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I am reading about Signing vs. Public-Key encryption and a question pop up in mind that is it possible to transform any signature scheme to a public-key encryption scheme?

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No, we know no general way to build a public-key encryption scheme from a signature scheme.

And accordingly, we have some signature schemes with no known way to build a public-key encryption scheme from the same primitives. An example is the eXtended Merkle Signature Scheme, built from symmetric hash and PRF.

[It was not asked, but: I learned from Maeher's comment that we can build a signature scheme from any public key encryption scheme. And I can't tell from memory a public-key encryption scheme that does not have a related signature scheme built from similar primitives or underlying problem(s)].

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    $\begingroup$ You can build signature schemes from any public key encryption scheme. PKE implies one-way functions and we know how to construct signature schemes using nothing but one-way functions. $\endgroup$
    – Maeher
    Nov 10, 2020 at 15:15
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No, although some signature schemes are built from public key encryption schemes, others are not.

For example, the Lamport Signature scheme.

A few of the signature schemes submitted to the NIST post-quantum project are also hash based, like SPHINCS+.

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  • $\begingroup$ This answers the non-trivially-equivalent question: are all signature schemes built from a public-key encryption scheme? $\endgroup$
    – fgrieu
    Nov 10, 2020 at 14:00

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