weierstrass

A general way to study these is the Weierstrass elliptic functions [2], which are inverses of this mapping.

This doesn't really explain why older cryptographic standards mandating Weierstrass curves allow cofactors greater than 1, admittedly.

The Montgomery ladder doesn't have certain kinds of weaknesses common in naive implementations of Weierstrass form addition.

It could be as well a Weierstrass function that it will not change anything to the law of your process.

This will sound quite interesting, but it's a really bad approximation of the Weierstrass function!

Condition (2) will be met for any realistic system---it just prevents you from choosing some pathological perturbation function like the Weierstrass function.

Which, like the Weierstrass function, illustrates that functions that look simple may be so weird that our standard intuition does not apply.

Also consider that Newton invented calculus using infinitesimals, which made sense to him spatially but didn't find rigorous footing until Weierstrass [2].

I can fit Montgomery ladder in 10 lines, and can't do that with a secure Weierstrass implementation.

A Weierstrass frequency distribution is very far from that.

Bernstein just warns against all the (admittedly many) pitfalls in implementations, and that the Weierstrass normal form makes it easier to run into problems than the normal form he proposes.

I believe that ECC (using Edwards or Montgomery forms; debatable when using general Weierstrass form) is easier to get right with respect to solid implementations than RSA is.

Case in point: "the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function."

If that were true, everybody and their little sister would understand the Bolzano–Weierstrass theorem by now.

How many jobs in the past hundred years required or would have been made easier if the people doing them understood the Bolzano–Weierstrass theorem?

Or in Weierstrass's words to Kovalevskaya, [...] it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician.