椭圆曲线密码学导论

Hacker News 上怎么说

• throwaway81523 · 2021 年

  实现 ECC 的时候极容易踩坑。如果你真想自己手写实现，有本 Menezes 和 Vanstone 等人写的书（好像叫《Guide to Elliptic Curve Cryptography》），里面对各种操作都有非常详尽的指导。我以前折腾过一个 P256 的实现。不过这本书现在有点过时了，而且没记错的话（IIRC），里面压根没提 Curve25519。

  It is very easy to make mistakes in ECC implementation. If you really want to code it yourself, there's a book by Menezes, Vanstone et al ("Guide to Elliptic Curve Cryptography, I think) that gives very explicit instructions about how to do everything. I used to hack on a P256 implementation. But, it is a bit out of date by now, and IIRC it doesn't say anything about Curve25519.

• throwaway81523 · 2026 年

  用 secp 曲线搞东西很难做到不出错。这也是大家转向 curve25519 及类似方案的原因之一。Hankerson、Menezes 和 Vanstone 写的《Guide to Elliptic Curve Cryptography》基本上就是手把手教你如何严谨地搞定 secp* 的运算。要是能有一些形式化验证（formal verification）来进一步确保正确性就更好了……

  It's very hard to get stuff right with the secp curves. That's one of the reasons for the move to curve25519 and similar. The book "Guide to Elliptic Curve Cryptography" by Hankerson, Menezes, and Vanstone is mostly very careful step by step instruction of how to do secp* arithmetic properly. It would still be useful to have some formal verification to help the assurance…

• throwaway81523 · 2021 年

  ……而 RSA 的难点全在模幂运算上，像 GMP 这类库早就把模幂优化到极致了（所以直接调 GMP 就完事）。如果你出于某种原因非要自己实现 ECC，Hankerson、Menezes 和 Vanstone 写的《Guide to Elliptic Curve Cryptography》会非常有帮助。不过我们大多数人最好还是直接用现成的实现。

  …RSA, where the work is all in the modular exponentiation operation, which is already carefully optimized in libraries like GMP (so just call GMP and you are done). Hankerson, Menezes, and Vanstone's "Guide to Elliptic Curve Cryptography" is very helpful if you are trying to implement ECC for some reason. Most of us are probably better off using an existing implementation.

• ReidZB · 2014 年

  ……可以去看看 Goldreich 的《Foundations of Cryptography》那几卷。内容非常硬核、非常枯燥，但干货满满。如果要专门研究椭圆曲线密码学，Menezes 等人出版了《Guide to Elliptic Curve Cryptography》，虽然前面提到的那两份参考资料（Hoffstein 等人以及 Koblitz）里也讨论了椭圆曲线。不过，我还是挺欣赏这种专门聚焦 ECC 的做法，毕竟这个领域庞大到值得写几整本书。作为参考……

  …see Goldreich's Foundations of Cryptography volumes. They are very dense, very dry works, but packed with information. For a specific focus on elliptic curve cryptography, Menezes et al. have published Guide to Elliptic Curve Cryptography, though elliptic curves are also treated in the above two references (Hoffstein et al. and Koblitz). Still, I appreciate the focus on ECC, which is large enough to merit entire books. For a reference…

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