编辑“︁Curve25519”︁（章节）

== 數學屬性 ==
所用的曲線是{{math|<var>y</var><sup>2</sup> {{=}} <var>x</var><sup>3</sup> + 486662<var>x</var><sup>2</sup> + <var>x</var>}}，蒙哥馬利曲線，在由素數{{math|2<sup>255</sup> − 19}}定義的素數場的二次擴展上，並且使用基點{{math|<var>x</var> {{=}} 9}}。這個基點的[[階_(群論)|階數]]是<math>(2^{252} + 27742317777372353535851937790883648493)</math><ref name=":2">{{cite conference |last=Bernstein |first=Daniel J. |authorlink=Daniel J. Bernstein |title=Curve25519: New Diffie-Hellman Speed Records |year=2006 |url=https://cr.yp.to/ecdh/curve25519-20060209.pdf |conference=Public Key Cryptography |series=Lecture Notes in Computer Science |volume=3958 |pages=207–228 |location=New York |publisher=Springer |isbn=978-3-540-33851-2 |doi=10.1007/11745853_14 |mr=2423191 |editor1-first=Moti |editor1-last=Yung |editor2-first=Yevgeniy |editor2-last=Dodis |editor3-first=Aggelos |editor3-last=Kiayias |editor4-first=Tal |display-editors=3 |editor4-last=Malkin |access-date=2019-10-22 |archive-date=2020-01-22 |archive-url=https://web.archive.org/web/20200122064214/http://cr.yp.to/ecdh/curve25519-20060209.pdf |dead-url=no }}</ref>.

該協議使用壓縮橢圓點（僅''X''座標），因此它允許在[[橢圓曲線迪菲-赫爾曼金鑰交換|ECDH]]中高效地使用[[橢圓曲線的純量乘法|Montgomery梯子]]，僅使用''XZ''座標。<ref>{{cite web|title=EFD / Genus-1 large-characteristic / XZ coordinates for Montgomery curves|last1=Lange|first1=Tanja|author1-link=Tanja Lange|url=https://www.hyperelliptic.org/EFD/g1p/auto-montgom-xz.html|website=EFD / Explicit-Formulas Database|accessdate=8 February 2016|archive-date=2017-03-09|archive-url=https://web.archive.org/web/20170309075702/http://hyperelliptic.org/EFD/g1p/auto-montgom-xz.html|dead-url=no}}</ref>

Curve25519的構造使其避免了許多潛在的實現缺陷。<ref>{{Cite web|title = SafeCurves: Introduction|url = https://safecurves.cr.yp.to|website = safecurves.cr.yp.to|accessdate = 2016-02-08|archive-date = 2017-09-05|archive-url = https://web.archive.org/web/20170905203243/http://safecurves.cr.yp.to/|dead-url = no}}</ref> 根據設計，它不受定時攻擊的影響，並且它接受任何32字節的字符串作為有效的公鑰，並且不需要驗證。

該曲線在雙有理幾何上等同於{{tsl|en|Ed25519}}簽名方案中使用的扭曲Edwards曲線。<ref>{{cite conference | last1=Bernstein | first1=Daniel J. | authorlink1=Daniel J. Bernstein | last2=Lange | first2=Tanja | title=Faster addition and doubling on elliptic curves | pages=29–50 | url=https://eprint.iacr.org/2007/286 | doi=10.1007/978-3-540-76900-2_3 | series=Lecture Notes in Computer Science | publisher=Springer | location=Berlin | conference=Advances in cryptology—ASIACRYPT | year=2007 | isbn=978-3-540-76899-9 | mr=2565722 | volume=4833 | editor1-first=Kaoru | editor1-last=Kurosawa | access-date=2018-12-10 | archive-date=2018-12-15 | archive-url=https://web.archive.org/web/20181215232612/https://eprint.iacr.org/2007/286 | dead-url=no }}</ref>