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Like Thomas suggests in the comments, a Merkle tree would help here. As would a simple hash list. Calculate the hash tree/list over suitably sized blocks of the file and let the recipient know it. As data comes in it can be verified to match. With file size $n$, block size $b$ and SHA-256 as the hash, a hash list takes about $n/b \cdot 256$ bits. So with up ...

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