Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosyst...
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Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosystems, such as HFE of Patarin and UOV of Kipnis et al., is based on the postulated hardness of solving quadratic polynomial systems over a finite field. Lokshtanov et al. (2017) were the first to introduce a probabilistic algorithm that, in the worst-case, solves a Boolean polynomial system in time O*(2(delta n)), for some delta is an element of (0, 1) depending only on the degree of the system, thus beating the brute-force complexity O*(2(n)). Later, Bjorklund et al. (2019) and then Dinur (2021) improved this method and devised probabilistic algorithms with a smaller exponent coefficient delta. We survey the theory behind these probabilistic algorithms, and we illustrate the results that we obtained by implementing them in C. In particular, for random quadratic Boolean systems, we estimate the practical complexities of the algorithms and their probabilities of success as their parameters change. (C) 2021 Elsevier B.V. All rights reserved.
The theoretical complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O (d), O (d log d), or O (d(2)). In fact, the asymptotic comple...
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The theoretical complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O (d), O (d log d), or O (d(2)). In fact, the asymptotic complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3 <= d <= 7, it improves the global behavior of the removal for random points by more than 45%. (C) 2010 Elsevier B.V. All rights reserved.
The magic word 'complexity' has been buzzing around in science, policy and society for quite some time now. There seems to be a common feel for a 'new way' of doing things, for overcoming the limits of...
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The magic word 'complexity' has been buzzing around in science, policy and society for quite some time now. There seems to be a common feel for a 'new way' of doing things, for overcoming the limits of tradition. From the combined perspective of critical complexity thinking and environment and health practice we want to contribute to the development of alternative routines that may help overcome the limitations of traditional environment and health science. On the one hand traditional environment and health science is too self-confident with respect to potential scientific insight in environment and health problems: complexity condemns us to limited and ambiguous knowledge and the need for simplification. A more modest attitude would be more realistic from that point of view. On the other hand from a problem solving perspective more boldness is required. Waiting for Godot (perfect undisputed knowledge) will not help us with respect to the challenges posed to society by environment and health problems. A sense of urgency is legitimate: the paralysis by traditional analysis should be resolved. Nevertheless this sense of urgency should not withhold us from investing in the problem solving quality of our endeavour;quality takes time, fastness from a quality perspective often leads us to a standstill. We propose the concept of critical complexification of environment and health practice that will enable the integration of relevant actors and factors in a pragmatic manner. We will illustrate this with practical examples and especially draw attention to the practical complexities involved, confronting us not only with fundamental questions, but also with fundamental challenges.
Fast Fourier Transforms (FFTs) are frequently employed in various applications such as image processing and speech recognition. Though FFT calculations can be speeded up considerably, real time processing requirements...
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Traditional medical approaches to moral issues found in the clinical setting can, if properly understood, enlighten our philosophical understanding of moral issues. Moral problem-solving, as distinct from ethical and ...
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