R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. ACM Symp. Theory of Computing, (Boston, MA), pp. 193--206, Apr. 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....searched in these algorithms is a parallelepiped, with its axes parallel to the basis vectors. In general, the development of closest point algorithms follows two main branches, inspired by two seminal papers: Pohst [63] in 1981 examined lattice points lying inside a hypersphere, whereas Kannan [46] in 1983 used a rectangular parallelepiped. Both papers later appeared in revised and extended versions, Pohst s as [30] and Kannan s (following the work of Helfrich [42] as [47] The Pohst and Kannan strategies are discussed in greater detail in Section III A. A crucial parameter for the ....

....plane algorithm was proposed by Klein [49] The other three methods all find the optimal (closest) point. Scanning all the layers in (13) and supplying each dimensional search problem with the same value of regardless of , yields the Kannan strategy. Variants of this strategy [12] 42] [46], 47] differ mainly in how the bounds are chosen for . In this context, a recent improvement by Blmer [14] seems particularly promising. Geometrically, the Kannan strategy amounts to generating and examining all lattice points within a given rectangular parallelepiped. The dimensional decoding ....

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R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. ACM Symp. Theory of Computing, Boston, MA, Apr. 1983, pp. 193--206.

.... is NPhard [1] Some other relevant results regarding approximate solutions for the LDP can be found in [2] 13] and [18] The fastest (best upper bound on the complexity) known algorithm for solving the LDP for a general lattice is due to Kannan [16] an improved version of his earlier work in [15]. Prior to [16] Helfrich [14] also made some improvements in the running time of some of the algorithms in [15] In [16] Kannan uses the same reduced basis as used in this paper, The reduced basis used by Kannan has an extra condition on the value of the G S coefficients i;j , i.e. j i;j ....

.... and [18] The fastest (best upper bound on the complexity) known algorithm for solving the LDP for a general lattice is due to Kannan [16] an improved version of his earlier work in [15] Prior to [16] Helfrich [14] also made some improvements in the running time of some of the algorithms in [15]. In [16] Kannan uses the same reduced basis as used in this paper, The reduced basis used by Kannan has an extra condition on the value of the G S coefficients i;j , i.e. j i;j j1=2for 1 j in. However, this condition does not affect the G S orthogonalization of the basis. and shows that ....

R. Kannan, "Improved algorithms on integer programming and related lattice problems," in Proc. 15th Annu. ACM Symp. on Theory of Computing, 1983, pp. 193--206.

....counterexamples) but it can also sometimes be proved, notably in the case of lattices arising from low density knapsacks. For exact SVP, the best algorithm known (in theory) is the recent randomized O(d) time algorithm by Ajtai et al. 6] which improved Kannan s superexponential algorithm [77, 79] (see also [67] For exact CVP, the best algorithm remains Kannan s super exponential algorithm [77, 79] with running time O(d log d) see also [67] for an improved constant) 3 Finding small roots of multivariate linear equations One of the early and most natural applications of lattice ....

....knapsacks. For exact SVP, the best algorithm known (in theory) is the recent randomized O(d) time algorithm by Ajtai et al. 6] which improved Kannan s superexponential algorithm [77, 79] see also [67] For exact CVP, the best algorithm remains Kannan s super exponential algorithm [77, 79], with running time O(d log d) see also [67] for an improved constant) 3 Finding small roots of multivariate linear equations One of the early and most natural applications of lattice reduction in cryptology was to find small roots of multivariate linear equations, where the equations are ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of 15th STOC, pages 193--206. ACM, 1983.

....basis B. Point lattices are pervasive structures in mathematics, and have been studied extensively. See [25] for example, for a survey of the field. In the area of combinatorial mathematics alone it is possible to phrase many different problems as questions about lattices. Integer programming [20], factoring polynomials with rational coefficients [27] integer relation finding [16] integer factoring [35] and diophantine approximation [36] are just a few of the areas where lattice problems arise. In some cases, such as integer programming existence problems, it is necessary to determine ....

R. Kannan, Improved algorithms for integer programming and related lattice problems, Proc. 15 Symp. Theory. of Comp. (1983), 193-206.

....of that lattice is of the form (v Gamma u; 1) where u is a closest vector to v, in the original lattice . Depending on the lattice, one should choose a coefficient different than 1 in (v; 1) For exact SVP or CVP, the best algorithms known (in theory) are Kannan s super exponential algorithms [63, 65], with running time 2 O(d log d) 3 Knapsacks Cryptology and lattices share a long history with the knapsack (also called subset sum) problem, a well known NP hard problem considered by Karp: given a set fa 1 ; a 2 ; an g of positive integers and a sum s = P n i=1 x i a i , where x ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of 15th STOC, pages 193--206. ACM, 1983.

....arg,nMin II x II 2 = arg, n Min II II 2 (7) where n is the signal constellation. This is equivalent of searching the closest vector in lattice A (in the 2NTdimensional Euclidean space R NT) to the received vector x. Till now many algorithms have been proposed to solve this well known problem [7] [8]. We modify the CPS algorithm, proposed in [6] to make this algorithm fast and low complex. just proportional to Ilwk, II 2, and thus the post detection SNRs are proportional to 1 llwk, II 2 wherew, is kith row of . Assuming that all components of s belong to the same constellation, s , with ....

R. Kannan, "Improved Algorithm for integer programming and related lattice problems,' in Proc. Of the ACM Symposium on Theory of Computing, pp. 1932.

.... by Babai [2] namely s X i=1 (w i u i ) 2 # 2 s min ( s X i=1 (z i u i ) 2 , z = z 1 , z s ) # L ) Moreover, because in our applications the dimension s is fixed we can also use algorithms which find the closest vector in a lattice in polynomial time (see [11, 1]) The following result can be interpreted as a statement about short vectors in a certain two dimensional lattice, and thus has some links with Lemma 1. However, as usual with two dimensional lattices, continued fractions provide stronger statements and shorter proofs. Lemma 2. There exists a ....

R. Kannan, Improved algorithms for integer programming and related lattice problems, Proc. 15th ACM Symp. on Theory of Comput., Boston, MA, May 25-27, 1983, 193--206.

....vector will not be much shorter than the length of many other vectors (24 vs. 40 for p = 197, h = 24 see the appendix) Shortest vector algorithms of the rst type will therefore nd the special vector in this case. However, the best shortest vector algorithm known currently is the one of Kannan [15], and its performance is no better, in our application, than the brute force attack sketched in 7.3. The proximity of the special vector length (24) to lengths of many other vectors (40) should prevent the ecient short vector algorithms of the second type from nding the special vector. 15 The ....

Kannan, R., \Improved algorithms for integer programming and related lattice problems ", Proceedings of the Fifteenth Annual Symposium on Theory of Computing, ACM, pp. 193-206, 1983.

....version of the algorithm, in which step 5 is omitted. The computational complexity in the training algorithm is, for high d , dominated by the so called closest point problem, the search for the closest lattice point of the training data, in step 3. Algorithms have been developed by Kannan [38] and Agrell and Eriksson [39] It has been theoretically proved that the problem is NP hard, see, e.g. 40] but the complexity is nevertheless not overwhelming. To indicate the order of magnitude, we mention that with one implementation of the algorithm in [39] the average time to find the ....

R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. Annual ACM Symposium on Theory of Computing, pp. 193--206, Boston, MA, Apr. 1983.

.... have numerous theoretical and practical applications in computational number theory and cryptography: Factoring polynomials with rational coefficients [LLL82] finding linear Diophantine approximations [Lag80] breaking various cryptosystems [Lag83,Sch95,VGT88] and integer linear programming [Kan83,Len83]. In 1982, Lenstra, Lenstra and Lovasz [LLL82] gave a powerful approximation reduction algorithm. It depends on a real approximation parameter 2]1; 2[ and is called LLL( It is a possible generalization of its 2 dimensional version, which is the famous Gauss algorithm. The celebrated LLL ....

....also O( 2= p 3) n 2 =2 log M ) The first formulation (i) is based on Proposition 17, and Lemmata 4, 16. The proof of the second formulation (ii) uses also Lemma 18 (which is proved under a very plausible heuristic) The next Lemma is an adaptation of ones used by Babai, Kannan and Schnorr [Bab86,Kan83,Sch87] when finding a shortest vector in a lattice with a Lovasz reduced basis on hand. Lemma 16. Let t 2]1; 2[ be a real parameter and L be a lattice generated by a basis b : b 1 ; b n ) which is not necessarily integral and whose vectors are of arbitrary 6 The naive bound is obtained ....

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R. Kannan. Improved algorithm for integer programming and related lattice problems. In 15th Ann. ACM Symp. on Theory of Computing, pages 193--206, 1983.

.... the reduction problem have numerous theoretical and practical applications in computational number theory and cryptography: Factoring polynomials with rational coecients [11] nding linear Diophantine approximations [9] breaking various cryptosystems [10, 14] 19] and integer linear programming [6, 12]. In 1982, Lenstra, Lenstra and Lov asz [11] gave a powerful approximation reduction algorithm. It depends on a real approximation parameter t 2]1; 2[ and is called LLL(t) It begins with the Gram Schmidt orthogonalizing process, then it aims to ensure, for each index i; 1 i p 1, a lower bound ....

....is p 2 log t M p. Daud e and Vall ee [4] exhibited an upper bound for the average number of iterations (in a simple natural model) which asymptotically equals (p 2 =2) log t n p. There is already a wide number of variations around the LLL algorithm (due for instance to Kannan or Schnorr [6, 13]) whose goal is to nd lattice bases with sharper Euclidean properties than the original LLL algorithm. Here, we choose the other direction, and we present two new variations around the LLL reduction that are a priori weaker than the usual LLL reduction. They are called Schmidt reduction and ....

Kannan, R. Improved algorithm for integer programming and related lattice problems. In 15th ACM Symp. on Theory of Computing (1983), pp. 193-206.

....searched in these algorithms is a parallelepiped, with its axes parallel to the basis vectors. In general, the development of closest point algorithms follows two main branches, inspired by two seminal papers: Pohst [38] in 1981 examined lattice points lying inside a hypersphere, whereas Kannan [27] in 1983 used a rectangular parallelepiped. Both papers later appeared in revised and extended versions, Pohst s as [20] and Kannan s (following the work of Helfrich [25] as [28] The Pohst and Kannan strategies are discussed in greater detail in Section III A. A crucial parameter for the ....

....one, but the error can be bounded. The other three methods all find the optimal (closest) point. Scanning all the layers in (13) and supplying each (n Gamma1) dimensional search problem with the same value of ae n Gamma1 regardless of un , yields the Kannan strategy. Variants of this strategy [10,25,27,28] differ mainly in how the bounds ae k are chosen for k = 1; n. Geometrically, the Kannan strategy amounts to generating and examining all lattice points within a given rectangular parallelepiped. The n dimensional decoding error vector x Gamma x consists, in the given recursive ....

R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. of the ACM Symposium on Theory of Computing, pp. 193--206, Boston, MA., Apr. 1983.

....of that lattice is of the form (v Gamma u; 1) where u is a closest vector to v, in the original lattice . Depending on the lattice, one should choose a coefficient different than 1 in (v; 1) For exact SVP or CVP, the best algorithms known (in theory) are Kannan s super exponential algorithms [63, 65], with running time 2 O(d log d) 3 Knapsacks Cryptology and lattices share a long history with the knapsack (also called subset sum) problem, a well known NP hard problem considered by Karp: given a set fa 1 ; a 2 ; an g of positive integers and a sum s = P n i=1 x i a i , where x ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of 15th STOC, pages 193--206. ACM, 1983.

....case. Let b denote the maximum number of bits required to specify the coefficients in a constraint or in the objective function. Then the algorithm performs O(2 d dn 8 d d p n log n log n) d O(d) b log n expected number of operations, each involving at most d O(1) b bit numbers. See [117] for an earlier result on integer programming in fixed dimensions. Geometric Optimization June 6, 2000 Abstract Linear Programming 10 4 Abstract Linear Programming In this section we present an abstract framework that captures linear programming, as well as many other geometric optimization ....

R. Kannan, Improved algorithms for integer programming and related lattice problems, Proc. 15th Annu. ACM Sympos. Theory Comput., 1983, pp. 193--206.

....geometry of numbers, the development of the LLL lattice reduction algorithm [59] had a deep impact in many areas of computer science, ranging from integer programming, to cryptography. Using the LLL reduction algorithm it was possible to solve integer programming in a fixed number of variables [60, 59, 44], factor polynomials over the rationals [59, 57, 72] finite fields [56] and algebraic number fields [58] disprove century old conjectures in mathematics [65] break the Merkle Hellman crypto system [74, 2, 11, 50, 51, 63] check the solvability by radicals [55] solve low density subset sum ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 193--206, Boston, Massachusetts, 25--27 Apr. 1983.

....fi = 1= n Delta) O(n) for 1 i n and for any two numbers fi; fl 2 A that occur in the computation. The integer programming problems can thus be transformed into equivalent problems in n dimensional integer lattices which can be represented by (n log Delta) O(1) bits. By a theorem of Kannan [13] each of those problems can be solved in time n O(n) log Delta) O(1) Hence the computation of all neighbors of A takes time (n log Delta) O(n) In the depth first search tree, each reduced principal ideal is represented in terms of the uniquely defined n Theta n integer transformation ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In 15th Annual ACM Symposium on Theory of Computing, pages 193--206, 1983.

....probabilistic variant of Sauer s lemma, that greatly simplifies Ajtai s original proof. 1. Introduction The Shortest Vector Problem (SVP) is a famous problem in mathematics that underlies the solution of many other important optimization and combinatorial problems, such as integer programming [19, 18, 14], polynomial factorization [18, 16, 20, 17] low density subset sum [15, 10, 8] cryptanalisys [21, 7, 13, 11, 5] just to say a few. In this paper we show that approximating the shortest vector in a lattice within any constant factor less than p 2 is NP hard for randomized reductions. We also ....

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 193--206, Boston, Massachusetts, 25--27 Apr. 1983.

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R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. ACM Symp. Theory of Computing, (Boston, MA), pp. 193--206, Apr. 1983.

No context found.

R. Kannan. Improved algorithms for integer programming and related lattice problems. In 15th Annual ACM Symposium on Theory of Computing, pages 193--206, 1983.

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R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proc. ACM Symp. Theory of Computing, (Boston, MA), pp. 193--206, Apr. 1983.

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R. Kannan. Improved Algorithms for Integer Programming and Related Lattice Problems. In Proceedings of the 15th Annual ACM Sympos. Theory of Computing, 1983; pages 193-- 206, .

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R. Kannan. Improved Algorithms for Integer Programming and Related Lattice Problems. 15th Annual ACM Sympos. Theory of Computing, pages 193---206, 1983.

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R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 193--206, Boston, Massachusetts, 25--27 Apr. 1983.

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R. Kannan, "Improved algorithms for integer programming and related lattice problems," in Proceedings of the ACM Symposium on the Theory of Computing, (Boston), pp. 193--206, April 1983.

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R. Kannan, "Improved algorithms for integer programming and related lattice problems," Proc. of 15th Annual ACM Symp. on Theory of Computing, pp. 193--206, Boston, MA, Apr. 1983.

No context found.

R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. 15 th Symp. Theory. of Comp., pages 193--206, 1983.

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