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I have seen Curve25519 and X25519, Curve448 and X448. I've seen a small note in this answer

(Historical note: Originally, X25519 was called Curve25519, but now Curve25519 just means the elliptic curve and X25519 means the cryptosystem.)

  • Is it a standard to say CurveABC is the Elliptic Curve and XABC is the cryptosystem?
  • Or, it is just a convention between the Elliptic Curve Cryptographers.
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  • $\begingroup$ Note: This is an example to show how to ask a question for this broad question $\endgroup$
    – kelalaka
    Oct 8, 2020 at 15:12

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It is an open standard by IETF.org

We can find the details in the mail archive of IETF, D. J. Bernstein's response;

It has become increasingly common for "Curve25519" to refer to an elliptic curve, while the original paper defined "Curve25519" as an X-coordinate DH system using that curve. "Ed25519" unambiguously refers to an Edwards-coordinate signature system using that curve.

Kenny and others in Toronto recommended changing terminology to clearly separate these three items. Let me suggest the following terminology:

  • "X25519" is the recommended Montgomery-X-coordinate DH function.
  • "Ed25519" is the recommended Edwards-coordinate signature system.
  • "Curve25519" is the underlying elliptic curve.

All relevant coordinate systems already have standard names in the literature, and I would suggest sticking to those names whenever it's necessary to discuss the coordinate systems per se: [...]

Bolds are mine.

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    $\begingroup$ Thanks to Squeamish Ossifrage for directing the link. $\endgroup$
    – kelalaka
    Oct 8, 2020 at 15:40
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    $\begingroup$ I thought before reading this answer, that X stood for the X as in DH Key eXchange, since X25519 is the cryptosystem developed using that particular curve Curve25519 !!! $\endgroup$
    – Vivekanand V
    Oct 8, 2020 at 19:01
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    $\begingroup$ @VivekanandV Schrödinger's cat. Until the author's intent is not known all are valid :). X X-coordinate, Ed Edwards-coordinate. consistent. But revealed. The cat is living :) $\endgroup$
    – kelalaka
    Oct 8, 2020 at 19:32
  • $\begingroup$ @kelalaka thanks for the answer, it is the kind of insight I was looking for in in the broad question you linked in your comment. $\endgroup$
    – ams
    Oct 9, 2020 at 13:45
  • $\begingroup$ @ams hope you got my point. A good question has your (single or very related) problem(s), what you see here and there (wiki, or the www) then your understanding/confusion,... correct tags and a good title ( not broad one, too), then post and wait. $\endgroup$
    – kelalaka
    Oct 9, 2020 at 17:08

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