp256k1, Bitcoin Core, Miniscript, and what it means to inspect the basement wiring instead of just admiring the house from the curb. We also previewed future arcs on hash functions, Merkle trees, and linear algebra, including why SHA-256 feels so different from elliptic-curve cryptography. And because this is still Magic Internet Math, we took a wild but fitting detour into Lewis Carroll, imaginary numbers, quaternions, and the stubborn human habit of confusing identity with ideas.Hell Money: https://hell.money/libsecp256k1: https://github.com/bitcoin-core/secp256k1Miniscript: https://miniscript.org/Bitcoin Amsterdam: https://www.bitcoin.amsterdam/Lugano Plan ₿ Forum: https://planb.lugano.ch/planb-forum/Midwest Bitcoin Summit: https://midwestbtc.com/mediaBitcoin Policy Institute: https://www.btcpolicy.org/Bitcoin Policy Summit: https://www.btcpolicysummit.org/Motivate the Math: https://podcasts.apple.com/us/podcast/motivate-the-math/id1790329241Lewis Carroll Society of North America: https://www.lewiscarroll.org/Alice's Adventures in Wonderland: https://www.simonandschuster.com/books/Alices-Adventures-in-Wonderland/Lewis-Carroll/Alices-Adventures-in-Wonderland/9781665925778Phil Spector: https://philspector.com/
- Sep 13, 20261h 17m
Kayfabe Math is Now Mainstream
We’re back because math keeps refusing to stay in the classroom. In this episode, I unpack two very different kinds of “math pop culture.” First, we revisit entropy, seed generation, and wallet security through a Bitcoin lens: dice rolls versus machine randomness, Bitkey’s firmware checks, Bitcoin Core as the most-reviewed software stack in the room, and why truly understanding how keys are made matters more than blindly outsourcing trust. The deeper theme running through that whole section is validation—how much of security is really just getting honest about what you can and cannot verify for yourself.From there, we turn to two recent rage-bait flashpoints: the anti-algebra Tucker Carlson clip and the hype around AI “solving” Navier-Stokes. I argue that abstraction is the toolkit that lets you reason your way out of the matrix, and that dismissing algebra as fake misses the entire point of mathematical thinking. On the AI side, we get into proof formalization, peer review, incentive problems in mathematics, and whether machine-generated breakthroughs actually advance math—or mostly advance our need for better audit trails. We close on a more optimistic note: AI may compress old workflows, but it also increases the value of high-agency humans who can validate, reason, and build beyond the rails.AnchorWatch: https://www.anchorwatch.com/Presidio Bitcoin: https://www.presidiobitcoin.org/Block: https://block.xyz/Bitkey: https://bitkey.world/Bitkey Wallet GitHub: https://github.com/Bitkey-Wallet/Bitcoin Core: https://bitcoincore.org/libsecp256k1 GitHub: https://github.com/bitcoin-core/secp256k1BitDevs: https://bitdevs.org/OpenAI Navier–Stokes announcement: https://openai.com/index/navier-stokes-solution/Clay Mathematics Institute: https://www.claymath.org/Algebraic Number Theory and Fermat’s Last Theorem: https://www.routledge.com/Algebraic-Number-Theory-and-Fermats-Last-Theorem/Stewart-Tall/p/book/9781032610931Hell Money podcast: https://play.fountain.fm/show/W8pjfavWchv8diXjikgCWavlake: https://wavlake.com/Sir Andrew Wiles: https://www.maths.ox.ac.uk/people/andrew.wilesOpen Ordinals: https://ordinals.org/Casey Rodarmor’s blog: https://rodarmor.com/blog/
- Sep 2, 20262h 1m
Legalize MATH
Magic Internet Math is back. After the summer that set the Bitcoin world on fire — the Cold Card entropy disaster — Brian and Rob Hamilton (AnchorWatch) reunite for a two-hour deep dive on the event Rob spent three sleepless weeks at the center of, and what it taught them about AI, entropy, censorship, and self-custody.The spine of it: when funds started moving off Cold Cards, Rob loaded the firmware into the LLMs. Claude and Codex hedged and downgraded; Kimi K3 — a Chinese open-weight model — instantly printed the vulnerability. That contrast becomes the episode's thesis: the machines see more than they let on, and the open models will say what the censored ones won't. From there it opens into the math of entropy (why you roll your own dice), a love letter to libsecp256k1, and why sovereignty requires actually knowing the math — because they can censor the models, but they can't ban the math.In this episode:• The Cold Card entropy disaster — MK3 (~20–32 bits) vs MK4 (~70 bits), "cannon fodder for the next wave," and Rob's pinned-tweet triage guide for single-sig and multisig holders• Using LLMs to find the vulnerability — Claude and Codex censored and hedging, Kimi K3 printing it instantly: "the machine sees more than it lets on"• Open-weight Chinese models as the workhorse — LLMs as "slot machines" that spit out vulnerabilities; 100 scans at $100 beating one $10k elite scan• Why you roll your own dice: y² = x³ + 7, 2²⁵⁶ points, equal probability, and XOR (addition mod 2) to combine entropy the machine can't fake• "Cryptography is a weapon; entropy is the bullet" — brain wallets, hash functions, and why crypto books assume good entropy• Bitcoin Core's four entropy sources, Alex Waltz's call-trace deep dive, ralo mcfluid, and why "Satoshi's keys are the AD-IQ proof" the RNG is sound• Custody done right — River and owning your stack, Prime Trust, "only the paranoid survive," and "the soundest sleepers are the first to get wrecked"• FROST and Frostsnap (Lloyd & Nick), nonces, and the two places you actually need randomness• Agent-orchestration reality — lazy downstream agents, "any ambiguity in delegation will be used against you," front-loading the architecture, and interviewing the CTO to grill yourself on the gaps• AI-psychosis jokes, wellness checks, and why knowing the math keeps you from anthropomorphizing the machine• A love letter to libsecp256k1 — the red team dog-piled it and found nothing but doc nits; "if you've contributed to libsecp, you never buy a beer at a conference again"• Rob's first Bitcoin Core and Knots contributions (fail-open commit-verification bugs), OpenSats funding the red-team tooling, and a shout-out to Lawrence, the #5 Core contributor• The fork drama, the right to exit, Byzantine generals, and the "whale fall" of Blake-algorithm altcoin miners descending on the new chain• Why sovereignty requires the math — "you can't be a sovereign individual if you don't understand enough" — and going to the Bitcoin store to find the manager who doesn't exist️ Brian Hirschfield × Rob Hamilton Magic Internet Math — a first-principles podcast about mathematics, Bitcoin, and meaning⚡ Follow:Brian on X: https://x.com/Fundamentals21mRob Hamilton on X: https://x.com/Rob1HamAnchorWatch: https://anchorwatch.comMagic Internet Math: https://magicinternetmath.com#Bitcoin #Cryptography #Entropy #ColdCard #AI #OpenWeights #KimiK3 #secp256k1 #SelfCustody #MagicInternetMath #Mathematics #RobHamilton #AnchorWatch
- Jul 19, 20261h 23m
Quantum Computers Don't Exist w/ Brandon Black
In this episode, Rob and I sit down with Brandon Black (aka Rearden Code) for a deep dive into the quantum-computing debate around Bitcoin. We unpack why Brandon thinks the more useful framing is not “post-quantum” but “post-secp256k1,” and why Bitcoiners should care less about sci-fi narratives and more about understanding the actual cryptographic assumptions that protect coins today. Along the way, we revisit the now-legendary conference panel on quantum risk, talk through exposed pubkeys, xpub leakage, multisig, Taproot, Lightning force closes, and the difference between a theoretical break and an economically viable attack.We also spend a lot of time sizing risk correctly. Brandon helps us separate slow, visible cryptographic degradation from the far less likely “everything breaks tomorrow” scenario, and we explore how incentives change depending on whether the attacker wants profit, strategic advantage, or chaos. The conversation eventually zooms all the way out to math, physics, and engineering: Shor’s algorithm may exist on paper, but building a machine that can actually matter for Bitcoin is still a very different question. If you want a rigorous, grounded conversation about quantum threats, elliptic curves, MuSig2, FROST, and how to think clearly instead of panicking, this is a great one.The legendary panel: https://www.youtube.com/watch?v=wbJkODcrz7QMagic Internet Math: https://magicinternetmath.com/Brandon Black speaker profile: https://sv23.adoptingbitcoin.org/speakers/brandonblack/James O'Beirne: https://jameso.be/BIP 360: https://github.com/bitcoin/bips/blob/master/bip-0360.mediawikiBIP 327 (MuSig2): https://github.com/bitcoin/bips/blob/master/bip-0327.mediawikiBIP 340 (Schnorr Signatures for secp256k1): https://github.com/bitcoin/bips/blob/master/bip-0340.mediawikiBIP 341 (Taproot): https://github.com/bitcoin/bips/blob/master/bip-0341.mediawikiBIP 32 (Hierarchical Deterministic Wallets): https://github.com/bitcoin/bips/blob/master/bip-0032.mediawikiBIP 39 (Mnemonic code for generating deterministic keys): https://github.com/bitcoin/bips/blob/master/bip-0039.mediawikiFROST Protocol (RFC 9591): https://www.rfc-editor.org/rfc/rfc9591.html
- Jun 29, 20261h 10m
Perfect Number Day and the Road to Schnorr
We celebrated June 28 as “Perfect Number Day” and finally told one of my favorite origin stories behind this whole project: a late-night drive home from a Phish show with Kayla, where a quick conversation about why 6 and 28 are perfect numbers turned into a full-blown family math obsession. We revisited how that curiosity led us to trial-and-error the next perfect number, fire up a Raspberry Pi, write Python code, hit the limits of the machine, and eventually discover just how deep number theory goes—from the Pythagoreans to Euclid, Mersenne primes, and the still-open questions around perfect numbers.From there, we closed out our elliptic curve study guide by turning to Schnorr signatures, Taproot, MuSig, FROST, libsecp256k1, and the practical realities of Bitcoin development. We talked through why Schnorr is cleaner than ECDSA, how signature aggregation improves privacy and scalability, why wrench-attack resistance matters, and why the ecosystem around libsecp256k1 deserves far more attention. We also reflected on the tiny, careful pace of high-stakes Bitcoin cryptography work, the importance of maintainers like Jonas Nick, Pieter Wuille, Tim Ruffing, and Andrew Poelstra, and why understanding these mathematical foundations is the whole point of Magic Internet Math.Magic Internet Math: https://magicinternetmath.com/Magic Internet Math Podcast: https://podcasts.apple.com/us/podcast/magic-internet-math/id1868224151Phish Tour Archives: https://phish.com/tour-archives/BIP 340 — Schnorr Signatures for secp256k1: https://bips.dev/340/BIP 341 — Taproot: SegWit version 1 spending rules: https://bips.dev/341/BIP 342 — Validation of Taproot Scripts: https://bips.dev/342/BIP 327 — MuSig2 for BIP340-compatible Multi-Signatures: https://bips.dev/327/RFC 6979 — Deterministic Usage of DSA and ECDSA: https://www.rfc-editor.org/info/rfc6979/RFC 9591 — FROST Threshold Schnorr Signatures: https://www.rfc-editor.org/info/rfc9591/libsecp256k1: https://github.com/bitcoin-core/secp256k1rust-bitcoin: https://rust-bitcoin.org/AnchorWatch: https://www.anchorwatch.com/Blockstream: https://blockstream.com/Blockstream Research: https://research.blockstream.com/Jonas Nick: https://blog.blockstream.com/author/jonas/Andrew Poelstra and the Blockstream Research team: https://blog.blockstream.com/blockstream-research-the-focus/Pieter Wuille (sipa): https://github.com/sipaDaniel Shanks — Solved and Unsolved Problems in Number Theory: https://bookstore.ams.org/chel-297/SHAttered (the SHA-1 collision project): https://shattered.io/sha1-collision/
- Jun 24, 202658 min
My Daughter, the Mathematician (Fathers Day Special)
In this Father’s Day episode of Magic Internet Math, I welcomed my daughter Kayla to the mic for her first appearance on the show. We started with her recent work explaining group theory in just three minutes, then used that as a jumping-off point to talk about why ideas like groups, inverses, isomorphisms, elliptic curves, and point addition matter so much in Bitcoin and cryptography. We also got honest about the difference between intuition and rigor: I’ve spent a lot of time building conceptual bridges for this audience, while Kayla is bringing the mathematician’s instinct to stop, verify the ground beneath her feet, and ask what has actually been proved.From there, we traced the path that led her into math in the first place, from early logic puzzles and Waldorf-school mental math to an unusually rich high-school calculus experience, AP Calc BC, Penn State, linear algebra, analysis, and her current summer research in number theory. Along the way, we talked about why good teachers matter, why “math person” does not mean “numbers person,” how linear algebra suddenly makes everything click, and why Gauss keeps showing up everywhere once you start taking mathematics seriously.Magic Internet Math: https://magicinternetmath.com/BTC Prague: https://btcprague.com/Programming Bitcoin by Jimmy Song: https://jimmysong.org/books/programming-bitcoin/Understanding Cryptography by Christof Paar and Jan Pelzl: https://link.springer.com/book/10.1007/978-3-642-04101-3Benedict Gross (Harvard profile): https://people.math.harvard.edu/~gross/Harvard Math E-222 Abstract Algebra archive: https://legacy-www.math.harvard.edu/archive/122_fall_03/index.htmlLinear Algebra Done Right by Sheldon Axler: https://linear.axler.net/index.htmlPenn State Department of Mathematics: https://science.psu.edu/mathWaldorf Education (AWSNA): https://www.waldorfeducation.org/
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About Magic Internet Math
This podcast exists to liberate Bitcoin holders from second-class citizenship by teaching the mathematics that underlies their convictions. We operate on a simple premise: if you don't understand the math of Bitcoin, you cannot truly know what you know—you're dependent on others' authority, forever vulnerable to doubt and manipulation. Mathematics is the primary pathway to conviction in your own reasoning. Through accessible, conversational exploration of Bitcoin's mathematical foundations—treating math as the liberal art it was always meant to be—we equip listeners with genuine understanding rather than borrowed beliefs. We reject the deliberate demoralization campaign that convinced generations they're not capable of mathematical thinking.
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