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I would like to map integer values to points on Ed25519, and then back again.

Is there a technique that takes advantage of the specific structure of Ed25519?

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You might want to take a look at Elligator 2, but it has some limitations. – CodesInChaos May 5 '14 at 20:58
@CodesInChaos: Thank you! This looks very promising. – Chris May 6 '14 at 10:20

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