6 ms·

There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman

• cgio 1y ago
  The 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.
  • techNoob123 1y ago
    lol same - but i needed this comment to take the bait
  • bawolff 1y ago
    For historical context, TLS only really started to use the eliptic curve version of diffie-helman in the mid 2010s. Prior to that, plain diffie helman was more popular. (E.g. here is a thread from a decade ago complaining about lack of support https://security.stackexchange.com/questions/59459/how-widely-deployed-is-tls-with-cipher-ecdhe https://security.stackexchange.com/questions/59459/how-widel... ). ECC also used to have patents which restricted adoption back in the day. There also used to be a lot of conspiracy theories that NSA backdoored nist curved ( e.g. https://m.slashdot.org/story/191445 https://m.slashdot.org/story/191445 ). Probably fud, but it slowed down adoption of eliptic curves.
    • tgma 1y ago
      I believe the patent issue was by far the dominant friction for adoption in 2000s. On the NIST curve problem, well, maybe FUD, but evidently, they indeed backdoored the elliptic curve-based random number generator, so I would say some distrust is warranted. Irrespective of the curve issue, ed25519/x25519 is superior and has other nice properties like not catastrophically breaking if you can't generate a unique random "k" for ECDSA as PlayStation discovered the hard way[1]. [1]: https://youtu.be/DUGGJpn2_zY?t=2142 https://youtu.be/DUGGJpn2_zY?t=2142
      • xyzzyz 1y ago
        > like not catastrophically breaking if you can't generate a unique random "k" for ECDSA How does that work? I thought that this vulnerability was independent of the curve that is used. Ed25519 has a bunch of other nice properties related to resistance to programming errors, but I never heard of this one.
        • cesarb 1y ago
          > How does that work? I thought that this vulnerability was independent of the curve that is used. The trick is that this value in Ed25519 is a deterministic hash of the private key and the message, which means that for the same message, the same value will always be chosen, and since the private key is part of the hash, for different messages the value is basically random unless you already know the private key. Yes, the same trick can be used with any curve, and it will avoid the vulnerability; but for Ed25519, it's always been part of the specification so everyone does it that way, while for ECDSA it's a later addition (RFC 6979) and many implementations still do it the old way.
    • phkahler 1y ago
      >> Probably fud, but it slowed down adoption of eliptic curves. Not FUD at all. NIST revoked their recommendation to use those curves because of it. The facts are that a "backdoor" exists whether anyone knows what it is or not, and NSA could not explain how those particular numbers were chosen so out of caution we must assume they know the backdoor and not use those curves.
      • tgma 1y ago
        I believe you are mixing up NIST ECC curves with Dual_EC_DRBG[1] which is a random number generator based on elliptic curves which is widely considered to be backdoored and they revoked its recommendation (there are much better ways to construct an RNG so it was stupid to use in practice for reasons other than backdoor too.) The P-series curves are still a NIST recommendation and widely deployed in TLS. The B-series fell out of fashion due to practical reasons. [1]: https://en.wikipedia.org/wiki/Dual_EC_DRBG https://en.wikipedia.org/wiki/Dual_EC_DRBG
• shiandow 1y ago
  What insanity led someone to use 0 for the terminal object?
  • pwdisswordfishz 1y ago
    It makes some sense in the category of rings, but basically only there.
    • cionx 1y ago
      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.)
      • mti 1y ago
        The relevant category is really abelian group schemes (because DH necessarily works in a cyclic group), so 0 is quite reasonable.
• eximius 1y ago
  So, 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.
• 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.
• glitchc 1y ago
  It's a nice article, just the title needs a small tweak to add the word practical, as in "There is no practical DH..."
• ogogmad 1y ago
  I 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.
• deleted 1y ago
  [deleted]
• vitalnodo 1y ago
  Just wondering — has anyone come across a course or resource that explores how cryptographic systems are built up from smaller building blocks? Like, using something like SageMath for algebraic structures, a prover like Lean to verify properties — to get a feel for how things actually fit together. There's something cool about trying to reimplement known standards just to understand them better (with the usual "don’t roll your own crypto" warning, of course). By the way, it would be nice to have a kind of sandbox for the more formal path to explore how to build my own SP-network, prove that the Decisional Diffie-Hellman assumption holds or something. I've seen cryptopals and cryptohack.
  • moi2388 1y ago
    That’d indeed be nice to do.
  • cmrx64 1y ago
    the types of arguments and proofs used in cryptography are notoriously difficult to mechanize, a little bit of lean isn’t going to go get anywhere close to stating a security property. eg CryptHOL needs a ton of infrastructure, and in still very limited: https://eprint.iacr.org/2017/753.pdf https://eprint.iacr.org/2017/753.pdf
    • nioj 1y ago
      There is also SSProve which has similar goal to CryptHOL: https://dl.acm.org/doi/full/10.1145/3594735 https://dl.acm.org/doi/full/10.1145/3594735 https://github.com/SSProve/ssprove https://github.com/SSProve/ssprove
  • oleganza 1y ago
    When I learned crypto 5-10 years ago, it turned out that a lot of "building blocks" are mostly hacks. Looking back from 2020s we see that some of the standards that we use for the last 20-30 years can in principle be thrown out of the window (they can't for compatibility reasons, though) and replaced with much cleaner and more universal replacements. If we do not talk about modern exotic stuff (post-quantum crypto, zkSNARKS, homomorphic encryption), the 99% of everyday cryptography is based on two building blocks: 1. Symmetric crypto for ciphers and hash functions. 2. Algebraic group with "hard discrete log problem" for key exchange, signatures, asymmetric encryption and simple zero-knowledge proofs. Historically, these two categories are filled with a zoo of protocols. E.g. AES is a block cipher, but SHA(1,2) is a hash function. Today, you can roughly achieve everything of the above with two universal building blocks: - Keccak for all of symmetric crypto: it is suited both for encryption, hashing, duplex transcripts for ZK protocols etc. - Ristretto255 group based on Curve 25519: for diffie-hellman, signatures, key derivation, threshold schemes, encryption and more. The problem is that none of the described features is implemented in a turnkey standard, and we are still stuck using older crypto. Heck, even Git is using SHA-1 still. Then, after you have your building blocks, there are more hairy stuff such as application-specific protocols: TLS, Signal, PAKE/OPAQUE, proprietary hardware security schemes for full disk encryption and access controls etc.
• jhanschoo 1y ago
  The author seems to be a busy person, but I would appreciate it if they would consider modifying their blog publishing software so that the math and definitions could be rendered in text and KaTeX instead of raster images.
  • thaumasiotes 1y ago
    What would that help with? I know KaTeX supposedly lets you copy-and-paste the text, but that won't get you text that is fit for any purpose.
    • willvarfar 1y ago
      (hopefully modern screen reading software is clever enough to OCR and summarise images as it does text. And if not, then that seems to be something that is now within reach of modern AI and just needs application.)
      • thaumasiotes 1y ago
        It wouldn't really matter; the image includes a lot of structure that isn't present in the text that KaTeX provides. A text-only screen reader will never produce anything but gibberish if you feed it KaTeX output. KaTeX will also style the text so that it visually resembles the image, stopping this problem from occurring for non-blind readers. You could have a screen reader that parsed all of the layout, but it seems like you might just as well OCR the image at that point.
      • James_K 1y ago
        The image does have LaTeX source as it's alt text, so should be at least somewhat understandable for screen readers.
    • deleted 1y ago
      [deleted]
    • red_trumpet 1y ago
      Better typography helps with reading. The same reason someone would complain if this was typed in Comic Sans. Also the images don't scale well, so if you look at this with a high resolution display, either the images are too small or not sharp.
• rco8786 1y ago
  Random fun fact. Whit Diffie is cousin to the late country music singer Joe Diffie. Their Wiki pages make no mention of this, though they do mention the other’s page in case users end up on the wrong one.
• tromp 1y ago
  The plot at the bottom is of Y^2 = X^3 + X^2, not Y^2 = X^3 - X^2 as stated.
• red_trumpet 1y ago
  I don't get this article. They make a big fuss about looking for the right category, because apparently the category of groups cannot explain the difficulty of breaking Diffie-Hellman, since there is an isomorphism <g> -> Z/nZ. But the same isomorphism exists in the category of algebraic varieties! So was all in vain? To me this sounds a bit like an a-posteriori justification of why people use elliptic curves to do cryptography. This feels weird to me, as elliptic curves are a well-studied subject in algebraic geometry, whose history reaches back over 150 years to Clebsch. So when Victor Miller and Neal Koblitz proposed to do cryptography with them, they already knew very well about elliptic curves (both have a background in algebraic geometry of finite fields).
  • graybanana 1y ago
    I think the author's point is that while you might a priori think there are lots of groups out there that might be good candidates for DH, it turns out that elliptic curves are a strictly better choice than every alternative, and the reasons for this were definitely no fully known at the time Miller and Koblitz proposed ECC. There was a period during which there was lots of interest in using abelian varieties of higher dimension (arising as Jacobians of curves of higher genus), with dimensions g=3 and g=4 being particularly attractive because then you could work over a very computationally friendly base field like Fp with p = 2^61-1. But it turns out the discrete logarithm problem (and therefore DH) is strictly easier in these settings (one can exploit Weil restrictions to get an algorithm that is still exponential-time but strictly better than O(p^(g/2)). But this wasn't known until the 2000's. That leaves g=1 and g=2 as the best choices, and the group law is faster and simpler for g=1, and as far as I know nobody is really working on the g=2 case anymore (but there was a lot of activity in this area 10-20 years ago).
• cryptothrow101 1y ago
  I find it very hard to follow the overall reasoning, in particular due to the inconsistency around whether one should only consider algebraic structures up to isomorphism or not. For any n, there is only one finite cylic group of size n, up to isomorphism (namely Z/nZ). As the author mentions, there are concrete choices of finite cyclic groups (like Z/nZ) that make Diffie-Hellman insecure. Consequently, the security comes entirely from the choice of the _representation_ of the group elements (since the algebraic structure is the same, i.e., it is just a relabeled Z/nZ). In a similar spirit, people sometimes argue that Diffie-Hellman in some group G must be equally secure as in another group H, because the groups are isomorphic. This is unsound. In order to make such an argument sound, one needs to prove (or at least mention in case it is trivial) that one can _efficiently_ compute the isomorphism and its inverse.
• cdelsolar 1y ago
  wtf did i just read