RT @isislovecruft@twitter.com

just moratorium new papers and findings until the 13th month of March, please i’m begging you

🐦🔗: twitter.com/isislovecruft/stat

RT @isislovecruft@twitter.com

this has been an absolutely wild week in cryptography, there’s a pandemic going on, please, can we just have a break, please

🐦🔗: twitter.com/isislovecruft/stat

RT @Para_Charlie@twitter.com

Mittelerde, Belagerung von Gondor, 3019 D.Z.

Bote: "Mein Herr Denethor, die dritte Welle des Angriffs beginnt, und so wie es aussieht hat der Feind sein Heer zur Hälfte mit Bergtrollen aufgefüllt!"

Denethor: "Ich verstehe... öffnet die Hälfte der Stadttore und die Blumenläden."

🐦🔗: twitter.com/Para_Charlie/statu

RT @mjg59@twitter.com

ANYWAY just a reminder that if you find a website that has Google Ads on it and hosts sexist, racist, homophobic or transphobic content (even if it's in the comments), you can report it at support.google.com/adsense/con

🐦🔗: twitter.com/mjg59/status/13673

RT @anked@twitter.com

unfassbar frech, wie die Bundesregierung versucht, das Recht auf freie Meinungsäußerung durch bei Messengern, eMaildiensten, Audiochats (Clubhouse), Videochats (Skype u Co) etc einzuschränken. Meinungsfreiheit umfasst auch das Recht auf Anonymität! 🤬🤬🤬 twitter.com/golem/status/13670

🐦🔗: twitter.com/anked/status/13670

RT @mjg59@twitter.com

Racism, sexism, homophobia and transphobia seek to remove power from people based on who they are. Free software seeks to empower people regardless of who they are. We don't achieve free software's goals while tolerating intolerance.

🐦🔗: twitter.com/mjg59/status/13673

RT @linzsports@twitter.com

Stop glamorizing “the grind” and start glamorizing getting 7+ hours of sleep at night, having healthy relationships, feeling safe at work, taking sick days, being paid a living wage, working hard when you’re at work, boundaries, and self-caring your way to success.

🐦🔗: twitter.com/linzsports/status/

RT @SchmiegSophie@twitter.com

Indeed, if you take the elliptic curve point addition formulas, apply them to the nodal curve, and look closely, you can reformulate them as the multiplication of two numbers of the underlying field.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

That point of infinity? The curve given by y²=x²(x-1), the nodal curve. You can generalize the notion of a Jacobian to this curve and guess what? The Jacobian is none other than the multiplicative group

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

Elliptic curves are governed by an invariant, the j-invariant. The interesting thing is, this j-invariant, when seen as a variable, forms itself a curve, something called the (coarse) moduli space. The problem is, by default that space is missing a point at infinity to be proper

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

Epilog: there is even more to this story: namely how the elliptic curves are friends with the multiplicative group, but not the additive one.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

That leaves: one dimensional abelian varieties. We can classify those. It turns out they are the Jacobians variety of a curve of genus 1, and are isomorphic to that curve as a variety.
A curve of genus 1, better known under their common name, elliptic curves.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

So let's look at the abelian varieties instead. Turns out, they have one very important invariant, their dimension. Higher dimensional abelian varieties have been proposed, but in the end just add a lot of complexity with no benefits.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

Ugh. The additive group has a very efficient dlog algorithm. You might have heard of it, it's called "division". The multiplicative group works, but allows for index calculus and the number field sieve for dlog, because of how it relates to numberfields.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

It kinda makes sense, since everything is a polynomial over a finite field anyways, so why not lean into it?

Those objects are part of the category of algebraic varieties. The plot thickens.

So we need the group objects in that group. They're called algebraic groups.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

We want a compact representation of the elements, and a fast algorithm to evaluate the group operation. This is really the only non canonical part of this story, but how about using tuples of numbers for the elements and represent the group operation as a polynomial in those?

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

The "other object" thing is what's called a category (groups form a category, sets form a category, rings form a category, everything is a category). Groups are the group objects of the category of sets, i.e. they are groups of which we only know that they're also sets.

🐦🔗: twitter.com/SchmiegSophie/stat

RT @SchmiegSophie@twitter.com

We made the very thing we want to use to hide information invisible to ourselves!

So what do we do? Well we clearly need more than just a group, we need a group, but it also has to be an other object. A group object.

🐦🔗: twitter.com/SchmiegSophie/stat

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