More Like this

1. Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity

   https://doi.org/10.4230/LIPIcs.ISAAC.2021.54  

   Allender, Eric; Gouwar, John; Hirahara, Shuichi; Robelle, Caleb (November 2021, Leibniz international proceedings in informatics)

   Ahn, Hee-Kap; Sadakane, Kunihiko (Ed.)

   A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤^{NC^0}_m reductions. In this paper, we improve this, to show that the complement of MKTP is hard for the (apparently larger) class NISZK_L under not only ≤^{NC^0}_m reductions but even under projections. Also, the complement of MKTP is hard for NISZK under ≤^{P/poly}_m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK_L is the non-interactive version of the class SZK_L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP). 

   more » « less
2. One-Way Functions and a Conditional Variant of MKTP

   https://doi.org/10.4230/LIPIcs.FSTTCS.2021.7  

   Allender, Eric; Cheraghchi, Mahdi; Myrisiotis, Dimitrios; Tirumala, Harsha; Volkovich, Ilya (November 2021, Leibniz international proceedings in informatics)

   Bojanczy, Mikolaj; Chekuri, Chandra (Ed.)

   One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs. 

   more » « less
3. The Non-Hardness of Approximating Circuit Size

   https://doi.org/10.1007/978-3-030-19955-5_2  

   Allender, Eric; Ilango, Rahul; Vafa, Neekon (January 2021, Theory of computing systems)

   The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under "local" reductions computable in TIME(n^0.49) . The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC^0) is closely related to many of the longstanding open questions in complexity theory. All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function. Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed, but this approach establishes only hardness of extremely good approximations of the form 1+o(1), and these improved hardness results are not yet known to hold for MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform AC^0 m-reductions, implying MKTP is not in AC^0[p] for any prime p. It was still open if similar circuit lower bounds hold for MCSP. One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond 1 + o(1) to omega(1), as KT-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed. More speci cally, we prove that PARITY does not reduce to the problem of computing superlinear approximations to KT-complexity or circuit size via AC^0-Turing reductions that make O(1) queries. This is signi cant, since approximating any set in P/poly AC^0-reduces to just one query of a much worse approximation of circuit size or KT-complexity. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in earlier work (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC^0 reductions. It may also be a step toward con rming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform AC0 m-reductions. 

   more » « less
4. On One-way Functions and Kolmogorov Complexity

   Liu, Yanyi; Pass, Rafael (January 2020, IEEE Symposium on Foundations of Computer Science (FOCS))

   null (Ed.)

   We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial t(n) ≥ (1 + ε)n, ε > 0, the following are equivalent: • One-way functions exists (which in turn is equivalent to the existence of secure private-key encryption schemes, digital signatures, pseudorandom generators, pseudorandom functions, commitment schemes, and more); • t-time bounded Kolmogorov Complexity, Kt, is mildly hard-on-average (i.e., there exists a polynomial p(n) > 0 such that no PPT algorithm can compute Kt, for more than a 1 − 1/p(n) fraction of n-bit strings). In doing so, we present the first natural, and well-studied, computational problem characterizing the feasibility of the central private-key primitives and protocols in Cryptography. 

   more » « less
5. Constraining the Clustering Transition for Colorings of Sparse Random Graphs

   Anastos, Michael; Frieze, Alan; Pegden, Wesley (March 2018, The Electronic journal of combinatorics)

   Let Ωq denote the set of proper [q]-colorings of the random graph Gn,m,m=dn/2 and let Hq be the graph with vertex set Ωq and an edge {σ,τ} where σ,τ are mappings [n]→[q] iff h(σ,τ)=1. Here h(σ,τ) is the Hamming distance |{v∈[n]:σ(v)≠τ(v)}|. We show that w.h.p. Hq contains a single giant component containing almost all colorings in Ωq if d is sufficiently large and q≥cdlogd for a constant c>3/2. 

   more » « less