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38
Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems.
 Math. Programming
, 1993
"... We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of the L3algorithm of Lenstra, Lenstra, Lov'asz (1982). We present a variant of the L3 algorithm with "deep insertions" and a practical algorithm for block KorkinZolotarev reduct ..."
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Cited by 212 (6 self)
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We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of the L3algorithm of Lenstra, Lenstra, Lov'asz (1982). We present a variant of the L3 algorithm with "deep insertions" and a practical algorithm for block KorkinZolotarev 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 SPARC 1+ computer.
An improved lowdensity subset sum algorithm
 in Advances in Cryptology: Proceedings of Eurocrypt '91
"... Abstract. The general subset sum problem is NPcomplete. 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 sh ..."
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Cited by 83 (14 self)
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Abstract. The general subset sum problem is NPcomplete. 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 LagariasOdlyzko algorithm would solve almost all subset sum problems of density < 0.6463... in polynomial time if it could invoke a polynomialtime algorithm for finding the shortest nonzero 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 nonzero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms. Key words, subset sum problems; knapsack cryptosystems; lattices; lattice basis reduction. Subject classifications. 11Y16. 1.
Analysis of PSLQ, An Integer Relation Finding Algorithm
 Mathematics of Computation
, 1999
"... Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In th ..."
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Cited by 71 (29 self)
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Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In this paper we define the parameterized integer relation construction algorithm PSLQ(r), where the parameter rcan be freely chosen in a certain interval. Beginning with an arbitrary vector X = (Xl,..., Xn) _ K n, iterations of PSLQ(r) will produce lower bounds on the norm of any possible relation for X. Thus PS/Q(r) can be used to prove that there are no relations for × of norm less than a given size. Let M x be the smallest norm of any relation for ×. For the real and complex case and each fixed parameter rin a certain interval, we prove that PSLQ(r) constructs a relation in less than O(fl 3 + n 2 log Mx) iterations.
A fortran90 based multiprecision system
 ACM Transactions on Mathematical Software
, 1995
"... The author has developed a new version of his Fortran multiprecision computation system that is based on the Fortran90 language. With this new approach, a translator program is not required — translation of Fortran code for multiprecision is accomplished by merely utilizing advanced features of For ..."
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Cited by 67 (17 self)
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The author has developed a new version of his Fortran multiprecision computation system that is based on the Fortran90 language. With this new approach, a translator program is not required — translation of Fortran code for multiprecision is accomplished by merely utilizing advanced features of Fortran90, such as derived data types and operator extensions. This approach results in more reliable translation and also permits programmers of multiprecision applications to utilize the full power of the Fortran90 language. Three multiprecision datatypes are supported in this system: multiprecision integer, real and complex. All the usual Fortran conventions for mixed mode operations are supported, and many of the Fortran intrinsics, such as SIN, EXP and MOD, are supported with multiprecision arguments. This paper also briefly describes an interesting application of this software, wherein new numbertheoretic identities have been discovered by means of multiprecision computations.
MPFUN: A Portable High Performance Multiprecision Package
, 1990
"... The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especi ..."
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Cited by 48 (4 self)
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The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especially on vector supercomputers, (3) advanced algorithms, including FFTbased multiplication and quadratically convergent algorithms for π and transcendental functions, and (4) extensive selfchecking and debug facilities that permit the package to be used as a rigorous system integrity test. Converting application programs to run with these routines is facilitated by an automatic translator program. This paper describes the routines in the package and includes discussion of the algorithms employed, the implementation techniques, performance results and some applications. Notable among the performance results is that this package runs up to 40 times faster than another widely used package on a RISC workstation, and it runs up to 400 times faster than the other package on a Cray supercomputer.
Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 47 (35 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multilevel implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.
Experimental Evaluation of Euler Sums
, 1993
"... In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent ..."
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Cited by 41 (10 self)
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In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.
Multiprecision translation and execution of Fortran programs
 ACM Transactions on Mathematical Software
, 1993
"... This paper describes two Fortran utilities for multiprecision computation. The first is a package of Fortran subroutines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision. This package is in some cases over 200 times ..."
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Cited by 33 (13 self)
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This paper describes two Fortran utilities for multiprecision computation. The first is a package of Fortran subroutines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision. This package is in some cases over 200 times faster than that of certain other packages that have been developed for this purpose. The second utility is a translator program, which facilitates the conversion of ordinary Fortran programs to use this package. By means of source directives (special comments) in the original Fortran program, the user declares the precision level and specifies which variables in each subprogram are to be treated as multiprecision. The translator program reads this source program and outputs a program with the appropriate multiprecision subroutine calls. This translator supports multiprecision integer, real and complex datatypes. The required array space for multiprecision data types is automatically allocated. In the evaluation of computational expressions, all of the usual conventions for operator precedence and mixed mode operations are upheld. Furthermore, most of the Fortran77 intrinsics, such as ABS, MOD, NINT, COS, EXP are supported and produce true multiprecision values.
A Polynomial Time, Numerically Stable Integer Relation Algorithm
, 1991
"... Let x =(x1,x2, ···,xn) be a vector of real numbers. x is said to possess an integer relation if there exist integers ai not all zero such that a1x1 + a2x2 + ·· · + anxn =0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the ai given x. The most efficient of these ..."
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Cited by 24 (6 self)
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Let x =(x1,x2, ···,xn) be a vector of real numbers. x is said to possess an integer relation if there exist integers ai not all zero such that a1x1 + a2x2 + ·· · + anxn =0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the ai given x. The most efficient of these existing integer relation algorithms (in terms of run time and the precision required of the input) has the drawback of being very unstable numerically. It often requires a numeric precision level in the thousands of digits to reliably recover relations in modestsized test problems. We present here a new algorithm for finding integer relations, which we have named the “PSLQ ” algorithm. It is proved in this paper that the PSLQ algorithm terminates with a relation in a number of iterations that is bounded by a polynomial in n. Because this algorithm employs a numerically stable matrix reduction procedure, it is free from the numerical difficulties that plague other integer relation algorithms. Furthermore, its stability admits an efficient implementation with lower run times on average than other algorithms currently in use. Finally, this stability can be used to prove that relation bounds obtained from computer runs using this algorithm are numerically accurate.