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Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.25
We present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from Lenstra, Lenstra, Lovász reduction to Korkine–Zolotareff reduction. Let λ(L) be the length of a shortest nonzero element of a lattice L. We present an algorithm which for k∈N finds a nonzero lattice vector b so that |b|2⩽(6k2)nkλ(L)2. This algorithm uses O(n2(kk+o(k))+n2)log B) arithmetic operations on O(n log B)-bit integers. This holds provided that the given basis vectors b1,…,bn∈Zn are integral and have the length bound B. This algorithm successively applies Korkine–Zolotareff reduction to blocks of length k of the lattice basis. We also improve Kannan's algorithm for Korkine-Zolotareff reduction.
Performance and storage requirements of topology-conserving maps for robot manipulator control
(1989)
A new programming paradigm for the control of a robot manipulator by learning the mapping between the Cartesian space and the joint space (inverse Kinematic) is discussed. It is based on a Neural Network model of optimal mapping between two high-dimensional spaces by Kohonen. This paper describes the approach and presents the optimal mapping, based on the principle of maximal information gain. It is shown that Kohonens mapping in the 2-dimensional case is optimal in this sense. Furthermore, the principal control error made by the learned mapping is evaluated for the example of the commonly used PUMA robot, the trade-off between storage resources and positional error is discussed and an optimal position encoding resolution is proposed.
We propose two improvements to the Fiat Shamir authentication and signature scheme. We reduce the communication of the Fiat Shamir authentication scheme to a single round while preserving the e±ciency of the scheme. This also reduces the length of Fiat Shamir signatures. Using secret keys consisting of small integers we reduce the time for signature generation by a factor 3 to 4. We propose a variation of our scheme using class groups that may be secure even if factoring large integers becomes easy.
It is well known that artificial neural nets can be used as approximators of any continous functions to any desired degree. Nevertheless, for a given application and a given network architecture the non-trivial task rests to determine the necessary number of neurons and the necessary accuracy (number of bits) per weight for a satisfactory operation. In this paper the problem is treated by an information theoretic approach. The values for the weights and thresholds in the approximator network are determined analytically. Furthermore, the accuracy of the weights and the number of neurons are seen as general system parameters which determine the the maximal output information (i.e. the approximation error) by the absolute amount and the relative distribution of information contained in the network. A new principle of optimal information distribution is proposed and the conditions for the optimal system parameters are derived. For the simple, instructive example of a linear approximation of a non-linear, quadratic function, the principle of optimal information distribution gives the the optimal system parameters, i.e. the number of neurons and the different resolutions of the variables.
Coster, Matthijs J. ; Joux, Antoine ; LaMacchia, Brian A. ; Odlyzko, Andrew M. ; Schnorr, Claus Peter ; Stern, Jacques
The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short nonzero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463 . . . in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408 . . . if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.
It is well known that artificial neural nets can be used as approximators of any continuous functions to any desired degree and therefore be used e.g. in high - speed, real-time process control. Nevertheless, for a given application and a given network architecture the non-trivial task remains to determine the necessary number of neurons and the necessary accuracy (number of bits) per weight for a satisfactory operation which are critical issues in VLSI and computer implementations of nontrivial tasks. In this paper the accuracy of the weights and the number of neurons are seen as general system parameters which determine the maximal approximation error by the absolute amount and the relative distribution of information contained in the network. We define as the error-bounded network descriptional complexity the minimal number of bits for a class of approximation networks which show a certain approximation error and achieve the conditions for this goal by the new principle of optimal information distribution. For two examples, a simple linear approximation of a non-linear, quadratic function and a non-linear approximation of the inverse kinematic transformation used in robot manipulator control, the principle of optimal information distribution gives the the optimal number of neurons and the resolutions of the variables, i.e. the minimal amount of storage for the neural net. Keywords: Kolmogorov complexity, e-Entropy, rate-distortion theory, approximation networks, information distribution, weight resolutions, Kohonen mapping, robot control.
We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of theL3-algorithm of Lenstra, Lenstra, Lovász (1982). We present a variant of theL3-algorithm with "deep insertions" and a practical algorithm for block Korkin—Zolotarev reduction, a concept introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
We call a distribution on n bit strings (", e) locally random, if for every choice of e · n positions the induced distribution on e bit strings is in the L1 norm at most " away from the uniform distribution on e bit strings. We establish local randomness in polynomial random number generators (RNG) that are candidate one way functions. Let N be a squarefree integer and let f1, . . . , f be polynomials with coe±- cients in ZZN = ZZ/NZZ. We study the RNG that stretches a random x 2 ZZN into the sequence of least significant bits of f1(x), . . . , f(x). We show that this RNG provides local randomness if for every prime divisor p of N the polynomials f1, . . . , f are linearly independent modulo the subspace of polynomials of degree · 1 in ZZp[x]. We also establish local randomness in polynomial random function generators. This yields candidates for cryptographic hash functions. The concept of local randomness in families of functions extends the concept of universal families of hash functions by Carter and Wegman (1979). The proofs of our results rely on upper bounds for exponential sums.