H. W. Lenstra, Jr., Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), no. 4, 538--548. Home/Search Document Not in Database Summary Related Articles Check |
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Fast Reduction of Ternary Quadratic Forms - Eisenbrand, Rote (Correct)
....A basic question here is: Given a form F , what is the minimal nonzero value #(F) min F (x 1 , x d ) x#=0 of the form which is attained at an integral vector This problem will be of central interest in this paper. Problem 1. Given a form F , compute #(F ) At least since Lenstra s [9] polynomial algorithm for integer programming in fixed dimension, the study of quadratic forms has also become a major topic in theoretical computer science. Here, one is interested in the lattice variant of Problem 1, which is: Given a basis of an integral lattice, find a shortest nonzero vector ....
H. W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538--548, 1983.
An Automaton-Theoretic Approach to Safety and Security in.. - Lanotte (2003) (Correct)
....the general case of polynomial formulae on real variables. For exponential formulae, quantifier elimination exists for some cases (see [73] In the case of integer variables, if one considers linear formulae without quantifier on integer variables, then the satisfiability is decidable (see [68]) If one deals with first order logic on linear formulae with integer variables (called Presburger Arithmetic) 51] o#ers a quantifier elimination algorithm which is given by introducing equivalences modulo a natural value. When one tackles formulae with mixed integer and real variables, ....
.... # id 2 .#) is equivalent to a existentially quantified formula # k .# # in # S ( # ) where # # is simple. 2.5. DECIDABILITY OF SATISFIABILITY 35 But, for any # S ( # ) where # # is simple, it is decidable whether is equal to empty set. More precisely, the problem is NP . In [68] it is proved that there exists an algorithm exponential on number of integer variables, but polynomial in the number of equations. # BD ( # id) it is decidable whether # is satisfiable. Proof. Let # be in # BD ( # id 1 ) and # id 2 be the vector of quantified identifiers of #; # is ....
Lenstra H.W.: Integer Programming with a Fixed Number of Variables. Mathematics of Operations Research 8 1983, 538--548.
Parameterized Complexity for the Skeptic - Downey (2003) (15 citations) (Correct)
....[DF99a] The O(k) in the exponent hides evil, and the derandomization method at present seems far from practical. One final technique we have not discussed in [DF99a] is the use of INTEGER PROGRAMMING in the design of FP algorithms. This is discussed in Niedermeier [Nie02] Theorem 6. 4 (Lenstra [Le83]) The integer programming feasibility problem can be solved with O(p L) arithmetical operations in Z of O(p L) bits in size, where p is the number of variables, and L the number of bits of the input. Niedermeier [Nie02] gave one example of the use of this method for establishing parametric ....
H. Lenstra, "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, 8 (1983), 538-548.
On the Complexity of Decoding Lattices Using the.. - Banihashemi, Khandani (1998) (1 citation) (Correct)
....Korkin Zolotarev (K Z) and more recently Lenstra, Lenstra, and Lov asz (L ) see, e.g. 12, pp. 147 164] After the introduction of the L reduced basis, which can be computed in polynomial time, reduction theory has found many applications in a variety of areas (see, e.g. 2] 13] 16] [19] [21] 24, pp. 71 74] However, it can be shown that for the decoding of lattices, the K Z reduced basis is a more powerful tool than the L reduced basis [6] In the following, we explain the K Z reduced basis, which is used in our decoding algorithms. Let be a lattice with ordered basis and ....
H. W. Lenstra, "Integer programming with a fixed number of variables," Math. of Operations Res., vol. 8, pp. 538--548, Nov. 1983.
The Two Faces of Lattices in Cryptology - Nguyen, Stern (2001) (7 citations) (Correct)
.... forms developed by Lagrange [86] Gauss [55] Hermite [68] Korkine and Zolotarev [82, 83] among others, and to Minkowski s geometry of numbers [103] With the advent of algorithmic number theory, the subject had a revival in 1981 with Lenstra s celebrated work on integer programming (see [89, 90]) which was, among others, based on a novel lattice reduction technique (which can be found in the preliminary version [89] of [90] Lenstra s reduction technique was only polynomial time for fixed dimension, which was however enough in [89] That inspired Lov asz to develop a polynomial time ....
.... of numbers [103] With the advent of algorithmic number theory, the subject had a revival in 1981 with Lenstra s celebrated work on integer programming (see [89, 90] which was, among others, based on a novel lattice reduction technique (which can be found in the preliminary version [89] of [90]) Lenstra s reduction technique was only polynomial time for fixed dimension, which was however enough in [89] That inspired Lov asz to develop a polynomial time variant of the algorithm, which computes a so called reduced basis of a lattice. The algorithm reached a final form in the seminal ....
[Article contains additional citation context not shown here]
H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538--548, 1983.
The Two Faces of Lattices in Cryptology - Nguyen, Stern (2001) (7 citations) (Correct)
.... forms developed by Lagrange [86] Gauss [55] Hermite [68] Korkine and Zolotarev [82, 83] among others, and to Minkowski s geometry of numbers [103] With the advent of algorithmic number theory, the subject had a revival in 1981 with Lenstra s celebrated work on integer programming (see [89, 90]) which was, among others, based on a novel lattice reduction technique (which can be found in the preliminary version [89] of [90] Lenstra s reduction technique was only polynomial time for fixed dimension, which was however enough in [89] That inspired Lov asz to develop a polynomial time ....
.... geometry of numbers [103] With the advent of algorithmic number theory, the subject had a revival in 1981 with Lenstra s celebrated work on integer programming (see [89, 90] which was, among others, based on a novel lattice reduction technique (which can be found in the preliminary version [89] of [90] Lenstra s reduction technique was only polynomial time for fixed dimension, which was however enough in [89] That inspired Lov asz to develop a polynomial time variant of the algorithm, which computes a so called reduced basis of a lattice. The algorithm reached a final form in the ....
[Article contains additional citation context not shown here]
H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Technical report, Mathematisch Instituut, Universiteit van Amsterdam, April 1981. Report 81-03.
Lattices and Cryptography: an Overview - Stern (1998) (2 citations) (Correct)
....a so called reduced basis of a lattice and provides a partial answer to SVP since it runs in polynomial time and approximates the shortest vector within a factor of 2 n=2 . Actually, a reduction algorithm of the same flavor had already been included in Lenstra s work on integer programming (cf. Len83] circulated around 1979) and the lattice reduction algorithm reached a final form in the paper [LLL82] of Lenstra, Lenstra and Lov asz, from which the name LLL algorithm comes. Further refinements of the LLL algorithm were proposed by Schnorr ( Sch87, Sch88] who has improved the above factor ....
H. W. Lenstra. Integer programming with a fixed number of variables. Math. Oper. Res., 8:538--548, 1983.
Lattice Reduction: a Toolbox for the Cryptanalyst - Joux, Stern (1994) (20 citations) (Correct)
....of algorithmic number theory, the subject had a revival around 1980, when Lov asz found a polynomial time algorithm that computes a so called reduced basis of a lattice. Actually, a reduction algorithm of the same flavor had already been included in Lenstra s work on integer programming (cf. Len83] circulated around 1979) and the lattice reduction algorithm reached a final form in the paper [LLL82] of Lenstra, Lenstra and Lov asz, from which the name LLL algorithm comes. Further refinements of the LLL algorithm were proposed by Schnorr ( Sch87, Sch88] The relevance of those algorithms ....
H. W. Lenstra. Integer programming with a fixed number of variables. Math. Oper. Res., 8:538--548, 1983.
Lattice Reduction in Cryptology: An Update - Nguyen, Stern (2000) (12 citations) (Correct)
.... forms developed by Lagrange [71] Gauss [44] Hermite [55] Korkine and Zolotarev [67, 68] among others, and to Minkowski s geometry of numbers [85] With the advent of algorithmic number theory, the subject had a revival around 1980 with Lenstra s celebrated work on integer programming (see [74]) which was, among others, based on a novel but non polynomial time lattice reduction technique. That algorithm inspired Lov asz to develop a polynomial time algorithm that computes a so called reduced basis of a lattice. It reached a final form in the seminal paper [73] where Lenstra, Lenstra ....
.... were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69] In particular, their relevance to The technique is however polynomial time for fixed dimension, which was enough in [74]. cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA cryptosystem [100] The success of reduction algorithms at breaking various cryptographic schemes over the past twenty years (see ....
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H. W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538--548, 1983.
Solving a System of Diophantine Equations with Lower and .. - Aardal, Hurkens, Lenstra (1998) (4 citations) Self-citation (Lenstra) (Correct)
....of a full dimensional polytope. Such a description could also be obtained by for instance deriving the Hermite normal form of the matrix A, but that typically creates large (but polynomially bounded) numbers. A full dimensional polytope is useful if one wants to apply the algorithm of Lenstra [12], or of Lov asz and Scarf [14] 3 Basis reduction and its use in integer programming We begin by giving the definition of a lattice and a reduced basis. Definition 1 A subset L ae IR is called a lattice if there exists a basis b 1 ; b 2 ; b l of IR L = f l j=1 ff j b j : ff ....
....l condition (5) is violated, then replace b j by b j Gamma d jk cb k , where d jk c = d jk Gamma e. Interchange: If condition (6) is violated for an index j; 1 j l, then interchange vectors b j Gamma1 and b j . Basis reduction was introduced in integer programming by H.W. Lenstra, Jr. [12], who showed that the problem of determining if there exists a vector x 2 ZZ such that Ax d can be solved in polynomial time when n is fixed. Before this result was published, only the cases n = 1; 2 were known to be polynomially solvable. The idea behind Lenstra s algorithm can be explained ....
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H.W. Lenstra, Jr. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research 8, 538--548.
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H. W. Lenstra, Jr., Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), no. 4, 538--548.
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H. W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538--548, 1983.
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H.W. Lenstra Jr., Integer programming with a fixed number of variables, Math. Op. Res. 8 (1983), 538-548.
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H. W. Lenstra. Integer Programming with a Fixed Number of Variables. Math. Oper. Res., 8:538---548, 1983.
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H. W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538--548, 1983.
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H.W. Lenstra, Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. 8 (1983), 538--548. 18
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H.W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538-- 548, 1983.
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H.W. Lenstra, Integer programming with a fixed number of variables, Mathematics of Operations Research, Vol. 8, pp. 538-548, 1983
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H.W. Lenstra Jr., Integer programming with a fixed number of variables, Mathematics of Operations Research, 8(1983), 538--548.
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