Geo Pereira

Researcher and Security Engineer. Enjoy cryptography, computer security, embedded systems and math.

[Aran Nayebi @aran_nayebi]

FWIW, I think this moves up my AI timelines a bit. I think the next milestone will be "Artificial *Grothendieck* Intelligence" (AGrI): defining new general mathematical structures to solve the hardest of open problems as special cases, like the Riemann Hypothesis or P vs. NP.

What impressed me about the OpenAI planar unit-distance result is not just that it solved a hard problem, but the particular way it seems to have done so.

For decades, the expert intuition was that the best constructions should look roughly grid-like. That intuition was *not* obviously silly; it was held by extremely serious mathematicians (of the likes of Erdos!). And yet the model found a new family of constructions that defeated it, based on literature in other areas of mathematics.

This feels like one of those cases where the "vague idea" is natural, but the solution lives in a huge space of possible design choices: which symmetries to preserve, which to break, which parameters to introduce, which ugly cases to try, which seemingly-unmotivated configurations to keep exploring.

Humans tend to navigate that space with aesthetic priors. We get embarrassed by ugly constructions. We avoid paths that do not look conceptually clean early on. The model seems much more willing to "fearlessly" plough through the design space until something works.

I imagine a lot of open problems in mathematics (and theoretical computer science!) may have a similar flavor, and would not be surprised if many of them start to fall soon.

But for the "very big" problems, maybe extensive search through constructions in the vast existing literature is not enough. Maybe what is needed for those problems is closer to Grothendieck-style mathematics: inventing the right ambient language in which the original problem becomes a special case of a more general structure.

That's what I mean by Artificial Grothendieck Intelligence (AGrI). Not merely AI that proves theorems, but AI that invents the new mathematical objects in which the theorems become *inevitable*.

And why stop at one AGrI? You could imagine simulating something like the IHES school: manager agents dividing a research program into subprograms, subagents pursuing lemmas for hours or days, other agents distilling the resulting abstractions, checking them, and communicating the useful pieces back upward.

One reason Grothendieck's IHES school was so successful is that its abstractions were relatively human-compressible. Once you adopted the relative perspective, the ideas could propagate through the community.

But maybe that constraint has also been a bottleneck. Maybe many longstanding open problems, like those in number theory which Grothendieck felt was the hardest nut to crack, have solutions that are checkable in principle, but whose motivating abstractions are not human-compressible.

In fact, I would wager that many, if not all, of these longstanding, open human conjectures live in PSPACE, but PSPACE is massive! I could imagine the AGrIs of the future might easily find non-human compressible abstractions that can be checked in PSPACE, but are infeasible for any human to check manually.

Thus, the next frontier may be mathematics that is machine-discovered, machine-compressible, and machine-checkable — beautiful, in a different way to the machines, but not necessarily in the human way.

I can't wait to see what open problems get solved next. What an exciting time to be alive.

[Peter Kutas @kutasp]

Very interesting paper!

[Geo Pereira @gmathcrypto]

@IIHDrBirmingham Interesting, thank you!

[Geo Pereira @gmathcrypto]

@IIHDrBirmingham @anneke_vdw @IIHUK @ehutton Are popular GLP-1 medications other than Exenatide (say semaglutide or tirzepatide) also believed to be promissing at reducing ICP?

[Geo Pereira @gmathcrypto]

@IMSR_UoB @GreenJournal @IIHDrBirmingham @DrMollan @O_Grech Wondering if a similar study with a more heterogenous population (such as inclusion of non-obese women and men) would give even more precise clues on the crucial IIH markers. Any thoughts?

[IACR @IACR_News]

#ePrint SIDH with masked torsion point images: TB Fouotsa https://t.co/4f0DsjFpmu

[Peter Kutas @kutasp]

Nice paper by Tomoki. 2 years ago with a had discussion about what happens when you only reveal the subgroup generated by torsion point images and concluded similarly that if you use prime powers you can recover them with pairings. Nice countermeasure to use many primes 1/2

[American Mathematical Society @amermathsoc]

Congrats to the '22 Fields Medalists, Maryna Viazovska, James Maynard, June Huh, and Hugo Duminil-Copin! Viazovska is the second woman to ever win a Fields Medal. All four recipients have made tremendous contributions to mathematics, and we look forward to their future work.

[Deirdre Connolly¹ @durumcrustulum]

Also, SIKE is not dead! 💫

[National Institute of Standards and Technology @NIST]

As we prepare for the possible assault of a future quantum computer, NIST has chosen the first group of tools for our post-quantum cryptographic standard:

🔒 CRYSTALS-Kyber
🔒 CRYSTALS-Dilithium
🔒 FALCON
🔒 Sphincs+

Learn more: https://t.co/9dXynKmZLO

[Geo Pereira @gmathcrypto]

@isogenies This lecture by Benedict Gross has been the best I was able to find so far:
https://t.co/aa6HLP4FkW

[Geo Pereira @gmathcrypto]

Hat to collaborators @PatrickLonga, @pbarreto, Javad Doliskani, David Jao, and many others.

[Geo Pereira @gmathcrypto]

Very happy to hear that my implementation works on compressed SIKE are going to be part of Debian distros!

[Christian Paquin @chpaquin]

This is great news! Bringing post-quantum cryptography closer to real-life systems

[Geo Pereira @gmathcrypto]

@realdemarchi @virvett It's curious that most of the time I catch myself trying to fix the code not because this could be the most efficient approach but because I usually commit myself on the endeavour of understanding how the old code was designed and how it should work.

[Diego F. Aranha 🕷️ @dfaranha]

O mundo dá voltas. TSE anunciou hoje que migrará assinaturas digitais para padrão ICP-Brasil, utilizando a curva elíptica E-521 (+EdDSA?). Conheço essa curva de algum lugar :) @pbarreto @geovandro @jeff_ricardini

https://t.co/Jpchxf2gxY

https://t.co/fl7hZJXG89

[Geo Pereira @gmathcrypto]

@susurrusus Congrats John, all the best in the new journey!

[PQCrypto Conf @PQCryptoConf]

Registration for PQCrypto21 (July 20 - 22) is open and free. https://t.co/vFFa9Kv51G Due to COVID-19 pandemic, PQCrypto2021 will be held in a virtual conference. Check out the list of accepted papers and join us. We strongly encourage your active attendance.

[Patrick Longa @PatrickLonga]

The $IKE challenges are announced by C. Costello at the NIST 3rd PQC conference. Two prizes of $5K and $50K to be awarded for the breaking of the mini-instances $IKEp182 and $IKEp217, resp. Happy cryptanalyzing! #SIKE 💪https://t.co/G2axtjvdYU