Polynomial Reduction Research Articles

On the Power of Secure Two-Party Computation

Ishai, Kushilevitz, Ostrovsky and Sahai (STOC 2007; SIAM J Comput 39(3):1121–1152, 2009) introduced the powerful “MPC-in-the-head” technique that provided a general transformation of information-theoretic MPC protocols secure against passive adversaries to a ZK proof in a “black-box” way. In this work, we extend this technique and provide a generic transformation of any semi-honest secure two-party computation (2PC) protocol (with mild adaptive security guarantees) in the so-called oblivious-transfer hybrid model to an adaptive ZK proof for any $$\textsf {NP}$$ language, in a “black-box” way assuming only one-way functions. Our basic construction based on Goldreich–Micali–Wigderson’s 2PC protocol yields an adaptive ZK proof with communication complexity proportional to quadratic in the size of the circuit implementing the $$\textsf {NP}$$ relation. Previously such proofs relied on an expensive Karp reduction of the $$\textsf {NP}$$ language to Graph Hamiltonicity [Lindell and Zarosim (TCC 2009; J Cryptol 24(4):761–799, 2011)]. As an application of our techniques, we show how to obtain a ZK proof with an “input-delayed” property for any $$\textsf {NP}$$ language without relying on expensive Karp reductions that is black box in the underlying one-way function. Namely, the input-delayed property allows the honest prover’s algorithm to receive the actual statement to be proved only in the final round. We further generalize this to obtain a “commit-and-prove” protocol with the same property where the prover commits to a witness w in the second message and proves a statement x regarding the witness w in zero-knowledge where the statement is determined only in the last round. This improves a previous construction of Lapidot and Shamir (Crypto 1990) that was designed specifically for the Graph Hamiltonicity problem and relied on the underlying primitives in a non-black-box way. Additionally, we provide a general transformation to construct a randomized encoding of a function f from any 2PC protocol that securely computes a related functionality (in a black-box way) from one-way functions. We show that if the 2PC protocol has mild adaptive security guarantees (which are satisfied by both the Yao’s and GMW’s protocol), then the resulting randomized encoding can be decomposed to an offline/online encoding.

Simplification of tensor expressions in computer algebra

Computer algebra is widely used in various fields of mathematics, physics and other sciences. The simplification of tensor expressions is an important special case of computer algebra. In this paper, we consider the reduction of tensor polynomials to canonical form, taking into account the properties of symmetry under permutations of indices, the symmetries associated with the renaming of summation indices, and also linear relations between tensors of a general form. We give a definition of the canonical representation for polynomial (multiplicative) expressions of variables with abstract indices, which is the result of averaging of the original expression by the action of some finite group (the signature stabilizer). In practice, the proposed algorithms demonstrate high efficiency for expressions made of Riemann curvature tensors.

Cooperative Temporal Data Dissemination in SDN-Based Heterogeneous Vehicular Networks

Heterogeneous network resources are expected to cooperate with each other to support temporal data services in vehicular networks. However, it is challenging to implement an efficient data scheduling strategy due to the following factors: first, there are different time constraints on services, which are imposed by the application requirements of both temporal data quality and transmission delay; second, the heterogeneity of wireless interfaces further complicates the transmission task assignment in dynamic vehicular environments. Therefore, this paper proposes an software-defined network-based architecture to enable unified management on heterogeneous network resources. Then, we formulate the cooperative temporal data dissemination (CTDD) problem by considering the property of temporal data, the heterogeneity of wireless interfaces, and the delay constraints on service requests. Further, we prove the NP-hardness of the CTDD by constructing a polynomial-time reduction from a well know NP-hard problem, classical knapsack problem. On this basis, we design a heuristic algorithm called priority-based task assignment (PTA), which synthesizes dynamic task assignment, broadcast efficiency, and service deadline into priority design. Accordingly, PTA is able to adaptively distribute broadcast tasks of each request among multiple interfaces, so as to improve overall system performance. Last but not least, we build the simulation model and implement the proposed algorithm. The comprehensive simulation results show the superiority of the proposed algorithm under a wide range of scenarios.

Computational Complexity of Cycle Discrepancy

The irregularities or the deviations from the state of absolute uniformity are the core focus of the area of discrepancy theory. Cycle discrepancy is about the quantification of the least possible deviation of bi-labeling of graph vertices from an ideal bi-labeling which divides each cycle of a graph in two equal parts. This paper shows that it is NP-hard to compute the cycle discrepancy of a given graph. A polynomial time reduction of Hamiltonian problem to the problem of computing cycle discrepancy is used to establish this result.

Reconstruction of Full Rank Algebraic Branching Programs

An algebraic branching program (ABP) A can be modelled as a product expression X 1 amp;middot; X 2 … X d , where X 1 and X d are 1 × w and w × 1 matrices, respectively, and every other X k is a w × w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly( m )). The polynomial computed by A is the entry of the 1 × 1 matrix obtained from the product ∏ k=1 d X k . We say A is a full rank ABP if the w 2 ( d − 2) + 2 w linear forms occurring in the matrices X 1 , X 2 , …, X d are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m -variate polynomial f of degree at most m , the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs “no full rank ABP exists” (with high probability). The running time of the algorithm is polynomial in m and β, where β is the bit length of the coefficients of f . The algorithm works even if X k is a w k−1 × w k matrix (with w 0 = w d = 1), and w = ( w 1 , …, w d − 1 ) is unknown . The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMM w, d , the (1, 1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to w ∈ N d −1 . At its core, the algorithm exploits a connection between the irreducible invariant subspaces of the Lie algebra of the group of symmetries of a polynomial f that is equivalent to IMM w, d and the “layer spaces” of a full rank ABP computing f . This connection also helps determine the group of symmetries of IMM w, d and show that IMM w, d is characterized by its group of symmetries.

The design and scheduling of chemical batch processes: Computational complexity studies

Summary The pharmaceutical industry is quite restrictive concerning quality and safety, the manufacturing disruptions often lead to drug shortages in despite of the high costs involved. Due to the minimization of equipment costs, the design and scheduling of chemical batch processes (DSCBP) is a well-known problem, and various sub-problems are selected from the literature because they were successively enlarging the design and schedule policies: single machine or multiple machines (S or M) in each stage; and single product campaigns or multiple products campaigns (SPC or MPC). In this paper, four problems are studied (by combinatorics: SS, MS, SM, and MM) and it is shown that they are all NP-hard in strong sense through polynomial reduction. This study can support innovative algorithms and methodologies for solving DSCBP problems, in a way to improve equipments sizing and configurations design, and thereby contributing to curb disruptions within pharmaceutical supply chains (PharmSC).

MPF Problem over Modified Medial Semigroup Is NP-Complete

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

A certain family of subgroups of ℤ𝑛⋆ is weakly pseudo-free under the general integer factoring intractability assumption

AbstractLet {\mathbb{G}_{n}} be the subgroup of elements of odd order in the group {\mathbb{Z}^{\star}_{n}}, and let {\mathcal{U}(\mathbb{G}_{n})} be the uniform probability distribution on {\mathbb{G}_{n}}. In this paper, we establish a probabilistic polynomial-time reduction from finding a nontrivial divisor of a composite number n to finding a nontrivial relation between l elements chosen independently and uniformly at random from {\mathbb{G}_{n}}, where {l\geq 1} is given in unary as a part of the input. Assume that finding a nontrivial divisor of a random number in some set N of composite numbers (for a given security parameter) is a computationally hard problem. Then, using the above-mentioned reduction, we prove that the family {((\mathbb{G}_{n},\mathcal{U}(\mathbb{G}_{n}))\mid n\in N)} of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble {(\mathcal{U}(\mathbb{G}_{n})\mid n\in N)} is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function {\nu\colon D\to N} (where {D\subseteq\{0,1\}^{*}}) and a polynomial-time samplable probability ensemble {(\mathcal{G}_{d}\mid d\in D)} (where {\mathcal{G}_{d}} is a distribution on {\mathbb{G}_{\nu(d)}} for each {d\in D}) such that the family {((\mathbb{G}_{\nu(d)},\mathcal{G}_{d})\mid d\in D)} of computational abelian groups is weakly pseudo-free.

On the complexity of computation maximal exponent of periodicity of word equations and expressible relations (note)

Denote by SWE the satisfiability problem for word equations. Consider the problem of computability of the maximal exponent of periodicity for word equations MaxExpProblem(e,k) in which given positive integer k and a word equation e one is to check whether maximal exponent of periodicity of a solution of a word equation is at least k. We present both a deterministic polynomial time reduction of the MaxExpProblem(e,k) problem to the satisfiability problem for word equations and a deterministic polynomial time reduction in the opposite direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete. As a simple consequence we get that computation of maximal exponent of periodicity of a solution set for a word equation is in PSWE (SWE is an oracle) and, consequently in PSPACE.Let ϕ be an existential boolean formula built on word equations expressing a relation on words by means of its unknowns. Consider the problem of checking the maximal exponent of periodicity of a component of a relation CMaxExpProblem(ϕ,k) in which given positive integer k and a formula ϕ one is to check whether maximal exponent of periodicity of a component of the relation is at least k.We give nondeterministic polynomial time reduction of the described above CMaxExpProblem(ϕ,k) problem to the satisfiability problem for word equations and deterministic polynomial time reduction in the other direction. Consequently, if the satisfiability problem is in NP then the other problem is NP-complete.Denote by SAT(WE) the satisfiability problem of closed existential formula built on word equations. As a simple consequence of our considerations we get that computation of maximal exponent of periodicity of a component of expressible relation is in PSAT(WE) (SAT(WE) is an oracle), and, consequently, in PSPACE. Another consequence of that is that if SWE is in NP, then the computability problem is in PNP. The above results hold also for semigroups with involution.If the expressible relation is defined by a formula using word equations and regular constraints we prove that the problem is PSPACE-complete. The same holds for free semigroups with involution and free groups.

Unique (optimal) solutions: Complexity results for identifying and locating–dominating codes

We investigate the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of an r-Locating–Dominating Code with bounded size” (U-LDCr), “Uniqueness of an Optimal r-Locating–Dominating Code” (U-OLDCr), “Uniqueness of an r-Identifying Code with bounded size” (U-IdCr), “Uniqueness of an Optimal r-Identifying Code” (U-OIdCr), for any fixed integer r≥1.In particular, we describe a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OLDCr, and from U-SAT to U-OIdCr; for U-LDCr and U-IdCr, we can do even better and prove that their complexity is the same as that of U-SAT, up to polynomials. Consequently, all these problems are NP-hard, and U-LDCr and U-IdCr belong to the class DP.

A Flow-Network-Based Polynomial-Time Approximation Algorithm for the Minimum Constrained Input Structural Controllability Problem

This paper deals with minimum cost constrained input selection (minCCIS) for state-space structured systems. Our goal is to optimally select an input set from the given inputs such that the system is structurally controllable when the set of states that each input can influence is prespecified and each input has a cost associated with it. This problem is known to be NP-hard. First, we give a flow-network-based novel necessary and sufficient graph theoretic condition for checking structural controllability. Using this condition, we propose a polynomial time reduction of the problem to a known NP-hard problem: the minimum-cost fixed-flow problem (MCFF). Subsequently, we show that approximation schemes of MCFF directly apply to the minCCIS problem. Using the special structure of the flow-network constructed for the structured system, we formulate a polynomial time approximation algorithm based on minimum weight bipartite matching problem and a greedy scheme for solving the MCFF problem on the system flow-network. The proposed algorithm gives a so-called $\Delta$ -approximate solution to MCFF, where $\Delta$ denotes the maximum in-degree of input vertices in the flow-network of the structured system.

Coding-Assisted Broadcast Scheduling via Memetic Computing in SDN-Based Vehicular Networks

This paper embarks the first study on exploiting the synergy between vehicular caching and network coding for enhancing the bandwidth efficiency of data broadcasting in heterogeneous vehicular networks by presenting a service architecture that exercises the software defined network concept. In particular, we consider the scenario where vehicles request a set of information and they could be served via heterogeneous wireless interfaces, such as roadside units and base stations (BSs). We formulate a novel problem of coding-assisted broadcast scheduling (CBS), aiming at maximizing the broadcast efficiency for the limited BS bandwidth by exploring the synergistic effect between vehicular caching and network coding. We prove the NP-hardness of the CBS problem by constructing a polynomial-time reduction from the simultaneous matrix completion problem. To efficiently solve the CBS problem, we employ memetic computing, which is a nature inspired computational paradigm for tackling complex problems. Specifically, we propose a memetic algorithm, which consists of a binary vector representation for encoding solutions, a fitness function for solution evaluation, a set of operators for offspring generation, a local search method for solution enhancement, and a repair operator for fixing infeasible solutions. Finally, we build the simulation model and give a comprehensive performance evaluation to demonstrate the superiority of the proposed solution.

User Association Under SINR Constraints in HetNets: Upper Bound and NP-Hardness

This letter studies the multi-slot user association problem (MSA) under signal-to-interference-plus-noise ratio (SINR) constraints in heterogeneous cellular networks. MSA consists of associating a set of users to a set of base stations during a set of slots while guaranteeing the SINR requirements of the associated users. To solve MSA, we study its single-slot version, called single-slot user association problem (SSA). We provide an upper bound of the optimal solution of MSA using SSA and prove its tightness. Next, we simplify the NP-hardness proof and show that both the SSA and MSA are NP-hard using a new and simple polynomial-time reduction. Finally, we present the simulation results to illustrate the tightness of the upper bound.

On NP-completeness of subset sum problem for Lamplighter group

In the paper “Knapsack problem for groups” A. Myasnikov, A. Nikolaev, and A. Ushakov stated a group version of well known Subset sum and Knapsack problems and many others. Actually they created a new direction of research as intersection of group theory and discrete optimization. They called the new direction as non-commututative optimization. Our work belongs to this new direction. In present paper we show that Subset sum problem is NP-complete for lamplighter group. Our result was obtained by polynomial reduction of well known Exact set cover problem which is strongly NP-complete to Subset sum problem for lamplighter group. As we expected, this result provides a convenient approach to prove NP-completeness of Subset sum problem for a wide class of groups.

A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction

A Weak Galerkin Finite Element Method for Elliptic Interface Problems with Polynomial Reduction

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the {\tau}-function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.

Practical attribute‐based signature schemes for circuits from bilinear map

Attribute-based signatures allow us to sign anonymously, in such a way that the signature proves that the signer's attributes satisfy some predicate, but it hides any other information on the signer's attributes beyond that fact. As well as any cryptographic primitive, one of the important goals of the research on this primitive is to construct a scheme that is expressive (supports a wide class of predicates), is practically efficient, and is based on well-studied cryptographic assumptions. The authors construct attribute-based signature schemes that support any Boolean circuit of unbounded depth and number of gates, are practically efficient, from the symmetric bilinear Diffie–Hellman assumption. Toward this end, they combine the Groth–Sahai proof system, which serve as an efficient proof system for algebraic equations, and the Groth–Ostrovsky–Sahai proof system, which are still inefficient, but can prove any NP language via a Karp reduction to circuit satisfiability.

Domination in some subclasses of bipartite graphs

A set D⊆V is called a dominating set of G=(V,E) if |NG[v]∩D|≥1 for all v∈V. The Minimum Domination problem is to find a dominating set of minimum cardinality of the input graph. In this paper, we study the Minimum Domination problem for star-convex bipartite graphs, circular-convex bipartite graphs and triad-convex bipartite graphs. It is known that the Minimum Domination Problem for a graph with n vertices can be approximated with an approximation ratio of lnn+1. However, we show that for any ϵ>0, the Minimum Domination problem does not admit a (1−ϵ)lnn-approximation algorithm even for star-convex bipartite graphs with n vertices unless NP ⊆ DTIME(nO(loglogn)). On the positive side, we propose polynomial time algorithms for computing a minimum dominating set of circular-convex bipartite graphs and triad-convex bipartite graphs, by making polynomial time Turing reductions from the Minimum Domination problem for these graph classes to the Minimum Domination problem for convex bipartite graphs.

Computational Aspects of the Colorful Carathéodory Theorem

Let \(C_1,\dots ,C_{d+1}\subset \mathbb {R}^d\) be \(d+1\) point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence \(p_1, \dots , p_{d+1}\) with \(p_i \in C_i\), for \(i = 1, \dots , d+1\), a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory. We define a novel notion of approximation that is compatible with the polynomial-time reductions to ColorfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed \(\varepsilon > 0\), outputs an \(\lceil \varepsilon d\rceil \)-colorful choice containing the origin in its convex hull in polynomial time. Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets \(C_1,\dots ,C_n\subset \mathbb {R}^d\) that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.

Model-order reduction of electromagnetic fields in open domains

We have developed several Krylov projection-based model-order reduction techniques to simulate electromagnetic wave propagation and diffusion in unbounded domains. Such techniques can be used to efficiently approximate transfer function field responses between a given set of sources and receivers and allow for fast and memory-efficient computation of Jacobians, thereby lowering the computational burden associated with inverse scattering problems. We found how general wavefield principles such as reciprocity, passivity, and the Schwarz reflection principle translate from the analytical to the numerical domain and developed polynomial, extended, and rational Krylov model-order reduction techniques that preserve these structures. Furthermore, we found that the symmetry of the Maxwell equations allows for projection onto polynomial and extended Krylov subspaces without saving a complete basis. In particular, short-term recurrence relations can be used to construct reduced-order models that are as memory efficient as time-stepping algorithms. In addition, we evaluated the differences between Krylov reduced-order methods for the full wave and diffusive Maxwell equations and we developed numerical examples to highlight the advantages and disadvantages of the discussed methods.