Lower Bounds on the Time/Memory Tradeoff of Function Inversion
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We study time/memory tradeoffs of function inversion: an algorithm, i.e., an inverter, equipped with an s-bit advice on a randomly chosen function f : [n] -> [n] and using q oracle queries to f, tries to invert a randomly chosen output y of f, i.e., to find x∈ f^-1(y). Much progress was done regarding adaptive function inversion - the inverter is allowed to make adaptive oracle queries. Hellman [IEEE transactions on Information Theory 80] presented an adaptive inverter that inverts with high probability a random f. Fiat and Naor [SICOMP 00] proved that for any s, q with s^3q = n (ignoring low-order terms), an s-advice, q-query variant of Hellmans algorithm inverts a constant fraction of the image points of any function. Yao [STOC 90] proved a lower bound of sq ≥ n for this problem. Closing the gap between the above lower and upper bounds is a long-standing open question. Very little is known for the non-adaptive variant of the question. The only known upper bounds, i.e., inverters, are the trivial ones (with s+q = n), and the only lower bound is the above bound of Yao. In a recent work, Corrigan-Gibbs and Kogan [TCC 19] partially justified the difficulty of finding lower bounds on non-adaptive inverters, showing that a lower bound on the time/memory tradeoff of non-adaptive inverters implies a lower bound on low-depth Boolean circuits. Bounds that, for a strong enough choice of parameters, are notoriously hard to prove. We make progress on the above intriguing question, both for the adaptive and the non-adaptive case, proving the following lower bounds on restricted families of inverters.
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