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Elliptic Curves

math.mit.edu

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Re: Elliptic Curves

#31

Post by wybiral »

dom0 wrote: post #28
nhaehnle wrote: post #24

Earlier quoted context omitted.

On the question of whether it's intuitive or not: NP-complete problems tend to have fairly little structure. A problem like boolean satisfiability has basically no structure: all you have is a soup of clauses. The "tricky" problems that have solutions in P get their solutions from clever exploitation of structure. At least, that's how it becomes more intuitive to me.

The structure of a problem specifies it; an unstructured problem is less well-specified, and that makes it intuitively more difficult to approach. E.g. summation of a column of decimal numbers is a highly structured problem that's very easy to solve. Parsing the speech of yelling drunkards is not so structured, and much harder to solve.

And now my new crypto startup will be the first to market this with YDAAS (Yelling Drunkards As A Service).

Re: Elliptic Curves

#32

Post by [deleted] »

ice109 wrote: post #4
stablemap wrote: post #3

Earlier quoted context omitted.

It believe mathematicians will continue to favor boards. Sitting in the audience I greatly prefer this; however, such lectures have less chance of being posted at all. For this topic we are lucky to have Silverman's book [1], which everyone seems to like. [1] https://www.math.brown.edu/~jhs/AECHome.html

can you explain to me why an elliptic curve over C is a quotient manifold (like the torus on page 7 of the first set of slides)?

[deleted]

Re: Elliptic Curves

#33

Post by tptacek »

nincompope wrote: post #29
tptacek wrote: post #21

Earlier quoted context omitted.

I'm going to push back a little on "new" and "weaknesses still being found". The underlying theory of curves and their hardness has been pretty stable for awhile --- since well before 2000, I think. More progress has been made against conventional multiplicative group Diffie-Hellman than has against curves. The complicating factor isn't the curve problems themselves, but rather implementation details , some of them p…

> More progress has been made against conventional multiplicative group Diffie-Hellman than has against curves. How is "none" more than "none"?

It isn't, but that's got nothing to do with my argument.

Re: Elliptic Curves

#34

Post by number-sequence »

lukejduncan wrote: post #5

I've seen a few links to resources on elliptic curves on HN, but I don't understand their importance or why they're trending with folks here. Can someone give me some context?

Yes, elliptic curves are huge in cryptography, but they are also mathematically significant. They are the degree three nonsingular algebraic plane curves with at least one rational point. This makes them essentially "one step up" from the conic sections. The conic sections, of course, are well understood including the parametrization of their rational points. However, elliptic curves are much more subtle! We do not even have proven algorithms for determining the size of any elliptic curve's set of rational points, and in specific the algebraic rank of that set. The Birch and Swinnerton-Dyer conjecture is one of the millennium prize problems, and it relates elliptic curves' algebraic ranks to their analytic ranks. Elliptic curves are also very related to modular forms, and this connection is part of the theory that allowed Andrew Wiles to prove Fermat's Last Theorem. In the study of the rational solutions to integer polynomial equations, i.e. Diophantine Analysis, elliptic curves are one of the next stepping stones that must be more thoroughly understood before we can have a more complete understanding of polynomials in general. Their applications in cryptography and integer factorization are huge in applied mathematics, but they are also incredibly important subjects to fields like algebraic geometry, diophantine analysis, and of course, number theory.

Re: Elliptic Curves

#35

Post by aswanson »

number-sequence wrote: post #34
lukejduncan wrote: post #5

I've seen a few links to resources on elliptic curves on HN, but I don't understand their importance or why they're trending with folks here. Can someone give me some context?

Yes, elliptic curves are huge in cryptography, but they are also mathematically significant. They are the degree three nonsingular algebraic plane curves with at least one rational point. This makes them essentially "one step up" from the conic sections. The conic sections, of course, are well understood including the parametrization of their rational points. However, elliptic curves are much more subtle! We do not e…

Comments like this make me really, really wish I had majored in mathematics.

Re: Elliptic Curves

#36

Post by 801699 »

lukejduncan wrote: post #5

I've seen a few links to resources on elliptic curves on HN, but I don't understand their importance or why they're trending with folks here. Can someone give me some context?

"... they're trending..."

they date back to the 80's.

to predict the future, some look to the past for ideas.

others monitor the present e.g. github commits for popular projects - reject anything that has no recent activity.

the context is per packet encryption, the space and time needed to do it.

this is not the approach taken by tls where one compromised packet can compromise the entire "encrypted stream". nor is the approach taken by dnssec where instead of encrypting some third party gives their blessing to (signs) the data being communicated.

elliptic curve crypto is not new. but encrypting each and every packet on the internet separately is "new" (or at least "different" from current practice).

this is my understanding. i could be wrong.

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