PPP-Completeness with Connections to Cryptography

Sotiraki, Katerina; Zampetakis, Manolis; Zirdelis, Giorgos

doi:10.1109/focs.2018.00023

Cited by 18 publications

(21 citation statements)

References 69 publications

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“…The class PPP is arguably less studied than other syntactically definable subclasses of TFNP, such as PLS (Polynomial Local Search) and PPA (Polynomial Parity Argument), and it is not known whether Pigeonhole Equal-Sums is complete in PPP. In fact, the first compete problem for the class was only identified relatively recently [SZZ18]. Our results for Pigeonhole Equal-Sums suggest that the problem is indeed simpler to solve than Equal-Sums.…”

Section: Related Workmentioning

confidence: 78%

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Allcock¹,

Hamoudi²,

Joux³

et al. 2021

Preprint

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Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time O(2 n/2 ) and O(2 n/3 ), respectively, based on the wellknown meet-in-the-middle technique. Here we introduce a novel dynamic programming data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value.Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to a same value modulo p. We use this data structure to give new O(2 n/2 ) and O(2 n/3 ) classical and quantum algorithms for Subset-Sum, not based on the meet-in-the-middle method. We then use the data structure in combination with variable time amplitude amplification and a quantum pair finding algorithm, extending quantum element distinctness and claw finding algorithms to the multiple solutions case, to give an O(2 0.504n ) quantum algorithm for Shifted-Sums, an improvement on the best known O(2 0.773n ) classical running time. We also study Pigeonhole Equal-Sums and Pigeonhole Modular Equal-Sums, variants of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For the former problem we give classical and quantum algorithms with running time O(2 n/2 ) and O(2 2n/5 ), respectively. For the more general modular problem we give a classical algorithm which also runs in time O(2 n/2 ).

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Section: Related Workmentioning

confidence: 78%

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Allcock¹,

Hamoudi²,

Joux³

et al. 2021

Preprint

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show abstract

“…Our starting point is the discrete logarithm problem in "general groups" suggested in [SZZ18]. Given the order s ∈ Z, s > 1, we denote by G = [s] = {0, .…”

Section: Our Resultsmentioning

confidence: 99%

“…A priori, the exact computational complexity of Index is unclear. Sotiraki et al [SZZ18] showed that it lies in the class PPP by giving a reduction to the PPP-complete problem Pigeon, where one is asked to find a preimage of the 0 n string or a non-trivial collision for a function from {0, 1} n to {0, 1} n computed by a Boolean circuit given as an input. No other upper or lower bound on Index was shown in [SZZ18].…”

Section: Our Resultsmentioning

confidence: 99%

“…PWPP: formalizes the weak pigeonhole principle and captures, e.g., the complexity of breaking collisionresistant hash functions and solving problems related to integer lattices [SZZ18].…”

Section: Introductionmentioning

confidence: 99%

“…Despite being a prominent total search problem, DLP was not extensively studied in the context of TFNP so far. Only recently, Sotiraki, Zampetakis, and Zirdelis [SZZ18] presented a total search problem motivated by DLP. They showed that it lies in the complexity class PPP and asked whether it is complete for the complexity class PPP.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Search Complexity of Discrete Logarithm

Hubáček¹,

Václavek²

2021

Preprint

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In this work, we study the discrete logarithm problem in the context of TFNP -the complexity class of search problems with a syntactically guaranteed existence of a solution for all instances. Our main results establish that suitable variants of the discrete logarithm problem are complete for the complexity class PPP, respectively PWPP, i.e., the subclasses of TFNP capturing total search problems with a solution guaranteed by the pigeonhole principle, respectively the weak pigeonhole principle. Besides answering an open problem from the recent work of Sotiraki, Zampetakis, and Zirdelis (FOCS'18), our completeness results for PPP and PWPP have implications for the recent line of work proving conditional lower bounds for problems in TFNP under cryptographic assumptions. In particular, they highlight that any attempt at basing average-case hardness in subclasses of TFNP (other than PWPP and PPP) on the average-case hardness of the discrete logarithm problem must exploit its structural properties beyond what is necessary for constructions of collision-resistant hash functions.Additionally, our reductions provide new structural insights into the class PWPP by establishing two new PWPP-complete problems. First, the problem Dove, a relaxation of the PPP-complete problem Pigeon. Dove is the first PWPP-complete problem not defined in terms of an explicitly shrinking function. Second, the problem Claw, a total search problem capturing the computational complexity of breaking claw-free permutations. In the context of TFNP, the PWPP-completeness of Claw matches the known intrinsic relationship between collision-resistant hash functions and claw-free permutations established in the cryptographic literature.

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The Classes PPA-k: Existence from Arguments Modulo k

Hollender

2019

Web and Internet Economics

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The complexity classes PPA-k, k ≥ 2, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with k thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-k in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.

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PPP-Completeness with Connections to Cryptography

Cited by 18 publications

References 69 publications

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

On Search Complexity of Discrete Logarithm

The Classes PPA-k: Existence from Arguments Modulo k

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