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Elliptic Curve Arithmetic

A set of mathematical operations defined on a group of points on a 2D elliptic curve. Bitcoin protocol uses predefined curve secp256k1. Here’s the simplest possible explanation of the operations: you can add and subtract points and multiply them by an integer. Dividing by an integer is computationally infeasible (otherwise cryptographic signatures won’t work). The private key is a 256-bit integer and the public key is a product of a predefined point G (“generator”) by that integer: A = G * a. Associativity law allows implementing interesting cryptographic schemes like Diffie-Hellman key exchange (ECDH): two parties with private keys a and b may exchange their public keys A and B to compute a shared secret point C: C = A * b = B * a because (G * a) * b == (G * b) * a. Then this point C can be used as an AES encryption key to protect their communication channel.

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