The essential cryptographic properties of a hash function are that it is both oneway and collisionfree.

The most basic attack we might MOUNT on a hash function is to choose inputs to the hash function at random until EITHER we FIND some input that will give us the target OUTPUT value we are looking for (thereby contradicting the oneway property), or we find two inputs that produce the same output (thereby contradicting the collisionfree property).

Suppose the hash function PRODUCES an nbit long output. If we are trying to find some input which will produce some target output value y, then since each output is equally likely we expect to have to try 2n possible input values.

The essential cryptographic properties of a hash function are that it is both oneway and collisionfree.

The most basic attack we might mount on a hash function is to choose inputs to the hash function at random until either we find some input that will give us the target output value we are looking for (thereby contradicting the oneway property), or we find two inputs that produce the same output (thereby contradicting the collisionfree property).

Suppose the hash function produces an nbit long output. If we are trying to find some input which will produce some target output value y, then since each output is equally likely we expect to have to try 2n possible input values.