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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 636 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
SALEM NUMBERS OF NEGATIVE TRACE
"... Abstract. We prove that, for all d ≥ 4, there are Salem numbers of degree 2d and trace −1, and that the number of such Salem numbers is ≫ d / (log log d) 2. As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d − 1isalso ≫ d / (log log d) 2. 1. ..."
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Cited by 6 (5 self)
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Abstract. We prove that, for all d ≥ 4, there are Salem numbers of degree 2d and trace −1, and that the number of such Salem numbers is ≫ d / (log log d) 2. As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d − 1isalso ≫ d / (log log d) 2. 1.
A Survey of GcdSum Functions
, 2010
"... We survey properties of the gcdsum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcdsum function and for the function defined by the harmonic mean of the gcd’s. ..."
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Cited by 5 (2 self)
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We survey properties of the gcdsum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcdsum function and for the function defined by the harmonic mean of the gcd’s.
Securing RSA against fault analysis by double addition chain exponentiation
 CTRSA 2009. Volume 5473 of LNCS
, 2009
"... Abstract. Fault Analysis is a powerful cryptanalytic technique that enables to break cryptographic implementations embedded in portable devices more efficiently than any other technique. For an RSA implemented with the Chinese Remainder Theorem method, one faulty execution suffices to factorize the ..."
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Cited by 4 (0 self)
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Abstract. Fault Analysis is a powerful cryptanalytic technique that enables to break cryptographic implementations embedded in portable devices more efficiently than any other technique. For an RSA implemented with the Chinese Remainder Theorem method, one faulty execution suffices to factorize the public modulus and fully recover the private key. It is therefore mandatory to protect embedded implementations of RSA against fault analysis. This paper provides a new countermeasure against fault analysis for exponentiation and RSA. It consists in a selfsecure exponentiation algorithm, namely an exponentiation algorithm that provides a direct way to check the result coherence. An RSA implemented with our solution hence avoids the use of an extended modulus (which slows down the computation) as in several other countermeasures. Moreover, our exponentiation algorithm involves 1.65 multiplications per bit of the exponent which is significantly less than the 2 required by other selfsecure exponentiations. 1
On the binomial convolution of arithmetical functions
, 806
"... Abstract. Let n = ∏ p pνp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ ( ∏ ( νp(n) dn p νp(d) f(d)g(n/d), where ..."
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Cited by 2 (1 self)
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Abstract. Let n = ∏ p pνp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ ( ∏ ( νp(n) dn p νp(d) f(d)g(n/d), where
DUALITY AND A FARKAS LEMMA FOR INTEGER PROGRAMS
"... Abstract. We consider the integer program max{c ′ x  Ax = b, x ∈ N n}. A formal parallel between linear programming and continuous integration on one side, and discrete summation on the other side, shows that a natural duality for integer programs can be derived from the Ztransform and Brion and Ve ..."
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Cited by 2 (0 self)
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Abstract. We consider the integer program max{c ′ x  Ax = b, x ∈ N n}. A formal parallel between linear programming and continuous integration on one side, and discrete summation on the other side, shows that a natural duality for integer programs can be derived from the Ztransform and Brion and Vergne ’ s counting formula. Along the same lines, we also provide a discrete Farkas lemma and show that the existence of a nonnegative integral solution x ∈ N n to Ax = b can be tested via a linear program. 1.
Algorithmic problems in twisted groups of Lie type
"... This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups. The two main problems under consideration are constructive recognition and constructive membership testing. We also consider problems of generatin ..."
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This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups. The two main problems under consideration are constructive recognition and constructive membership testing. We also consider problems of generating and conjugating Sylow and maximal subgroups. The algorithms are motivated by, and form a part of, the Matrix Group Recognition Project. Obtaining both theoretically and practically efficient algorithms has been a central goal. The algorithms have been developed with, and implemented
A REMARK ON AN INEQUALITY FOR THE PRIME COUNTING FUNCTION
, 2005
"... Abstract. We note that the inequalities 0.92 x log(x) do not hold for all x ≥ 30, contrary to some references. These estimates on π(x) came up recently in papers on algebraic number theory. < π(x) < 1.11 x log(x) 1. Chebyshev’s estimates for π(x) Let π(x) denote the number of primes not greater than ..."
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Abstract. We note that the inequalities 0.92 x log(x) do not hold for all x ≥ 30, contrary to some references. These estimates on π(x) came up recently in papers on algebraic number theory. < π(x) < 1.11 x log(x) 1. Chebyshev’s estimates for π(x) Let π(x) denote the number of primes not greater than x, i.e., π(x) = ∑ 1. One of the first works on the function π(x) is due to Chebyshev. He proved (see [2]) in 1852 the following explicit inequalities for π(x), holding for all x ≥ x0 with some x0 sufficiently large: c1 x x < π(x) < c2 log(x) log(x), p≤x c1 = log(2 1/2 3 1/3 5 1/5 30 −1/30) ≈ 0.921292022934, c2 = 6 5 c1 ≈ 1.10555042752. This can be found in many books on analytic number theory (see for example [1], [3], [8] and [9]). But it seems that this result is sometimes cited incorrectly: it is claimed that the estimates are valid for all x ≥ 30. For example, in [6], page 21 we read that c1 x log(x)
Some symmetric identities involving a sequence of polynomials
"... In this paper we establish some symmetric identities on a sequence of polynomials in an elementary way, and some known identities involving Bernoulli and Euler numbers and polynomials are obtained as particular cases. 1 ..."
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In this paper we establish some symmetric identities on a sequence of polynomials in an elementary way, and some known identities involving Bernoulli and Euler numbers and polynomials are obtained as particular cases. 1