github.com/openai/math · October 2026

OpenAI math

On 6 October 2026 OpenAI published a repository of manuscripts produced by an unreleased internal model, each answering a stated open problem. Its README counts 719 manuscripts in 372 families, classified by mathematical discipline, with about 42% of the top-line results checked in Lean. This page lists every family, with its manuscripts, page counts and Lean status, and marks the ones whose mathematics touches a cryptosystem. The marking is this site's reading, not OpenAI's.

Cryptographic relevance

Each family was screened by keyword over its title, description and manuscript abstracts, and every hit was read. Direct: a result about the cryptanalytic problem itself. Indirect: an algorithm or bound that cryptanalysis or a security proof of that family uses. Underlying: the pure mathematics behind the construction, with no bearing on its security. Tangential: a neighbouring field. Open a marked family for the reason. The remaining families are statistical physics, geometry, analysis and the like; lattice models in physics are not cryptographic lattices and are not counted.

All families

Click a column heading to sort, a family to list its manuscripts. Chips in one group combine; All clears the group. Manuscript links open the PDF in the repository.

Group by discipline

Method and limits

Source: The repository's README, CONTENTS.md and overview.tex as of 14 October 2026. The family numbers, titles, descriptions and disciplines are the catalogue's own. The repository's history records three manuscripts withdrawn and fourteen revised on 7 October 2026; the counts here are after those changes.
Pages: Counted from each manuscript PDF with pypdf. The repository notes that one writeup, the zero-free region for the Riemann zeta function, was edited by people.
Lean: A manuscript is marked when the repository's formalization index names it. 172 of the 719 manuscripts are named; the README's 42% counts top-line results, not manuscripts.
Relevance: Keywords screened: cryptography, cipher, encryption, RSA, Diffie–Hellman, discrete logarithm, elliptic curve, lattice, LWE, NTRU, knapsack, subset sum, factoring, hash function, pseudorandom, key exchange and distribution, Shor, quantum algorithm and circuit, code-based, isogeny, supersingular, multivariate, Gröbner, syndrome, finite field, primitive root, sieve, shuffle, permutation. Lattice hits in probability, mathematical physics, algebra and group theory are spin systems, percolation, congruence lattices or lattices in Lie groups, and were set aside. No family addresses Module-LWE, NTRU, the shortest-vector problem, code-based or hash-based schemes, or any NIST post-quantum standard.
Not assessed: Whether the manuscripts are correct. The Nature and Science coverage of the release records that mathematicians dispute several of them.