M. Gruber, C.G. Lekkerkerker, "Geometry of Numbers", North-Holland, 1987  Home/Search   Document Not in Database   Summary   Related Articles   Check

....discriminant of k and r 1 (resp. r 2 ) the number of real (resp. imaginary ) places of k. We particularly write # n (k) for # n,1 (k) Newman ( N, XI] and Icaza ( I] also considered # n (k) based on Humbert s reduction theory. Tables below show the known explicit values of # n (k) cf. BCIO] [G L], N] n 2 3 4 5 6 7 8 # n 2 # 3 # 2 # 2 # 8 64 3 # 64 2 1 2 3 7 11 2 3 5 # 2 (Q( # d) # 2 2 # 6 2 # 21 3 # 22 2 2 2 # 6 3 2 2 # 5 By using the Voronoi theory, Coulangeon [Co] proved that # n (k) is an algebraic number for all n if the class number of k is equal to ....

P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, 2nd ed., North-Holland, 1987.

....In section 4, analysis and comments are made on the feasibility of an automatic method and some possible directions to extend these results to any number of parameters and variables. 2 Some arithmetical and geometrical notions We rst recall some basic notions dealing with geometry of numbers [12, 14] and enumerative combinatorics [15, 16] Then some more specic concepts, closely related to the content of the paper, are introduced. A linearly parameterized convex polytope in the set Z of integer d uples is dened by: j A Delta x B Delta N cg where N is a symbolic p vector of ....

P.M. Gruber and C.G. Lekkerkerker. Geometry of Numbers. North-Holland, Amsterdam, 1987.

....of all the vertices which had to be integral. Hence it yields more complex computations than the powerful mathematical results used in this paper, but it applies with any number of parameters. 2. Definitions and assumptions We first recall some basic notions dealing with geometry of numbers [17, 23, 6] and enumerative combinatorics [25, 26] Then some more specific concepts, closely dedicated to the scope of the paper, are introduced. 2.1. Some basic notions A convex polyhedron P is a subset of R that is the intersection of a finite number of closed halfspaces. A bounded convex polyhedron ....

..... The set of all the integral points is called standard lattice denoted Z and any lattice L consists of the points G Delta z, z 2 Z , i.e. L = G Delta Z . So we get the most general lattice L by subjecting the standard lattice Z to an arbitrary non singular transformation G [17]. Hence, without loss of generality, only the standard lattice Z is considered in the following. The enumerator of any set S is the number of points of S Z . The intersection between a lattice L and the convex hull of a finite number of points z 1 , z m 1 in R is called a ....

P.M. Gruber and C.G. Lekkerkerker. Geometry of Numbers. North-Holland, Amsterdam, 1987.

....Hermite s constant: 1 (L) 1 (L ) fl d : A more difficult transference theorem (see [9] ensures that for all 1 r d: r (L) d Gammar 1 (L ) d: Both these transference bounds are optimal up to a constant. More information on lattice theory can be found in numerous textbooks, such as [65, 131, 92]. 2.2 Algorithmic problems In the rest of this section, we assume implicitly that lattices are rational lattices (lattices in Q ) and d will denote the lattice dimension. The most famous lattice problem is the shortest vector problem (SVP) given a basis of a lattice L, find u 2 L such that ....

M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North-Holland, 1987.

....useful values in analysis and transformation of scientific programs: they are illustrated with many examples and compared with related works [9, 22, 11, 12, 21, 16] 2 The polytope model 2. 1 Definitions and assumptions We first recall some basic notions dealing with geometry of numbers [10, 17] and enumerative combinatorics [19, 20] Then some more specific concepts, closely dedicated to the scope of the paper, are introduced. In the following, Q denotes the set of rational numbers. A convex polyhedron P is a subset of Q that is the intersection of a finite number of closed ....

....whose columns are the vectors g 1 ; g d , called the dilatation matrix. Any lattice L consists of the points G Delta z, z 2 Z , i.e. L = G Delta Z . So we get the most general lattice L by subjecting the standard lattice Z to an arbitrary non singular transformation G [10]. Hence, without loss of generality, only the standard lattice Z is considered in the following. The enumerator of any set S is the number of points of S L . The intersection between a lattice L and the convex hull of a finite number of points z1 , zm 1 in Q is called a ....

P.M. Gruber and C.G. Lekkerkerker. Geometry of Numbers. North-Holland, Amsterdam, 1987.

.... for the upper bound) log(d) 2e o(1) fl d 1:744d 2e (1 o(1) Minkowski proved more generally: Theorem 1 (Minkowski) For all d dimensional lattice L and all r d : i (L) fl d vol(L) r=d More information on lattice theory can be found in numerous textbooks, such as [53, 108, 76]. 2.2 Algorithmic problems In the rest of this section, we assume implicitly that lattices are rational lattices (lattices in Q ) and d will denote the lattice dimension. The most famous lattice problem is the shortest vector problem (SVP) which was apparently first stated by Dirichlet in ....

M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North-Holland, 1987.

....space and a given linear parallelization. This rate is parameterized by the size parameter and the execution instant. In section 6, conclusions and comparisons with related works are given. 2: Definitions and assumptions We first recall some basic notions dealing with geometry of numbers [9, 17, 1] and enumerative combinatorics [18, 19] Then some more specific concepts, closely related to the content of the paper, are introduced. 2.1: Polyhedra and polytopes Let Q denote the set of rational numbers and Z the set of integers. A convex polyhedron is defined by a finite set of linear ....

....whose columns are the vectors g 1 ; g d , called the dilatation matrix. Any lattice L consists of the points G Delta z, z 2 Z , i.e. L = G Delta Z . So we get the most general lattice L by subjecting the standard lattice Z to an arbitrary non singular transformation G [9]. Hence, without loss of generality, only the standard lattice Z is considered in the rest of the paper. The enumerator of any set E is the number of points in E L . The intersection between a lattice L and a polytope is called a lattice polytope P of L . A lattice polytope P of ....

P.M. Gruber and C.G. Lekkerkerker. Geometry of Numbers. North-Holland, Amsterdam, 1987.

....cases we considered, our method was faster. 1 This includes among others [1, 4, 8, 12] 2 The Boneh Durfee Lattice In this section we review the lattice attack by Boneh and Durfee on low exponent RSA. For an introduction into lattice theory and lattice basis reduction, we refer to the textbooks [9, 17]. Descriptions of Wiener s RSA attack and the method of Coppersmith can be found in [20, 6] For a good overview of RSA attacks, we refer to a survey article of Boneh [2] Let d e . We assume that the size of e is in the order of the size of N . If e is smaller, the attack of Boneh and Durfee ....

M. Gruber, C.G. Lekkerkerker, \Geometry of Numbers", North-Holland, 1987

....a critical lattice for s = 1, 2, and 3. The consequence for us is that a bound exists that is speci c for lattice rules: N N CL (s; s s (s) 1.4) Clearly (s) 1. The only known values of (s) are (1) 2) 1 and (3) 18=19. In the literature on geometry of numbers [GL87], s) is known as the density of closest (or densest) lattice packing for the s dimensional octahedron. Nontrivial upper bounds for (s) s 4, appear to be unknown. Every lattice rule provides a lower bound for (s) Examination of our recent results in Table 2 establishes (4) 512 621 . ....

....d) We may restate this as follows: Q( d(Q( 1 = min h2 h6=0 jhj: 2. 2) This equation relates the location of points h 2 with the enhanced degree of Q( We may use classical terminology to reexpress the import of this equation in terms taken from the geometry of numbers [GL87]. De nition 2.2. Classical) A lattice L is admissible with respect to a region if all its elements (other than the origin) lie outside Such a lattice is conventionally known as an 32747 191 lattice. Applied to our region h 2 s; when jhj ; 2.3) we have the following de nition. De ....

P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 1987.

.... Input ports corresponding to the same argument of the function are grouped together and form an input port domain (IPD) they have in common that they receive data from the same output port domain (OPD) 1 This simplification can be extended easily to the more general case of Z polyhedra [10] [11]. B. Edge domain An edge domain is the ordered pair #v i ;v j #of node domains together with the ordered pair #p i ;p j # of port domains where p i is the OPD of v i and p j the IPD of v j .This ordered pair corresponds with a data dependency in a DG. The data dependency can be expressed using ....

P.M. Gruberr and C.G. Lekkerkerker, Geometry of Numbers, North-Holland, Amsterdam, 1987.

....j det(a 1 ; a d )j= det( 2 N is called the index of fa 1 ; a d g with respect to . This value equals the number of cosets of the lattice fz 1 a 1 Delta Delta Delta z d a d : z i 2 Zg in the additive group . For more information about lattices we refer to [GL87]. Let C = fa 1 ; a m g, a i 2 be a pointed cone. For h 2 H (C) the number g C (h) min ( m X i=1 i : h = m X i=1 i a i ; i 0; 1 i m ) is called the height of h. By (1.2) we have a trivial upper bound of g C (h) m. This bound can be improved easily, since by ....

P.M. Gruber and C.G. Lekkerkerker, Geometry of numbers, 2nd ed., NorthHolland, Amsterdam, 1987.

....) each of length l, and a number T , find a subset S ae f1; mg such that P i2S a i = T mod 2 l . The subset sum problem can be viewed as that of inverting the function f(a; S) a; P i2S a i mod 2 l(n) 2. 2 Lattices The fundamental concepts concerning lattices can be found in [8, 18, 19]. If a1 ; an are linearly independent vectors in IR n , then we say that the set f P n i=1 k i a i jk1 ; kn 2 ZZg is a lattice in IR n . We will denote this lattice by L(a1 ; an ) The set a1 ; an is called a basis of the lattice; its length is max 1in ka i k. The ....

P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers, North-Holland, 1987

....arise from symmetries of the conorm function. Again, this is false for the Selling parameters. There are several reasons for studying Voronoi cells of lattices. Besides the applications to packing, covering and quantizing problems (see for example Barnes Sloane 1983; Conway Sloane 1988a; Gruber Lekkerkerker 1987; Ryskov Baranovskii 1976, 1979) there are connections with the theory of tilings. Following Gruber Lekkerkerker (1987 p. 168) we define an n dimensional parallelotope (or parallelohedron if n = 3) be a convex body S which admits a lattice tiling (in other words there is a lattice L such that ....

....for studying Voronoi cells of lattices. Besides the applications to packing, covering and quantizing problems (see for example Barnes Sloane 1983; Conway Sloane 1988a; Gruber Lekkerkerker 1987; Ryskov Baranovskii 1976, 1979) there are connections with the theory of tilings. Following Gruber Lekkerkerker (1987 p. 168) we define an n dimensional parallelotope (or parallelohedron if n = 3) be a convex body S which admits a lattice tiling (in other words there is a lattice L such that the translates S u, uL cover R n while their interiors are disjoint) Voronoi (1907 1908) conjectured that every ....

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Gruber, P. M. and Lekkerkerker, C. G. 1987. "Geometry of Numbers", North-Holland, Amsterdam, 2nd ed.

.... a, b, c . T has no points of L in its interior. For non strictly convex bodies there might be lattice points in the boundary. This case can be avoided by assuming that T is chosen with minimal area. It follows conv a, b, c #L = a, b, c and so c b, a b is a basis of L (cf. [GL]) There are coe#cients #, #, # # 0 with # # # = 1 and #a #b #c = z. By the symmetry of K we know 2z a # bd (z #K) and assuming # 0 allows us to write 2z a = 2 1 #) z (# #) b (# #) c. For # 1 2 this is a convex combination of b, c and z with a positive ....

....# d 3 4 (d 1)#(# 1 2) d 1)#(# 1 2) 2) to all the vertices of P . Further we have p = y a 1 . y a d 1 x 2 a d , with x = 1 (d 1)#(# 1 2) and y = d 3 4 # # 2 (d 1) # 1) Let = L) be the covering radius of the lattice L (cf. [GL]) We can choose # such that (i) R, since (Z d ) # d 2, ii) x, y 0, iii) d 1)y x 2 1. Conditions (ii) and (iii) imply that p is contained in the interior of the simplex P and has equal distance R to all of its vertices. Thus the d 1 facets of P are contained in balls with ....

P.M. Gruber, C.G. Lekkerkerker, `Geometry of Numbers', North Holland, Amsterdam, 1987.

....because s becomes larger. 2.2.2 Lattice theory. Coppersmith s technique, like many public key cryptanalyses, is based on lattice basis reduction. We only review what is strictly necessary for this paper. Additional information on lattice theory can be 3 found in numerous textbooks, such as [6, 17]. For the important topic of lattice based cryptanalysis, we refer to the recent survey [12] We will call lattice any subgroup of some (Z n ; which corresponds to the case of integer lattices in the literature. Consequently, for any integer vectors b 1 ; b r , the set L(b 1 ; ....

M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North-Holland, 1987.

....show that these incremental construction algorithms are fast in practice. 2. Preliminaries. In this section we recall some basic definitions of the geometry of numbers and fix notation for our further discussion. A detailed account of the geometry of numbers is given by GRUBER and LEKKERKERKER [GL87]. Let E be a d dimensional Euclidean space. Its scalar product is denoted by ( E E R 0 and the associated norm by k k : p ( The d dimensional unit ball fx 2 E : kxk 1g ist denoted by B d . boris.hemkemeier math.uni dortmund.de) y ....

PETER M. GRUBER, CORNELIS G. LEKKERKERKER. Geometry of numbers. North-Holland, 1987.

....both theorems use elementary geometric arguments. The lower bound in Theorem 1.2 is shown by Q 0 . For the upper bound we have to use something like Minkowski s theorem, and indeed in one of the stages of the proof we argue similarly to the usual proof of Minkowski s theorem, see e.g. PA95] or [Gru95]. Both proofs have analogous global structure: we treat separately thin bodies K, showing that P (K) is much larger for them than for the optimal value. In these arguments, we estimate the contribution to P (K) of an individual lattice point and then we sum up over all lattice points. In the ....

....of C are at distance O( p ) from C Gammaf loat . Lemma 4.6 Let K be a body with vol(K) 4 Gamma and K Z 2 = f0g, where 0 is sufficiently small. Then there is a tile T with T Gammaf loat K T . Proof. This is similar to the usual proof of Minkowski s theorem (see e.g. [Gru95] or [PA95] Let K 0 = 1 2 K and consider the system of bodies K 0 Z 2 ; since K Z 2 = f0g these bodies are pairwise disjoint. We consider the smallest t 1 1 such that the boundaries of the bodies in t 1 K 0 Z 2 first touch; say that t 1 K 0 touches t 1 K 0 z 1 . Let K 1 = t 1 K 0 ....

P. M. Gruber. Geometry of numbers. In Handbook of Convex Geometry, NorthHolland 1995, pages 739--764.

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M. Gruber, C.G. Lekkerkerker, "Geometry of Numbers", North-Holland, 1987

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P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers. North-Holland, 1987.

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P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, North-Holland, Amsterdam, 1987.

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P. M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North Holland, second edition, 1987.

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P. M. Gruber. Geometry of numbers. In P. Gruber and J. Wills, editors, Handbook of Convex Geometry, volume B, chapter 3.1, pages 739--763. Elsevier Science Publishers B.V., 1993.

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Peter M. Gruber. Geometry of numbers. In P.M. Gruber and J.M. Wills, editors, Handbook of Convex Geometry, volume B, chapter 3.1, pages 739--763. Elsevier Science Publishers B.V., 1993.

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P. M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North Holland, second edition, 1987.

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P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, 2nd ed., North-Holland, 1987.

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