Re: [Cryptography] Popular curves (was: NSA and cryptanalysis)
----- Forwarded message from Jose Luis Gomez Pardo
On 3/09/13 18:13 PM, Phillip Hallam-Baker wrote: ....
Do we have an ECC curve that is (1) secure and (2) has a written description prior to 1 Sept 1993?
(Not answering your direct question.) Personally, I was happy to plan on using DJB's Curve25519. He's done the research and says it is good. Comments?
iang
Curve25519 was designed for elliptic Diffie-Hellman taking care of both security and efficiency aspects and seems very strong in both of them. Some comments on its usage for other purposes can be found in http://stackoverflow.com/questions/2515948/use-of-curve25519 This curve was originally written for x86 32-bit platforms and a 64-bit implementation can be found in the following links: https://code.google.com/p/curve25519-donna/ https://github.com/agl/curve25519-donna In addition to this curve and to the NIST curves, another source for elliptic curves that can be (according to the developers) freely used is: http://certivox.org/display/EXT/CertiVox+Standard+Curves where cuves over 384 and 512-bit prime fields can be found which are likely secure. Of course, in all these cases you have to trust the curve developers somewhat although you can also check these curves for possible vulnerabilities. Alternatively, one can build one's own curve and for this one needs to have access to an implementation of the SEA point counting algorithm. A little while ago I was writing a cryptography book that uses Maple to implement both cryptographic schemes and cryptanalytic algorithms and, for a while, I contemplated the idea of programming SEA in Maple. However, I soon discarded it because there are already some freely available excelent implementations in compiled languages and my Maple implementation would necessarily be much slower. Thus, for some computations in the examples in my book I ended using MIRACL, a C/C++ library with excellent support for ECC which was recently adquired by CertiVox and can be found in the following links: http://www.certivox.com/miracl/ https://github.com/CertiVox/MIRACL Using the SEA algorithm one can readily find elliptic curves of prime order (or with a very small cofactor) which, additionally, can be tested to ensure that they satisfy some important conditions such as not having small embedding degree (to prevent the MOV reduction attack) or not having trace one (anomalous curves) which makes them also vulnerable. Of course, if the curves are (pseudo)randomly generated, it is very unlikely that they suffer from these vulnerabilities. Methods for verifiably random generation of such curves can be found in: http://www.secg.org/download/aid-780/sec1-v2.pdf and some recommended elliptic curves generated using these methods (including curves over 384-bit and 521-bit prime fields) are available from: http://www.secg.org/download/aid-784/sec2-v2.pdf Of course, I don't know whether these curves are completely free from IP concerns but, according to the sources where these curves are published, this seems to be the case (I am far from expert in the IP subject but, as a mathematician, the idea of someone "owning" an elliptic curve in some sense, seems to me very strange). Jose Luis. _______________________________________________ The cryptography mailing list cryptography@metzdowd.com http://www.metzdowd.com/mailman/listinfo/cryptography ----- End forwarded message ----- -- Eugen* Leitl <a href="http://leitl.org">leitl</a> http://leitl.org ______________________________________________________________ ICBM: 48.07100, 11.36820 http://ativel.com http://postbiota.org AC894EC5: 38A5 5F46 A4FF 59B8 336B 47EE F46E 3489 AC89 4EC5
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