B. Helfrich, "Algorithms to construct Minkowski reduced and Hermit reduced lattice bases," Theoretical Computer Sci. 41, pp. 125--139, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 performance of these algorithms is the initial size of the search region. Some suggestions to this point were given in [62] 78] for the Pohst strategy and in [12] for the ....

....of Hermite, whereas there is always at least one Minkowski reduced basis for each of them. Minkowski reduction has received much attention, particularly in number theory [18, pp. 27 28] 28, pp. 83 84] Algorithms to compute a Minkowski reduced basis of an arbitrary lattice may be found in [1] [42]. Two types of reduction that are more widely used in practice are Korkine Zolotareff (KZ) reduction and Lenstra Lenstra Lovsz (LLL) reduction. One reason for their popularity is that with both of those criteria, the dimensional reduction problem can be recursively reduced to an dimensional ....

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B. Helfrich, "Algorithms to construct Minkowski reduced and Hermite reduced lattice bases," Theor. Comput. Sci., vol. 41, no. 2--3, pp. 125--139, 1985.

.... 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 j1=2for 1 j in. However, this ....

B. Helfrich, "Algorithms to construct Minkowski reduced and Hermite reduced lattice bases," Theor. Comput. Sci., vol. 41, pp. 125--139, 1985.

....of Koy Schnorr [KS01a,KS02] achieves the same apfa ( within O(n log n) steps. Finding very short lattice vectors requires additional search beyond LLLtype reduction. The algorithm of Kannan [K83] nds the shortest lattice vector in time n by a diligent exhaustive search, see [H85] for an n 2 o(n) time algorithm. The recent probabilistic sieve algorithm of [AKS01] runs in 2 average time and space, but is impractical as the exponent O(n) is about 30 n. Schnorr [S87] has generalized the LLL algorithm in various ways that repeatedly construct short bases of ....

B. Helfrich, Algorithms to construct Minkowski reduced and Hermite reduced bases. Theor. Comp. Sc. 41, pp. 125-139, 1985.

....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 reduction in ....

.... 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 either integer equations or modular equations. ....

B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced bases. Theoretical Computer Science, 41:125--139, 1985.

.... The recent LLLtype reduction of Koy Schnorr improves the O(n ) time to O(n log n) This algorithm has not yet been implemented, the best LLL type code has time bound O(n ) see [KS01a, KS01b, KS02] The algorithm of Kannan [K83] nds the shortest lattice vector in time n , Helfrich [H85] improved the time bound to n n=2 o(n) The recent probabilistic sieve algorithm of [AKS01] runs in 2 average time and space, but is impractical as the exponent O(n) is about 30 n. A lattice basis b1 ; bn is a BKZ basis of block size k if all blocks b i ; b i 1 ; b i k 1 of size k ....

B. Helfrich, Algorithms to construct Minkowski reduced and Hermite reduced bases. Theor. Comp. Sc. 41, pp. 125-139, 1985.

....to obtain a (1 ffl) approximation for CVP in n dimensions. CVP is a well studied problem from many points of view. For the problem of computing the closest vector exactly, Kannan obtained an n O(n) time deterministic algorithm [10] and the constant in the exponent was improved by Helfrich [9]. Recently, Blomer obtained an O(n ) time deterministic algorithm to compute the closest vector exactly [2] For the problem of approximating the closest vector, using the LLL algorithm [12] Babai obtained a (3= p 2) n approximation algorithm that runs in polynomial time [3] Using a 2 ....

B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced bases. Theoretical Computer Science, 41:125--139, 1985.

....this algorithm improves the approximation factor obtained by the LLL algorithm to 2 n(log log n) 2 = log n . Kannan [6] obtained a 2 O(n log n) time algorithm for exactly computing the shortest vector. The constant in the exponent of Kannan s algorithm was improved to about 1 2 by Helfrich [5]. On the hardness front, the shortest lattice vector problem for the L1 norm was shown to be NP complete by van Emde Boas [4] Recently, Ajtai [2] proved that the shortest lattice vector problem under the L 2 norm is NP hard under randomized reductions. Micciancio [11] showed that the ....

....defined by b 2 (2) b n (2) where b i (j) denotes the projection of b i onto the space which is the orthogonal complement of b 1 ; b i . Then, for i = 2; n, we output a i , the shortest lattice vector whose projection onto the space orthogonal to a 1 is b i , as in [5], to satisfy j 1;i j 1=2, where 1;i = ha 1 ; a i (i)i=kb i (i)k. Clearly, the running time of this algorithm is at most 2 O(n) Xi Corollary 11 There is a randomized algorithm that given an integer k 0 and an n dimensional lattice L with shortest vector between 1 and 2, finds a non zero ....

B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced bases. Theoretical Computer Science, 41:125--139, 1985.

....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 performance of these algorithms is the initial size of the search region. Some suggestions to this point were given in [37, 49] for the Pohst strategy and in [10] 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 ....

B. Helfrich, "Algorithms to construct Minkowski reduced and Hermite reduced lattice bases," Theoretical Computer Science, vol. 41, nos. 2--3, pp. 125--139, 1985.

....easily verified to be sympletic. Then A Q = n # i =1 Q(# i )Q(# # i ) 0. The contrary is trivial since by the classification theorem Q # = n #H 0 . 5 Computational methods for quadratic forms References: Lag59, Jac57a, Jac57b, Gun81, Fro94, Fro06, LT85, Bri56, Leb56] References: [Hay68, Cot74, BP71, BKP76, BK77, BG, LLL82, Sch84, Hel85a, Hel85b, Bab85, Kan83a, Kan83b, Kan86, Lag80, Sch86a, Sch86b, vEB81, BK84, Die75, BK79, Poh81, AG85, Kan, FP85, Kal83] 6 Exercises 22 6 Exercises 1 Show that the group of self isomorphisms of the bilinear form given by the ndimensional identity matrix is isomorphic to the group of n n orthogonal matrix over Z. 2 Prove that the following statements are equivalent: 1. #(#,#) 0 for all # # M , then # ....

B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced lattice bases. Theoretical Comp. Science, 41:125--139, 1985. 21

....easily verified to be sympletic. Then A Q = n # i =1 Q(# i )Q(# # i ) 0. The contrary is trivial since by the classification theorem Q # = n #H 0 . 5 Computational methods for quadratic forms References: Lag59, Jac57a, Jac57b, Gun81, Fro94, Fro06, LT85, Bri56, Leb56] References: [Hay68, Cot74, BP71, BKP76, BK77, BG, LLL82, Sch84, Hel85a, Hel85b, Bab85, Kan83a, Kan83b, Kan86, Lag80, Sch86a, Sch86b, vEB81, BK84, Die75, BK79, Poh81, AG85, Kan, FP85, Kal83] 6 Exercises 22 6 Exercises 1 Show that the group of self isomorphisms of the bilinear form given by the ndimensional identity matrix is isomorphic to the group of n n orthogonal matrix over Z. 2 Prove that the following statements are equivalent: 1. #(#,#) 0 for all # # M , then # ....

B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced lattice bases. In K. Mehlhorn, editor, 2nd Annual Symp. Theor. Aspects of Comp. Science, volume 182 of Lect. Notes in Comp. Sci. SpringerVerlag, 1985. 21

....to estimate jjt i jj. It follows from Corollary 9 that jjt i jj p ks i i (L(A) dft(A) 1 (A) So the assertion follows from Lemma 13 and Theorem 1. Now can prove Corollary 3. Given a basis C = c 1 ; c k ) 2 ZZ m Thetak of a k dimensional integral lattice . The algorithm of [He] determines a reduced basis D 2 ZZ m Thetak such that jjd i jj p (i 3) 4 i ( 1 i k : The algorithm performs at most mk O(k) log jCj arithmetic operations on integers of binary length O(k 2 (log k log jCj) Here, jCj denotes the maximum of the absolute values of the entries of ....

....of binary length O(k 2 (log k log jCj) Here, jCj denotes the maximum of the absolute values of the entries of C. Choose a rational approximation A to B of precision p = q 0 (B; c; p (k 3) 4) Then jjA Gamma Bjj 2 Gammaq 2 (B;c;s) Hence, if we apply the reduction algorithm of [He] to C = 2 p 1 A and determine the reducing transformation T = t 1 ; t k ) 2 GL(k; ZZ) then by Theorem 2 we have jjBt i jj (1 c) p (i 3) 4 i (L) 1 i k : The transformation T can be determined by applying each reducing elementary transformation not only to the basis but ....

B. Helfrich, Algorithms to construct Minkowski reduced and Hermite reduced lattice bases, Theoretical Computer Science 41, 125-139, 1985.

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B. Helfrich, "Algorithms to construct Minkowski reduced and Hermit reduced lattice bases," Theoretical Computer Sci. 41, pp. 125--139, 1985.

No context found.

B. Helfrich, "Algorithms to construct Minkowski reduced and Hermit reduced lattice bases," Theoretical Computer Sci. 41, pp. 125--139, 1985.

No context found.

B. Helfrich, "Algorithms to construct minkowski reduced and hermit reduced lattice bases," Theoretical Computer Sci. 41, pp. 125--139, 1985.

No context found.

B. Helfrich, Algorithms to construct Minkowski reduced and Hermite reduced lattice bases, Theoretical Computer Science 41, 125-139, 1985.

No context found.

B. Helfrich, Algorithms to Construct Minkowski Reduced and Hermite Reduced Bases. Theor. Comput. Sci. 41, pp. 125-139, 1985.

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