OpenAI and Anthropic each say an AI model has refuted the exponential time hypothesis, one of the load-bearing assumptions of computer science, and yet neither lab's paper mentions cryptography a single time.

That gap matters to anyone holding bitcoin. Earlier this month, an Ethereum Foundation researcher warned that AI-driven math could break the signatures that guard crypto wallets, and a result this big is exactly what that camp was bracing for. We read both papers to see what they actually claim.

The short version: the papers do not touch the math behind Bitcoin keys. Both labs say their algorithms are theoretical, the speedup only appears on problems with millions of variables, and the hypothesis they say fell is about a different puzzle than the one that protects your coins.

What happened with the exponential time hypothesis?

OpenAI announced on Monday, October 26, that an internal model had proved the hypothesis false. The company says the AI found its disproof on October 16 and that the result was formalized in Lean (a programming language that lets a computer check every step of a proof) the next day. The 67-page paper went up on GitHub that afternoon.

Hours later, Anthropic posted its own 49-page paper, titled "A 2^o(n) Time Algorithm for 3-SAT" and credited to Claude. Anthropic says the core algorithm was found on October 16 "by an internal Claude research model with 14M output tokens and no human input," and that it reached the result "independently and concurrently" with OpenAI. If you have followed Anthropic's plans to go public, this is a very different kind of filing.

The hypothesis, known as ETH, was proposed by computer scientists Russell Impagliazzo and Ramamohan Paturi of UC San Diego. It says 3-SAT, a classic logic puzzle where you try to set a list of true/false switches so every rule is satisfied, needs exponential time. In plain terms, each extra switch should multiply the work.

Both papers describe algorithms that grow more slowly than that. The best previous 3-SAT algorithms, as Anthropic's paper lists them, ran in roughly 1.328^n steps (deterministic) and 1.308^n (randomized), where n is the number of switches.

Did AI just break Bitcoin's cryptography?

No, and the labs' own fine print says why. Bitcoin signs transactions with ECDSA and, since Taproot, with Schnorr signatures over the secp256k1 curve. The Schnorr spec says its security rests on the elliptic curve discrete logarithm problem (the difficulty of working backward from a public key to the private key) and targets a 128-bit security level.

That is a separate problem from 3-SAT. You could translate a key search into a 3-SAT puzzle, but neither paper attempts it, and the translation does not stay small.

The standard method, the Tseytin transformation, adds a new variable for every logic gate in the circuit being converted. A circuit that checks a signature has a lot of gates, so the resulting puzzle would have vastly more variables than the 256 bits in a private key.

That is where Anthropic's caveats bite. Its paper calls the result "purely asymptotic" (true only as problems get enormously large) and says the approach seems unlikely to beat brute force "unless n > 10^7," meaning more than 10 million variables. It adds that the result "says nothing about solving formulas in practice."

OpenAI is similarly cautious. Its announcement says finding faster algorithms "and turning theoretical advances into practical improvements, will require significant further research." And the Lean-checked version of OpenAI's proof covers a slightly weaker bound than the best one in the paper.

Did anyone check for crypto breaks?

Yes, at least informally. MIT's Ryan Williams, who is helping run the outside review, told TechCrunch that one of the first things he asked OpenAI science lead Dan Roberts was whether the result affects any crypto systems. OpenAI was "reasonably confident" it did not imply any breaks, Williams said.

Williams also told the outlet that better algorithms of this kind could, in principle, mean that cryptographic systems relying on certain key sizes would have to be treated as broken. So the question is fair to ask. It just has a reassuring answer for now.

The answer comes with an asterisk, because the proofs themselves are still being checked. Impagliazzo, Paturi and Williams wrote in an open letter, as quoted by Papex, that "our involvement should not be taken as certification of the claim."

Anthropic says its Lean proof has been re-checked by a second, independently written kernel (the small core program that verifies Lean proofs). That makes errors less likely, but it doesn't replace human review of what the theorem actually says.

Why does the bunker mode debate care about this?

In early October, Ethereum Foundation researcher Justin Drake wrote that it was "reasonable to brace for the possibility that ECDSA breaks before Qday, in the worst case in months not years," according to Cointelegraph Magazine. He urged holders to move coins to fresh addresses whose public keys have never appeared onchain.

Coinbase's head of cryptography, Yehuda Lindell, said there was "no evidence whatsoever" that elliptic curve assumptions had been weakened. Johns Hopkins cryptographer Matthew Green took the gloomier side: "I think we might lose public key cryptography."

The ETH papers give each side something. Drake's camp argued that widely held hardness beliefs are falling to AI, and ETH was about as widely held as they come. Williams told TechCrunch the result "calls into question a lot of the intuitions" researchers had about hard problems.

Lindell's camp argued that proving theorems is not the same as breaking a specific cryptosystem. On that point, the papers back him up. Neither one contains an attack on elliptic curves, hash functions or any deployed scheme. (Some of Crypto Twitter, as Decrypt noted, first had to be told that this ETH is not Ethereum.)

What should bitcoin holders do now?

Nothing new, based on what was published. Trezor Bitcoin analyst Lucien Bourdon told Cointelegraph during the bunker mode fight that "there's nothing to do right now," warning that rushed migrations cause mistakes and give scammers an opening. Nothing in these papers changes that.

The usual habits still apply. Avoid reusing addresses, keep keys on a device you trust (readers have voted on their favorite hardware wallets), and be suspicious of anyone using this news to rush you into moving coins.

The bigger signal is about pace. AI labs are now competing on price and on research results at the same time, and two of them say they reached the same theoretical breakthrough on the same day. If a real cryptanalysis result ever lands, it may arrive the same way.

Exponential time hypothesis FAQ

What is the exponential time hypothesis?

It is a 1999 conjecture by Impagliazzo and Paturi that 3-SAT cannot be solved in subexponential time. OpenAI says hundreds of papers have used it to argue that other problems are hard too.

Who refuted the exponential time hypothesis?

OpenAI and Anthropic both released AI-generated proofs on October 26, 2026, with Lean formalizations. An outside group led by Impagliazzo, Paturi and Williams is reviewing OpenAI's claim and has said its involvement is not certification.

Does this mean P equals NP?

No. Anthropic's paper says its result "does not bear on the P vs NP question directly." A subexponential algorithm is faster than exponential but still far from the polynomial time P = NP would require.

Can AI now crack bitcoin private keys?

Not based on these papers. Neither mentions cryptography, and Bitcoin's signatures rely on the elliptic curve discrete logarithm problem, not 3-SAT. OpenAI told Williams it was "reasonably confident" no crypto breaks follow.