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This recent new curve mentions something that's new to me: twist security.

http://safecurves.cr.yp.to/bada55.html

Are the existing C25519/Ed25519 curves secure against this form of attack?

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up vote 4 down vote accepted

Yes.

There are several different kinds of twist attacks. Curve25519 is safe against all the ones SafeCurves looks at. Specifically, the cost of a combined attack is $2^{124.3}$, which is considered safe.

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