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NumGfun: a Package for Numerical and Analytic Computation with Dfinite Functions
"... This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast highprecision nu ..."
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This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast highprecision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.
A Note on the Space Complexity of Fast DFinite Function Evaluation
"... Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear di ..."
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Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2 −p in timeO(p(lgp) 3+o(1) ) and spaceO(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p lgp) bits of memory. 1.
Series of reciprocal powers of kalmost primes
"... Abstract. Sums over inverse integer powers s of semiprimes and kalmost primes are reduced to sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known c ..."
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Abstract. Sums over inverse integer powers s of semiprimes and kalmost primes are reduced to sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k = 2,...,6 and integer powers s = 2,..., 8. 1. Prime Zeta Function Definition 1. The prime zeta function P(s) is the sum over the reciprocal sth powers of the prime numbers pj = 2, 3, 5, 7,... [13, 12] (1) P(s) ≡
Evaluation of Modular Algorithms for Highprecision Evaluation of Hypergeometric Constants
"... Abstract—Many important wellknown constants such as pi and ζ(3) can be approximated by a truncated hypergeometric series. A modular algorithm based on rational number reconstruction was previously proposed to reduce space complexity of the wellknown binary splitting algorithm [1]. In this paper, ..."
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Abstract—Many important wellknown constants such as pi and ζ(3) can be approximated by a truncated hypergeometric series. A modular algorithm based on rational number reconstruction was previously proposed to reduce space complexity of the wellknown binary splitting algorithm [1]. In this paper, we examine some variations of this algorithm using Mersenne number moduli and Montgomery multiplication. Implementations of these variations are compared to existing methods and evaluated for their practicality. I.
Marc MEZZAROBBA
"... sous la direction de Bruno SALVY Génération automatique de procédures numériques pour les fonctions Dfinies Rapport de stage de Master 2 Master parisien de recherche en informatique févrierjuillet 2007 ..."
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sous la direction de Bruno SALVY Génération automatique de procédures numériques pour les fonctions Dfinies Rapport de stage de Master 2 Master parisien de recherche en informatique févrierjuillet 2007
Contemporary RSA 1024 Cryptosystem: A Comprehensive Review Article
"... Security strength of RSA Cryptography is an enormous mathematical integer factorization problem. Deducing the private key‘d ’ from its equation e. d ≡ (1 mod ψ) where ψ = (p1). (q1), £ n Є I+, such that n = p. q; is a world wide effort. This paper introduced very significant integer factoring alg ..."
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Security strength of RSA Cryptography is an enormous mathematical integer factorization problem. Deducing the private key‘d ’ from its equation e. d ≡ (1 mod ψ) where ψ = (p1). (q1), £ n Є I+, such that n = p. q; is a world wide effort. This paper introduced very significant integer factoring algorithms such as trial division, ρ method, ECM, and NFS and effort to factor RSA150 composite number ‘n ’ of 512 bits by using NFS. It is found that the 512 bit RSA number may be believed to safe from the intruder. However, this system is slow for large volume of data. The computation of c ≡ me mod n required O ((size e)(size n) * (size n)) and space O(size e + size n). Similarly, decryption process also has required O ((size d) (size n) * (size n)) and space O (size d + size n). Java ‘BigInteger ’ class is introduced to overcome this shortcoming and successfully applied is presented through this paper.