While sending an EC point we mainly have two options, send it compressed or uncompressed. The uncompressed has prefix
04 and compressed has
03 which indicates which $y$ to choose. Actually, it is costly, first find the $y$ and $-y$ from the curve equation;
$$E(K): y^2=x^3+ax+b$$ a cube and a square root is required. If the base field is prime then select according to the last bit of $y$ or $-y$. Select the one with
02 is encoded and select the one with
03 is encoded.
For 32-byte coordinates, this can save 31-byte of message size ( one byte increased, we send 65 for uncompressed due to prefix). If we consider the cost, it increases the computation, and sending 64-bye instead of 33-byte is not really a huge issue.
Now, Bernstein in their slides mentions that
Never send uncompressed $(x; y)$. Design protocols to compress one coordinate down to $1$ bit, or $0$ bits! Drastically limits possibilities for attacker to choose points.
How does sending compressed points in ECC can limit the attacker's capabilities?