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Assume that some entity really holds the private key corresponding to the recommended/dubious constants of Dual_EC_DRBG.

According to this presentation, they would be able to reconstruct the internal state from only 32 bytes of random output and thereby predict all future output as well.

Would the backdoor also allow recovering previous output of the RNG, or does the resistance against state compromise extensions still hold in that situation?

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up vote 3 down vote accepted

State update takes $s$ to $x(sP)$, so you would have to compute d.log.s to run the state backwards.

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Is that computationally hard, even knowing all (potentially) hidden curve parameters? –  lxgr Sep 12 '13 at 18:03
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As far as is known, yes. If the curve is in some sense weak and d.log.s are easy to compute, then you could break the generator without faking the generation of the Q point. –  K.G. Sep 12 '13 at 19:35
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