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Elliptic curve cryptography in Mathematica

I can find no resources for doing elliptic curve cryptography. I have used the finite field package, but I find it cumbersome and it does not seem to have any builtin methods for ECC. How can I get ...
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Square root of a value defined in a finite field?

I am trying to find the right way to compute the square root of a number defined in a finite field. For example, ...