There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
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There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
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Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#2The logical click bait on this one. I could not resist thinking it’s like saying there is no circle other than a round circle, but I had no idea if it’s equivalent and fair comment given my limited cryptography knowledge (as in non existent). So I had to read it, and while I got little, by osmosis, I feel falling for the bait was worth it.
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#3What insanity led someone to use 0 for the terminal object?
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#4The logical click bait on this one. I could not resist thinking it’s like saying there is no circle other than a round circle, but I had no idea if it’s equivalent and fair comment given my limited cryptography knowledge (as in non existent). So I had to read it, and while I got little, by osmosis, I feel falling for the bait was worth it.
lol same - but i needed this comment to take the bait
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#5So, addition under a particular elliptic curve is isomorphic to multiplication under integer modulo groups...
But for some reason the author keeps referring to the underlying group as "Diffie-Hellman" instead of the process operating on some group.
The result is interesting, the journey to get there was very confusing.
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#6Not a cryptographer, but my expectation of the admitted-to-be-clickbait title would be to see a proof or some claim of that sort that characterizes traditional finite-field integer DH as a special case of ECDH. However, --and please correct me if I am wrong as I am not a cryptographer or a math wiz anymore-- my understanding is you could characterize both as instantiations of an abstract protocol on a cyclic group with some properties, so in that sense, they are siblings derived from a more general idea, not derivable from each other.
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#7It's a nice article, just the title needs a small tweak to add the word practical, as in "There is no practical DH..."
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#8What insanity led someone to use 0 for the terminal object?
It makes some sense in the category of rings, but basically only there.
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#9I think the article's point is that the notion of "groups" per se is too coarse for DH, because it abstracts away the data structure you use to represent the elements of the group. However, the notion of Algebraic Groups doesn't have this weakness, because it comes with an implied data structure: The data structure is tuples over finite fields, and the group operations are expressible as polynomials, which importantly are also algorithms. Done.
It then turns out that if you search for the algebraic groups that you can use for DH, you end up looking for candidates within the 2 possible kinds of algebraic groups: linear algebraic groups and abelian varieties. Within the abelian varieties, it turns out that elliptic curves are both the simplest kind, while also being perfect for DH. Within the linear algebraic groups, the possible cyclic groups are just the additive and multiplicative ones: The additive ones are hopelessly insecure, while the multiplicative ones are just the unit groups of a field, which are special cases of elliptic curves anyway. So elliptic curves really are all you need for DH.
Re: There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
#10What insanity led someone to use 0 for the terminal object?
It makes some sense in the category of rings, but basically only there.
It’s also the standard notation for zero objects, i.e., for terminal objects that are also initial. This entails all abelian/additive/preadditive categories, such as categories of modules, vector spaces, or abelian groups. (But there are also counterexamples, such as the categories of groups and of pointed sets.)
But I’d agree that it’s not standard notation to use 0 for a terminal object in an arbitrary category. I’d guess that most people use 1 instead, so that for example 1 × X ≅ X. (The post talks about group objects in the category of algebraic varieties (over some field), in which case 1 seems to be more appropriate than 0.)