When you read about ECDSA used in Bitcoin, you probably learn about adding points and doubling it. But it is also possible to subtract any two points. You can also multiply or divide any point by any number, but you cannot multiply or divide two public keys without knowing at least one matching private key.

Halving a point is possible, but it does not break ECDSA security. It is a bit different than typical integer division when you can just divide everything by two and quickly come to zero and break everything. It is not the case when using ECDSA and doing calculations over finite field.

To better understand it, you can do some calculations on smaller finite fields, for example using modulo 67. You can start from 1 and multiply it by two. You will get 1,2,4,8,16,32,64 and then suddenly you will get 128-67=61. Here you can see that dividing 61 by two will give you 64.