Discrete Logarithm and Minimum Circuit Size

Rudow, Michael

doi:10.1016/j.ipl.2017.07.005

Cited by 9 publications

(4 citation statements)

References 3 publications

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“…In particular, this observation yields an alternate way to show that integer factorization being in ZPP MKTP implies that the discrete log over prime fields is in ZPP MKTP [Rud17]. Recall that an instance of the discrete log problem consists of a triple x = (g, z, p), where g and z are integers, and p is a prime, and the goal is to find an integer y such that g y ≡ z mod p, or report that no such integer exists.…”

Section: Nonisomorphic Casementioning

confidence: 93%

“…Whereas NP-complete problems all reduce one to another, even under fairly simple reductions, less is known about the relative difficulty of presumed NP-intermediate problems. Regarding MCSP and MKTP, factoring integers and discrete log over prime fields are known to reduce to both under randomized reductions with zero-sided error [ABK + 06, Rud17]. Recently, Allender and Das [AD14] showed that GI and all of SZK (Statistical Zero Knowledge) reduce to both under randomized reductions with bounded error.…”

Section: Introductionmentioning

confidence: 99%

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Minimum Circuit Size, Graph Isomorphism, and Related Problems

Allender¹,

Grochow²,

Melkebeek³

et al. 2018

SIAM J. Comput.

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We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest. * This work subsumes and significantly strengthens the earlier arXiv submission 1511.08189 [AGM15]. See the end of Section 1 for more details.

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Section: Nonisomorphic Casementioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Minimum Circuit Size, Graph Isomorphism, and Related Problems

Allender¹,

Grochow²,

Melkebeek³

et al. 2018

SIAM J. Comput.

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show abstract

“…In various settings, the power of MCSP to distinguish between functions with circuits of size θ and those requiring size θ + 1 is not needed. Rather, in [3,10,4,31,28,23], the reduction succeeds assuming only that reliable answers are given to queries on instances of the form (T, θ), where either the truth table T requires circuits of size ≥ θ = |T | .9 or T can be computed by circuits of size ≤ |T | .01 . This is an appropriate time to call attention to one such reduction to approximations to MCSP.…”

Section: Introductionmentioning

confidence: 99%

“…All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function [3,4,10,19,24,31]. 1 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).…”

mentioning

confidence: 99%