Merkle–Damgård padded block concatenated outside the compression function hash?

Question

I learned that the output of hash function from Merkle Damgard is

H(x) = ZB+1 = h(ZB || L) = h(ZB - 1 || XB || L) where XB = block of padded x and L = XB+1

and it is proven that H(x) is collision resistant. However, what if I tweak the H(x) by extracting L from the compression function h(x) s.t

H(x) = h(ZB - 1 || XB) || L

I feel like this makes H(x) to be not collision resistant, but I don't know how to design an attack for this hash function. Can someone give me an idea (perhaps hint) to design such attack?

NOTE: Before anyone screams at me about anything, I want to say YES this is a practice question from a cryptography textbook. That is why I asked for a HINT not an answer. Thanks :)

Answer 1

Why don't you look at it the other way; suppose you could find a collision with your modified H(x) = h(ZB - 1 || XB) || L; could you use that collision to find a collision in the original H?

Answer 2

I am new to this too but would like to give it a try (as a test to my understanding). The compression function h(x) is collision resistance to start with, and length(L) << length(X), so the adversary can find a collision by brute force. However the same attack requires the adversary to brute force the whole message space X to find collision, hence the new H(x) is not as secure as the original Merkle Damgard scheme.