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Is there any standard algorithm where I can create a hash H(x,k) where x being the data and k being the private key. And verify on other end with H(x,l) where l being the public key?

How it can be achieved?

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    $\begingroup$ Do "Digital Signatures" do what you need? $\endgroup$ – SEJPM Jul 4 at 15:14
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as SEJPM said you're looking for digital signature algorithms. If you want names and standards, there's ECDSA based on elliptic curves and DSA based on the RSA philosophy.

https://en.wikipedia.org/wiki/Digital_signature

Good luck!

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