5 ms·

Not 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

by tgma 1y ago

Not 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.

cmrx64 1y ago
isn’t that how the article ends? maybe I don’t understand what you mean
- tgma 1y ago
  As I see it, the article describes something in that vein, but then immediately flips and reasserts the title, which suggests integer DH is a special case of ECDH, which does not seem a convincing characterization to me (it could be my misunderstanding; maybe someone could further clarify if it is conclusive).
graybanana 1y ago
While not completely explained in the article, the author correctly notes that the non-singular points on a singular cubic curve over a finite field form a group (under the same elliptic curve group law) that is isomorphic (in an easily computable way) to either the multiplicative or additive group of a finite field. You can find more details in Silverman's book on elliptic curves, or if you don't have access to that, see Section 24.7 in Lecture 24 of https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-2021/ https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-20.... One could nitpick by pointing out that singular cubics are not elliptic curves (which are, by definition, smooth projective curves). One could also nitpick that the real reason you don't want to use DH in the multiplicative group of a finite field is that there is a known subexponential time algorithm that breaks DH in that setting (forcing > 3000-bit key sizes) whereas for elliptic curves the only known attacks take exponential time (and 256-bit keys are fine). But I don't think either of these nitpicks contradicts the author's thesis.
- mti 1y ago
  While not mentioned by the author, there is an alternate reason why finite field DH can be seen as a special case of ECDH, kind of: namely, there exists special elliptic curves (actual, smooth projective curves) that do have an efficiently computable endomorphism to a suitable form of the multiplicative group, via pairings. This is in essence the MOV attack against the XTR cryptosystem. That doesn't fit neatly in OP's framework, though, because the map isn't a morphism of algebraic groups (it is efficient for other reasons), and it is cheating a little bit, because the inverse map isn't efficiently computable. Another point that the author glosses over a bit is that higher dimensional abelian varieties offer other instances of DH that are genuinely different from ECDH, and that are occasionally useful (mostly the case of Jacobians of hyperelliptic curves of genus 2). There isn't really a trick to make an arbitrary hyperelliptic curve DH/abelian variety DH instance a special case of ECDH: if anything, the relationship would be in the reverse direction.

by tgma 1y ago

cmrx64 1y ago

isn’t that how the article ends? maybe I don’t understand what you mean

tgma 1y ago
As I see it, the article describes something in that vein, but then immediately flips and reasserts the title, which suggests integer DH is a special case of ECDH, which does not seem a convincing characterization to me (it could be my misunderstanding; maybe someone could further clarify if it is conclusive).

graybanana 1y ago

While not completely explained in the article, the author correctly notes that the non-singular points on a singular cubic curve over a finite field form a group (under the same elliptic curve group law) that is isomorphic (in an easily computable way) to either the multiplicative or additive group of a finite field. You can find more details in Silverman's book on elliptic curves, or if you don't have access to that, see Section 24.7 in Lecture 24 of https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-2021/ https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-20.... One could nitpick by pointing out that singular cubics are not elliptic curves (which are, by definition, smooth projective curves). One could also nitpick that the real reason you don't want to use DH in the multiplicative group of a finite field is that there is a known subexponential time algorithm that breaks DH in that setting (forcing > 3000-bit key sizes) whereas for elliptic curves the only known attacks take exponential time (and 256-bit keys are fine). But I don't think either of these nitpicks contradicts the author's thesis.

mti 1y ago
While not mentioned by the author, there is an alternate reason why finite field DH can be seen as a special case of ECDH, kind of: namely, there exists special elliptic curves (actual, smooth projective curves) that do have an efficiently computable endomorphism to a suitable form of the multiplicative group, via pairings. This is in essence the MOV attack against the XTR cryptosystem. That doesn't fit neatly in OP's framework, though, because the map isn't a morphism of algebraic groups (it is efficient for other reasons), and it is cheating a little bit, because the inverse map isn't efficiently computable. Another point that the author glosses over a bit is that higher dimensional abelian varieties offer other instances of DH that are genuinely different from ECDH, and that are occasionally useful (mostly the case of Jacobians of hyperelliptic curves of genus 2). There isn't really a trick to make an arbitrary hyperelliptic curve DH/abelian variety DH instance a special case of ECDH: if anything, the relationship would be in the reverse direction.