@mrxtraf
In the private key group (mod n) we can add, negate, and invert - this allows for multiplication and division.
In the public key group (elliptic curve mod p of size n) we can add, negate, and double only. This leads to multiplication by a scalar.
One public key corresponds to exactly one private key, and vice versa. The proof is very easy. Let G is the generator of secp256k1. Let P=k*G is a point on the curve. Let also P=k'*G. Then (k-k')*G=O => (k-k') divides n. But n is prime, hence k=k' (mod n).
That is, you can’t divide the public key by 10?
Give me any public key from which you know the private key, I will divide it by 10. And I will give in return the result in the form of a public key. And you yourself divide the private key by 10, get the public key from it and compare.
The multiplication (and division, which is multiplication with the inverse) is by scalar only. You cannot multiply two public keys without solving ECDLP first. And if you somehow can, then all coins are belong to you.