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1. Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

   https://doi.org/10.4230/LIPIcs.CCC.2023.17  

   van Melkebeek, Dieter; Mocelin Sdroievski, Nicollas (July 2023, Proceedings of the 38th Computational Complexity Conference (CCC 2023))

   Ta-Shma, Amnon (Ed.)

   A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS'21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a. We show that if every Arthur-Merlin protocol that runs in time n^c for c = O(log² a) can only compute f correctly on finitely many inputs, then AM is in NP. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM, as well as an unconditional mild derandomization result for AM. 

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2. Instance-Wise Hardness and Refutation versus Derandomization for Arthur-Merlin Protocols

   https://doi.org/10.1007/s00037-025-00279-2  

   van_Melkebeek, Dieter; Mocelin_Sdroievski, Nicollas (December 2025, computational complexity)

   Abstract A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settingsblackboxderandomization, i.e., derandomization through pseudorandom generators, has been shown equivalent to lower bounds for decision problems against circuits.Recently, Chen and Tell (FOCS'21) established nearequivalences in the BPP setting betweenwhiteboxderandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function$$f$$ $f$on the given instance. Follow-up works managed to obtain full equivalences in the BPP setting by exploiting acompressionproperty of classical pseudorandom generator constructions. In particular, Chen, Tell, and Williams (FOCS'23) showed that derandomization of BPP is equivalent toconstructivelower bounds against algorithms that go through a compression phase.In this paper, we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness-to-derandomization direction, consider a length-preserving function$$f$$ $f$computable by a nondeterministic algorithm that runs in time$$n^a$$ $n^a$. We show that if every Arthur-Merlin protocol that runs in time$$n^c$$ $n^c$for$$c=O(\log^2 a)$$ $c=O\left({log}^{2}a\right)$can only compute$$f$$ $f$correctly on finitely many inputs, then AM is in NP. We also obtain equivalences between constructive lower bounds against Arthur-Merlin protocols that go through a compression phase and derandomization of AM viatargetedgenerators. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs of proximity for nondeterministic computations. As a by-product of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). By-products in the average-case setting include the first uniform hardness vs. randomness trade-offs for AM, as well as an unconditional mild derandomization result for AM. 

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3. Leakage-Resilient Hardness vs Randomness

   https://doi.org/10.4230/LIPIcs.CCC.2023.32  

   Liu, Yanyi; Pass, Rafael (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ta-Shma, Amnon (Ed.)

   A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. The celebrated "hardness v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84), Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness assumptions under which e.g., prBPP = prP (so-called "high-end derandomization), or prBPP ⊆ prSUBEXP (so-called "low-end derandomization), and more generally, under which prBPP ⊆ prDTIME(𝒞) where 𝒞 is a "nice" class (closed under composition with a polynomial), but these hardness assumptions are not known to also be necessary for such derandomization. In this work, following the recent work by Chen and Tell (FOCS’21) that considers "almost-all-input" hardness of a function f (i.e., hardness of computing f on more than a finite number of inputs), we consider "almost-all-input" leakage-resilient hardness of a function f - that is, hardness of computing f(x) even given, say, √|x| bits of leakage of f(x). We show that leakage-resilient hardness characterizes derandomization of prBPP (i.e., gives a both necessary and sufficient condition for derandomization), both in the high-end and in the low-end setting. In more detail, we show that there exists a constant c such that for every function T, the following are equivalent: - prBPP ⊆ prDTIME(poly(T(poly(n)))); - Existence of a poly(T(poly(n)))-time computable function f :{0,1}ⁿ → {0,1}ⁿ that is almost-all-input leakage-resilient hard with respect to n^c-time probabilistic algorithms. As far as we know, this is the first assumption that characterizes derandomization in both the low-end and the high-end regime. Additionally, our characterization naturally extends also to derandomization of prMA, and also to average-case derandomization, by appropriately weakening the requirements on the function f. In particular, for the case of average-case (a.k.a. "effective") derandomization, we no longer require the function to be almost-all-input hard, but simply satisfy the more standard notion of average-case leakage-resilient hardness (w.r.t., every samplable distribution), whereas for derandomization of prMA, we instead consider leakage-resilience for relations. 

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4. On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds

   https://doi.org/10.1145/3593581  

   Chen, Lijie; Rothblum, Ron D.; Tell, Roei; Yogev, Eylon (April 2023, Journal of the ACM)

   The Exponential-Time Hypothesis ( \(\mathtt {ETH} \) ) is a strengthening of the \(\mathcal {P} \ne \mathcal {NP} \) conjecture, stating that \(3\text{-}\mathtt {SAT} \) on n variables cannot be solved in (uniform) time 2 ϵ · n , for some ϵ > 0. In recent years, analogous hypotheses that are “exponentially-strong” forms of other classical complexity conjectures (such as \(\mathcal {NP}\nsubseteq \mathcal {BPP} \) or \(co\mathcal {NP}\nsubseteq \mathcal {NP} \) ) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds . We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: (1) The Randomized Exponential-Time Hypothesis ( \(\mathsf {rETH} \) ) implies that \(\mathcal {BPP} \) can be simulated on “average-case” in deterministic (nearly-)polynomial-time (i.e., in time \(2^{\tilde{O}(\log (n))}=n^{\mathrm{loglog}(n)^{O(1)}} \) ). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length \(\tilde{O}(\log (n)) \) ); this significantly improves the state-of-the-art in uniform “hardness-to-randomness” results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. (2) The Non-Deterministic Exponential-Time Hypothesis ( \(\mathsf {NETH} \) ) implies that derandomization of \(\mathcal {BPP} \) is completely equivalent to circuit lower bounds against \(\mathcal {E} \) , and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of \(\mathsf {NETH} \) , and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if \(\mathtt {CircuitSAT} \) for circuits over n bits of size poly( n ) can be solved by probabilistic algorithms in time 2 n /polylog( n ) , then \(\mathcal {BPE} \) does not have circuits of quasilinear size. 

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5. Leakage Resilience, Targeted Pseudorandom Generators, and Mild Derandomization of Arthur-Merlin Protocols

   https://doi.org/10.4230/LIPIcs.FSTTCS.2023.29  

   van Melkebeek, Dieter; Mocelin Sdroievski, Nicollas (December 2023, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023))

   Bouyer, Patricia; Srinivasan, Srikanth (Ed.)

   Many derandomization results for probabilistic decision processes have been ported to the setting of Arthur-Merlin protocols. Whereas the ultimate goal in the first setting consists of efficient simulations on deterministic machines (BPP vs. P problem), in the second setting it is efficient simulations on nondeterministic machines (AM vs. NP problem). Two notable exceptions that have not yet been ported from the first to the second setting are the equivalence between whitebox derandomization and leakage resilience (Liu and Pass, 2023), and the equivalence between whitebox derandomization and targeted pseudorandom generators (Goldreich, 2011). We develop both equivalences for mild derandomizations of Arthur-Merlin protocols, i.e., simulations on Σ₂-machines. Our techniques also apply to natural simulation models that are intermediate between nondeterministic machines and Σ₂-machines. 

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