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Depending on your trust assumptions about this server, you might be able to use cryptographic accumulators which provide constant-sized (non)membership proofs. However, as far as I know, no efficient strong accumulator scheme has been developed yet. Most accumulator constructions rely on the RSA assumption where the server knows the factorization $n = pq$ ...


Assuming that you know that $h_0$ is the root hash of a Merkle tree for the file, you can be sure that $h_1$ is a hash of a section of the file if you know that it's one of the hashes of the sections one level below the root and you know its sibling hashes, i.e. you have values $(h_1^1, \dots, h_1^{m_1})$ such that $\mathscr{H}(h_1^1, \dots, h_1^{m_1}) = ...

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