[HN Gopher] The Animated Elliptic Curve
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       The Animated Elliptic Curve
        
       Author : syncsynchalt
       Score  : 114 points
       Date   : 2023-07-03 16:01 UTC (6 hours ago)
        
 (HTM) web link (curves.xargs.org)
 (TXT) w3m dump (curves.xargs.org)
        
       | slavapestov wrote:
       | Elliptic curves over finite fields have immediate applications to
       | cryptography, but elliptic curves over the complex numbers are
       | cool too. Just like the sine and cosine parametrize the circle,
       | elliptic functions parametrize elliptic curves.
       | 
       | Whereas trig functions are singly periodic with period 2pi,
       | elliptic functions are doubly periodic functions, meaning that
       | there are fixed complex numbers z and w such that f(x+z) = f(x+w)
       | = f(x). You can prove that for a given period lattice, the field
       | of such functions is generated by two such fundamental functions
       | ---the Weierstrass elliptic function and its derivative. The
       | addition law for elliptic curves over C can be derived from
       | something like the equivalent of the addition law for sine and
       | cosine.
       | 
       | Elliptic functions have applications to physics and are also
       | interesting in their own right. This all leads to the study of
       | fractional linear transformations, modular forms and even
       | Fermat's Last Theorem. It's all quite lovely.
        
       | 1aqp wrote:
       | An important point to note, that is not very obvious from the
       | text, is that it is (very, very) difficult to retrieve ka from
       | A=ka.P and kb from B=kb.P. For an attacker who has A and B, it's
       | close to impossible to recover P and ka.kb.P
        
         | bembo wrote:
         | Isn't P always the same? Or is it shared before the exchange?
         | 
         | Edit: just looked it up and the base point for curve25519 is
         | x=9 so no point in recovering it.
        
       | arcsincosin wrote:
       | This is wonderful. As part of a cryptography course in undergrad,
       | I spent a fair number of hours adding/multiplying points on small
       | curves by hand and eventually computationally, and seeing these
       | animations represents the "feel" of that process better than any
       | resource I have seen.
       | 
       | It seems that elliptic curves are used quite shrewdly in public
       | key exchanges, used as a sort of off-the-shelf arithmetic system
       | that renders some benefits (harder to solve the elliptic curve
       | discrete log problem) and drawbacks (more expensive to compute,
       | because of these point ops) compared to key exchanges relying on
       | the integer discrete log problem. There is generally much energy
       | spent on explaining the point operations but not much on
       | explaining this context: this is an alternate arithmetic inserted
       | into the Diffie Helman scheme that is secure and not so hard to
       | work with.
       | 
       | The finite fields in application (e.g. in X25519) are so large
       | compared to the toy examples that the technical details matter
       | more for understanding the performance of the algorithms than the
       | actual cryptographic method. Understanding (and convincing
       | yourself of) the core behavior of the key exchange itself is
       | perhaps best done with the number theory and group theory lens.
       | There was a discussion here on HN a couple days ago which largely
       | bashed the theoretical approach in favor of simply cargo culting
       | the tools, which was appalling. I hope they all enjoy this link!
       | 
       | Anyhow, the application of elliptic curves to cryptography is
       | clever and I would recommend everyone with an interest (and a
       | little background in algebra) read Koblitz's paper[1] and
       | Miller's earlier description [2] if they are looking for more
       | context for OP's wonderful presentation.
       | 
       | 1: Koblitz: Elliptic Curve Cryptosystems
       | https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-198...
       | 
       | 2: Miller: Use of Elliptic Curves in Cryptography
       | https://link.springer.com/chapter/10.1007/3-540-39799-X_31
        
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       (page generated 2023-07-03 23:00 UTC)