In Diffie Hellman key exchange using Elliptic Curve cryptography, how are the public keys calculated?

Question

While calculating the public key, $d * G = (X_a, Y_a)$ and similarly for $(X_b, Y_b)$, where $d$ is the private key. If calculating $d * G$ means adding $G$ to itself $d$ times, why can't the same be done to guess the private key '$d$' by an intruder - adding $G$ to itself until he gets the public key and hence find the private key? How is the calculation of public key efficient and guessing it computationally infeasible?

Answer 1

Current day elliptic curve cryptosystems use keys 200+ bits long. In this context it means that your group has size $2^{200} \approx 10^{60},$ so on average you will need to compute $G,2G,\ldots,dG$ until $d \geq 5\times 10^{59}$ until you hit the point $(X_a,Y_a)$ and discover the discrete logarithm $d.$