Also please share any cool math-related resources you know — I want to learn as much as I can!

@acciomath What kinds of math are you interested in? What have you studied so far?

@axiom I am interested in basically everything, to be honest. In school, I have only studied up to single-variable calculus (AP Calc BC, if you know what that is).

@acciomath I do know what that is. You should study abstract algebra and linear algebra. I recommend Artin's book, "Algebra". This field is the foundation of essentially all of modern mathematics. Once you have studied that you will have much more freedom to learn about other areas of math.

@axiom That looks quite interesting, thanks!

@axiom However, I do have a particular fascination with cryptography

@acciomath Me too! Unfortunately I don't have any reading recommendation to give you.

@axiom Ah well

@acciomath @axiom For , you have to also study Computer Science, both and part — alone is not enough, it doesn't account for the aspect of all the constructions.
Some basic and then would also be in order, but it's utility to cryptography is less obvious.

@amiloradovsky @acciomath What application of algtop to crypto are you thinking of? "HoTT for theorem provers" doesn't count.

@axiom @acciomath I'm thinking along the lines of approximating the discrete constructions by a continuous ones, asymptotics. And making some use of the "topological" theory of differential equations (non/existence of solutions has a topological nature). While algebraic topology serves as a tool for reasoning about the invariants.
I presume there is a more direct connection, but I cant give a concrete examples.

@amiloradovsky @acciomath I'm still not sure how exactly it pertains to cryptography. Elliptic curves?

@axiom @acciomath The motivational example is the "birthday paradox" — you can't compute the numbers in question without approximation of the discrete/combinatorial functions by a well-behaved continuous ones. Or, say, the binomial versus normal distribution.

@amiloradovsky @acciomath I'm still unconvinced, sorry. In my mind the birthday paradox is only a very small footnote in cryptography and it has nothing whatsoever to do with algebraic topology anyway.

@axiom @acciomath That's all I've got :)

BP was just an example, to illustrate the idea how "continuous" math may be useful to "discrete" one, assuming cryptography is an applied discrete math.
I'm sure there are more examples.

How AT is useful to the continuous math should be more or less obvious.
I don't know of a more direct connection between AT and DM, but I "expect" something to be there… — Of course that doesn't count as a proof.

P.S. And that's besides HoTT/UVF and ECC/ASC.

@acciomath @axiom It's not really a "paradox", but just a problem about calculating the probability of a collisions (in some random attributes).
For instance, it may be employed to show that 64-bit random identifiers are generally not enough, but 128-bit (UUIDs) are.

@axiom @acciomath e.g. Arto Salomaa's book, or Bruce Schneier's?

@gemlog @cosullivan That looks really neat! Thanks!

@acciomath @cosullivan The only other one that comes to mind off the top of my head is kahn academy, but that's fairly well known:

Also, you should know that many major universities present courses you can audit for nothing including lab notes.

search term is mooc or moocs

You can help yourself to better hits using your own specific search terms in the kind of maths you like.

class-central.com/subject/math

mooc-list.com/tags/math

@acciomath @cosullivan
You're welcome!
Remember: we're all in this together. Keep your stick on the ice ;-)

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