A taxonomy of complexity classes of functions

SUMMARYIn this paper, we investigate a relationship between the length-decreasing self-reducibility and the many-one-like reducibilities for partial multivalued functions. We show that if any parsimonious (manyone or metric many-one) complete function for NPMV (or NPMV g ) is length-decreasing self-reducible, then any function in NPMV (or NPMV g ) has a polynomial-time computable refinement. This result implies that there exists an NPMV (or NPMV g )-complete function which is not lengthdecreasing self-reducible unless P = NP.

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“…Since it is shown, in [12], that P = NP holds if and only if F C ⊆ c PF, we have the following corollary:…”

Section: Resultsmentioning

confidence: 78%

“…Then sat is ≤-complete for F C (see the proof of [12,Theorem 13] 2 and Proposition 2.6 that f 1 ≤ f 2 holds. Hence, we see that f 2 is ≤-complete for F C. By the assumption and the condition (ii) of Lemma 2, we have f 2 ∈ c PF.…”

Section: Resultsmentioning

confidence: 99%

On the Length-Decreasing Self-Reducibility and the Many-One-Like Reducibilities for Partial Multivalued Functions

Huh

Isobe

Koizumi

et al. 2013

IEICE Trans. Inf. & Syst.

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“…Note that in [7] they found NPMV-complete functions that directly compute discrete logarithms and integer factorization, where NPMV denotes the class of all partial, multivalued functions computed nondeterministically in polynomial time [15]. Therefore our results should be regarded as a set (or a language) counterpart of the results presented in [7].…”

Section: Introductionmentioning

confidence: 74%

NP-Complete Sets for Computing Discrete Logarithms and Integer Factorization

Hasegawa

Isobe

Shizuya

2013

IIS

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We explore an NP-complete set such that the problem of breaking a cryptographic scheme reduces to the complete set, where the reduction can be given in a straightforward form like the reduction from the graph isomorphism to the subgraph isomorphism. We construct such NP-complete sets Å DL and Å IF for the discrete logarithm problem modulo a prime and the integer factoring problem, respectively. We also show that the decision version of Diffie-Hellman problem reduces directly to Å DL with respect to the polynomial-time many-one reducibility. These are the first complete sets that have direct reductions from significant cryptographic primitives.

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“…We follow the definition presented in [20]. For a multivalued function ξ, a single-valued refinement of ξ is a single-valued function ξ 0 such that dom(ξ) = dom(ξ 0 ), and ξ(x) includes y whenever ξ 0 (x) = y, where dom denotes the domain of functions.…”

Section: Preliminariesmentioning

confidence: 99%

On the Complexity of Computing Discrete Logarithms over Algebraic Tori

Isobe

Koizumi

Nishigaki

et al. 2014

IEICE Trans. Inf. & Syst.

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SUMMARYThis paper studies the complexity of computing discrete logarithms over algebraic tori. We show that the order certified version of the discrete logarithm problem over general finite fields (OCDL, in symbols) reduces to the discrete logarithm problem over algebraic tori (TDL, in symbols) with respect to the polynomial-time Turing reducibility. This reduction means that if the prime factorization can be computed in polynomial time, then TDL is equivalent to the discrete logarithm problem over general finite fields with respect to the Turing reducibility. key words: algebraic tori, order certified discrete logarithm, Turing reduction

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A taxonomy of complexity classes of functions

Cited by 117 publications

References 15 publications

On the Length-Decreasing Self-Reducibility and the Many-One-Like Reducibilities for Partial Multivalued Functions

On the Length-Decreasing Self-Reducibility and the Many-One-Like Reducibilities for Partial Multivalued Functions

NP-Complete Sets for Computing Discrete Logarithms and Integer Factorization

On the Complexity of Computing Discrete Logarithms over Algebraic Tori

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