Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension sigma (M) of the reals with digits in their continued fraction expansion b...
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Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension sigma (M) of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of sigma (M) for M -> a were provided by Hensley (J. Number Theory 40:336-358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166-172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371-385, 1987) and Vall,e (J. Number Theory 72:183-235, 1998) considered the case of a fixed bound M when the denominator N tends to a. Later, Hensley (Pac. J. Math. 151(2):237-255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions;second, it deals with various probability settings (and not only the uniform probability);third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M (0);fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336-358, 1992) about the Hausdorff dimension sigma (M) to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vall,e (J. Th,or. Nr. Bordx. 12:531-570, 2000), and refined by Baladi and Vall,e (J. Number Theory 110:331-386, 2005).
For large N, we consider the ordinary continued fraction of x = p/q with 1 infinity. If c is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. ...
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For large N, we consider the ordinary continued fraction of x = p/q with 1 <= p <= q <= N, or, equivalently, Euclid's gcd algorithm for two integers 1 <= p <= q <= N, putting the uniform distribution on the set of p and qs. We study the distribution of the total cost of execution of the algorithm for all additive cost function c oil the set Z(+)* of possible digits, asymptotically for N -> infinity. If c is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing diophantine conditions oil the cost, we are able to control the speed of convergence in the local limit theorem. We use previous estimates of the first author and Vallee, and we adapt to our setting bounds of Dolgopyat and Melbourne on transfer operators. Our diophantine condition is generic (with respect to Lebesgue measure). For smooth enough observables (depending on the diophantine condition) we attain the optimal speed.
The paper "euclidean algorithms are Gaussian" [V. Baladi, B. Vallee, euclidean algorithm are Gaussian, J. Number Theory 110 (2005) 331-386], is devoted to the distributional analysis of three variants of Euc...
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The paper "euclidean algorithms are Gaussian" [V. Baladi, B. Vallee, euclidean algorithm are Gaussian, J. Number Theory 110 (2005) 331-386], is devoted to the distributional analysis of three variants of euclidean algorithms. The Central Limit Theorem and the Local Limit Theorem obtained there are the first ones in the context of the "dynamical analysis" method. The techniques developed have been applied in further various works (e.g. [V. Baladi, A. Hachemi, A local limit theorem with speed of convergence for euclidean algorithms and Diophantine costs, Ann. Inst. H. Poincare Probab. Statist. 44 (2008) 749-770;E. Cesaratto, J. Clement, B. Daireaux, L Lhote, V. Maume, B. Vallee, Analysis of fast versions of the Euclid algorithm, in: Proceedings of Third Workshop on Analytic Algorithmics and Combinatorics, ANALCO'08, SIAM, 2008;E. Cesaratto. A. Plagne, B. Vallee, On the non-randomness of modular arithmetic progressions, in: Fourth Colloquium on Mathematics and Computer Science. algorithms, Trees, Combinatorics and Probabilities, in: Discrete Math. Theor. Comput. Sci. Proc., vol. AG, 2006, pp. 271-288]). These theorems are proved first for an auxiliary probabilistic model, called "the smoothed model," and after, the estimates are transferred to the "true" probabilistic model. In this note, we remark that "the smoothed model" described in [V. Baladi, B. Vallee, euclidean algorithm are Gaussian, J. Number Theory 110 (2005) 331-386] is not adapted to this transfer and replaces it by an adapted one. However, the results remain unchanged. (c) 2009 Elsevier Inc. All rights reserved.
We develop a general framework for the analysis of algorithms of a broad euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of e...
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We develop a general framework for the analysis of algorithms of a broad euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. (C) 2002 Elsevier Science B.V. All rights reserved.
We develop a general framework for the analysis of algorithms of a broad euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of e...
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We develop a general framework for the analysis of algorithms of a broad euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. (C) 2002 Elsevier Science B.V. All rights reserved.
The first moments for the number of steps in different euclidean algorithms are considered. For these moments asymptotic formulae with new remainder terms are obtained using refined estimates for sums of fractional pa...
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The first moments for the number of steps in different euclidean algorithms are considered. For these moments asymptotic formulae with new remainder terms are obtained using refined estimates for sums of fractional parts and some ideas in Selberg's elementary proof of the prime number theorem.
Nowadays fast computing and lightweight cryptography play a crucial role in the field of cryptography. Whenever we concern about the cryptography, the aspect of discrete mathematics can't be omitted. Security in c...
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ISBN:
(纸本)9789380544199
Nowadays fast computing and lightweight cryptography play a crucial role in the field of cryptography. Whenever we concern about the cryptography, the aspect of discrete mathematics can't be omitted. Security in cryptography is completely depends upon the key and the computations. Generation of key is nothing but the implementation of graph theory, discrete logarithms,linear &abstract algebra etc. These branches of modern mathematics has been playing a great role for the implementation of algorithms such as elliptic curve cryptography, stream cipher, block cipher, wireless sensor network etc. This paper is written to literally represent the key concept of such branches of modern mathematics and their implementations in the field of cryptograph. This paper mainly deals the applications of Galois Field, primitive polynomials, primitive polynomials over Galois Field, Number theoretic functions, Congruence Calculus or modular arithmetic's, Residue Class Rings and Prime Fields. The paper focus on these key topics to developa mathematical tool, that are needed for the design and security analysis of a cryptosystems.
This paper presents VLSI implementation of an area efficient 8-error correcting (63,47) Reed-Solomon(RS) encoder and decoder for the CDPD (Cellular Digital Packet Data)communication systems[1]. We implement this RS de...
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ISBN:
(纸本)0780366778
This paper presents VLSI implementation of an area efficient 8-error correcting (63,47) Reed-Solomon(RS) encoder and decoder for the CDPD (Cellular Digital Packet Data)communication systems[1]. We implement this RS decoder using euclidean algorithms which is regular, simple and naturally suitable for VLSI implementation. Constant multipliers based on certain composite field are deployed in the encoder, which significantly decreases the encoder's area. Multipliers over certain composite field GF((2(n))(2)) adopted in this paper lowers the complexity of the multiplication of the decoder. The RS encoder and decoder can independently operates at a clock frequency of 30 MHz. This chip was fabricated in 0.6mum CMOS 1P2M technology with a supply of voltage of 5v, with die area 4mm x 4mm. The chip has been fully tested and stratifies the demand of the CDPD communication systems.
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