Deduplication can introduce high indexing overhead and many studies have focused on reducing the indexing overhead in deduplication. In this question, we study the indexing issues in deduplication. Suppose that we fix the chunk size as $4$KB$,$ use SHA$-256$ for chunk fingerprinting, and store the chunks in $64-$bit address space. Note that the data units are assumed to be in power of $2.$

C$)$ We now put the full fingerprint index on disk and deploy a Bloom filter to save disk I$/$O$.$ Suppose that the Bloom filter is configured with a false positive probability $0\mathrm{.}01.$ Also, consider a workload with M chunks before deduplication and the deduplication ratio is $4$:$1.$ Derive the expected number of queries issued to the fingerprint index to check if a chunk is duplicate. State any of your assumptions.