Questions about the NaCL crypto library

Question

I was looking for libraries to implement an encryption system and was interested in using the NaCl: Networking and Cryptography library specifically the box function. Obviously,

Answer 1

For example, it mentions that to compute the key it uses the sender's private key and the receiver's public key to compute the secret key. But it doesn't explain how. Can anyone shed any light on it?

There is a nice, understandable video, but in german language: https://www.youtube.com/watch?v=aC05R9xqbgE. This video explains it for both the "discrete logarithm" and the "elliptic curve" solution.

In the "elliptic curve" scenario, the public key P is a point (with x and y coordinates) on the curve. P is calculated by multiplication of a Generator G (defined by the curve) with a secret key K.To calulate the secret key S, just do a multiplication of the peer's point P (peer's public key) with the own secret key K (a scalar number). Both peers make this operation:

S_Alice = P_Bob mult K_Alice = G mult K_Bob mult K_Alice

S_Bob = P_Alice mult K_Bob = G mult K_Alice mult K_Bob

As the curve is a commutative group, the order of the multiplications does not matter, so from the above calculations you can see that in the end both Bob and Alice calculated an identical key.