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There are public-key encryption schemes based on many different mathematical hardness assumptions, like the hardness of Decisional Diffie-Hellman problem, the hardness of the Factoring problem, the hardness of the RSA problem, etc.

But my question is, are there any public-key encryption schemes based on the hardness of the Discrete Log problem? Does the hardness of the Discrete Log problem even imply the existence of secure public-key encryption?

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  • $\begingroup$ Elgamal encryption. $\endgroup$ – kelalaka Nov 11 '20 at 19:38
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    $\begingroup$ @kelalaka El Gamal is based on the hardness of the Decisional Diffie-Hellman (DDH) problem, which is a stronger hardness assumption than the hardness of the Discrete Log (DLog) problem. $\endgroup$ – Keshav Srinivasan Nov 11 '20 at 19:40
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It is a major open research question whether such a scheme exists, and how to construct one (see, for example, Open Problem 9.10). Of course, we do have schemes like (hashed) ElGamal, which are based on the conjectured hardness of the (computational or decisional) Diffie-Hellman problem. But it is unknown whether either of these problems is equivalent to the discrete logarithm problem (except, of course, in groups where discrete log is easy).

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  • $\begingroup$ Can you give some kind of proof that it’s an open problem, like a citation? And is the open problem merely about finding such a construction, or is it about whether the hardness of DLog implies the existence of secure public-key encryption? $\endgroup$ – Keshav Srinivasan Nov 11 '20 at 23:31
  • $\begingroup$ See my updated answer. $\endgroup$ – Chris Peikert Nov 11 '20 at 23:40

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