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The complexity of analog computation
 in Math. and Computers in Simulation 28(1986
"... We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP w ..."
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We ask if analog computers can solve NPcomplete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NPcomplete problem, 3SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by wellbehaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.
On The Computational Hardness Of Testing SquareFreeness Of Sparse Polynomials
, 1999
"... We show that deciding squarefreeness of a sparse univariate polynomial over ZZ and over the algebraic closure of a finite field IFq of p elements is NPhard. We also discuss some related open problems about sparse polynomials. ..."
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Cited by 11 (1 self)
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We show that deciding squarefreeness of a sparse univariate polynomial over ZZ and over the algebraic closure of a finite field IFq of p elements is NPhard. We also discuss some related open problems about sparse polynomials.
Cipher Based on Quasigroup String Transformations in Z ,” arXiv: cs.CR/0403043, 2004. Authorized licensed use limited to: Oklahoma State University
 at 13:53 from IEEE Xplore. Restrictions apply. AND KAK: MULTILEVEL INDEXED QUASIGROUP ENCRYPTION FOR DATA AND SPEECH 281
"... Abstract. In this paper we design a stream cipher that uses the algebraic structure of the multiplicative group ZZ ∗ p (where p is a big prime number used in ElGamal algorithm), by defining a quasigroup of order p − 1 and by doing quasigroup string transformations. The cryptographical strength of th ..."
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Cited by 7 (1 self)
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Abstract. In this paper we design a stream cipher that uses the algebraic structure of the multiplicative group ZZ ∗ p (where p is a big prime number used in ElGamal algorithm), by defining a quasigroup of order p − 1 and by doing quasigroup string transformations. The cryptographical strength of the proposed stream cipher is based on the fact that breaking it would be at least as hard as solving systems of multivariate polynomial equations modulo big prime number p which is NPhard problem and there are no known fast randomized or deterministic algorithms for solving it. Unlikely the speed of known ciphers that work in ZZ ∗ p for big prime numbers p, the speed of this stream cipher both in encryption and decryption phase is comparable with the fastest symmetrickey stream ciphers.
Factorization of Polynomials
 Computing, Suppl. 4
, 1982
"... Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squaref ..."
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Cited by 6 (0 self)
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Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree decomposition algorithms and Hensel lifting techniques are analyzed. An attempt is made to establish a complete historic trace for today's methods. The exponential worst case complexity nature of these algorithms receives attention. _______________ Appears in Computer Algebra, second edition, B. Buchberger, R. Loos, G. Collins, editors, Springer Verlag, Vienna, Austria, pp. 9511 (1982).  2  1. Introduction The problem of factoring polynomials has a long and distinguished history. D. Knuth traces the first attempts back to Isaac Newton's Arithmetica Universalis (1707) and to the astronomer Friedrich T. v. Schubert who in 1793 presented a finite step algorithm to compute the factors...
Complexity Issues in Dynamic Geometry
 IN PROCEEDINGS OF THE SMALE FEST 2000, HONGKONG
, 2000
"... This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from... ..."
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Cited by 5 (5 self)
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This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from...
The NodeDeletion Problem for Hereditary . . .
, 1980
"... We consider the family of graph problems called nodedeletion problems, defined as follows: For a fixed graph property l7, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies l7? We show that if l7 is nontrivial and hereditary on indu ..."
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We consider the family of graph problems called nodedeletion problems, defined as follows: For a fixed graph property l7, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies l7? We show that if l7 is nontrivial and hereditary on induced subgraphs, then the nodedeletion problem for n is NPcomplete for both undirected and directed graphs.
Online at: www.jus.org.uk A New Analog Optical Processing Scheme for Solving NPHard Problems
, 2012
"... Many reallife problems are, in general, NPhard, i.e., informally speaking, are difficult to solve. To be more precise, a problem P0 is NPhard means that every problem from the class NP can be reduced to this problem P0. Thus, if we have an efficient algorithm for solving one NPhard problem, we c ..."
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Many reallife problems are, in general, NPhard, i.e., informally speaking, are difficult to solve. To be more precise, a problem P0 is NPhard means that every problem from the class NP can be reduced to this problem P0. Thus, if we have an efficient algorithm for solving one NPhard problem, we can use this reduction to get a more efficient way of solving all the problems from the class NP. To speed up computations, it is reasonable to base them on the fastest possible physical process – i.e., on light. It is known that analog optical processing indeed speeds up computation of several NPhard problems. Each of the corresponding speedup schemes has its success cases and limitations. The more schemes we know, the higher the possibility that for a given problem, one of these schemes will prove to be effective. Motivated by this argument, we propose a new analog optical processing scheme for solving NPhard problems. c2013 World Academic Press, UK. All rights reserved.
Attacks and Comments on Several Recently Proposed Key Management Schemes
"... Abstract. In this paper, we review three problematic key management (KM) schemes recently proposed, including Kayam’s scheme for groups with hierarchy [10], Piao’s group KM scheme [14], Purushothama’s group KM schemes [17]. We point out the problems in each scheme. Kayam’s scheme is not secure to co ..."
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Abstract. In this paper, we review three problematic key management (KM) schemes recently proposed, including Kayam’s scheme for groups with hierarchy [10], Piao’s group KM scheme [14], Purushothama’s group KM schemes [17]. We point out the problems in each scheme. Kayam’s scheme is not secure to collusion attack. Piao’s group KM scheme is not secure. The hard problem it bases is not really hard. Purushothama’s scheme has a redundant design that costs lots of resources and doesn’t give an advantage to the security level and dynamic efficiency of it. We also briefly analyze the underlying reasons why these problem emerge.
An Efficient Scheme for Centralized Group Key Management in Collaborative Environments
, 2013
"... The increasing demand for online collaborative applications has sparked the interest for multicast services, which in many cases have to guarantee properties such as authentication or confidentiality within groups of users. To do so, cryptographic protocols are generally used and the cryptographic ..."
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The increasing demand for online collaborative applications has sparked the interest for multicast services, which in many cases have to guarantee properties such as authentication or confidentiality within groups of users. To do so, cryptographic protocols are generally used and the cryptographic keys, in which they rely, have to be managed (e.g. created, updated, distributed). The procedures to perform these operations are determined by the socalled Group Key Management Schemes. Many schemes have been proposed and some of them have been proven to be vulnerable. This is the case of the Piao et al. scheme, whose scalability/efficiency is very good but it is vulnerable to many attacks because its security is based on a weak mathematical problem, so it can be broken in polynomial time. Inspired by the concepts proposed in the Piao et al. scheme we have redesigned the protocol and we have founded it on a hard mathematical problem and tweaked some of the procedures. This way, we propose a new scheme that is efficient, collusion free, and provides backward and forward secrecy.