I recognize several people in this picture!

Is it because we localize / do some sort of K-theory construction on the finite sets, and the good definitions of localisation involve \infty-categories?

and the spheres are given by iterating the reduced suspension, so I can see these are somewhat related.

But why do we go from a 1-categorical space (only one 'B') to an \infty-categorical space when taking the ring completion (iterating the loops)?

But it is not clear to me why the ring completion of this would be colim_k Ω^kS^k=Ω^∞S, aka the sphere spectrum...

I know that 'B' can be seen as a 'delooping', and we have an adjunction between looping Ω and suspension Σ (not the same Σ as the symmetric groups!)

Oh I have been trying to understand this "the integers are just a pale shadow of a sphere spectrum" for a while, so any intuition on this would be helpful!

From the mathoverflow post, I get the part about finite sets being more fundamental than N, and why we'd want to work on ∐_{n≥0} BΣn directly.

And you solve the failure rate problem, which was a big source of trouble in the previous implementation. And the new algorithm is simpler to implement!

So very exciting result! (Although I had been spoiled already by @jonathan.isogeny.club talk at Bordeaux in May :))

Congrats!

In summary: the strengh of cubical arithmetic are quite different than the strength of Miller's algo. And all pairing based families have been optimised for Miller's algo. My hope is that we'll find new interesting families optimised for cubical arithmetic instead.

In the biextension paper above, we show that cubical pairings are faster than Miller's algo in the case where the embedding degree is odd (so non denominator elimination) and the curve has D=1, so close to a Montgomery model (see Table 4 in the end).

in cubical arithmetic, while this greatly speeds up Miller's algo when the points are in the special subgroups G1, G2.

- When using quartic or sextic twists, we cannot have a Montgomery model on both the curve and its twist. Cubical arithmetic on other models is slower than on Montgomery curve.

Cubical arithmetic works very well on x-only coordinates on Montgomery curves, which makes it ideally suited to pairings applications of isogeny based cryptography.

For pairing based cryptography, there are several drawbacks:
- we don't know how to do denominator elimination [...]