A New View on Worst-Case to Average-Case Reductions for NP Problems
This work addresses a foundational issue in computational complexity theory, offering an incremental improvement in protocol design for reductions.
The paper tackles the problem of constructing non-adaptive worst-case to average-case reductions for NP problems by improving upon the Bogdanov-Trevisan result, replacing their hiding and heavy samples protocol with a histogram verification protocol to achieve a public-coin approach.
We study the result by Bogdanov and Trevisan (FOCS, 2003), who show that under reasonable assumptions, there is no non-adaptive worst-case to average-case reduction that bases the average-case hardness of an NP-problem on the worst-case complexity of an NP-complete problem. We replace the hiding and the heavy samples protocol in [BT03] by employing the histogram verification protocol of Haitner, Mahmoody and Xiao (CCC, 2010), which proves to be very useful in this context. Once the histogram is verified, our hiding protocol is directly public-coin, whereas the intuition behind the original protocol inherently relies on private coins.