J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142186, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A form which is reduced in the Gaussian sense is not necessarily reduced in the sense of (1) The Gaussian notion of reduction was modi ed by Seeber [13] such that a reduced form satis es (1) with 3 = 3. Gau [5] showed later that 3 = 2. The reduction algorithm of Gau was modi ed by Lagarias [7] to produce so called quasi reduced forms. They satisfy the slightly weaker condition that the rst diagonal element is at most twice the cubic root of the determinant. Lagarias proved that his modi ed ternary form algorithm runs in polynomial time. However, a quasi reduced form is not necessarily ....

....3:03 3:04 3:05 3:06 3:07 3:08 3:09 3:10 3:11 3:12 3:13 3:14 3:15 3:16 3:17 3:18 3:19 3:20 3:21 3:22 3:23 3:24 3:25 3:26 3:27 3:28 3:29 3:30 3:31 3:32 3:33 3:34 3:35 3:36 3:37 3:38 3:39 3:40 We proceed as follows. First we show that the Gaussian ternary form algorithm, in the variant of Lagarias [7], requires O(M(s) log s) bit operations. This is achieved via a re nement of the analysis given by Lagarias. Then we prove that, given a quasi reduced ternary form, it takes at most O(M(s) log s) bit operations to compute an equivalent reduced form. Therefore, a ternary form can be reduced with ....

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J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142186, 1980.

....form which is reduced in the Gaussian sense is not necessarily reduced in the sense of (1) The Gaussian notion of reduction was modified by Seeber [13] such that a reduced form satisfies (1) with # 3 = 3. Gau [5] showed later that # 3 =2. The reduction algorithm of Gau was modified by Lagarias [7] to produce so called quasi reduced forms. They satisfy the slightly weaker condition that the first diagonal element is at most twice the cubic root of the determinant. Lagarias proved that his modified ternary form algorithm runs in polynomial time. However, a quasi reduced form is not ....

....3:03 3:04 3:05 3:06 3:07 3:08 3:09 3:10 3:11 3:12 3:13 3:14 3:15 3:16 3:17 3:18 3:19 3:20 3:21 3:22 3:23 3:24 3:25 3:26 3:27 3:28 3:29 3:30 3:31 3:32 3:33 3:34 3:35 3:36 3:37 3:38 3:39 3:40 We proceed as follows. First we show that the Gaussian ternary form algorithm, in the variant of Lagarias [7], requires O(M(s)log s) bit operations. This is achieved via a refinement of the analysis given by Lagarias. Then we prove that, given a quasi reduced ternary form, it takes at most O(M(s)logs) bit operations to compute an equivalent reduced form. Therefore, a ternary form can be reduced with ....

[Article contains additional citation context not shown here]

J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142--186, 1980.

....vector from the larger vector thereby reducing its length. This normalization step is analogous to the division with remainder in the euclidean algorithm for integers. The integer k in the repeat loop of algorithm GAUSS is the nearest integer to the number (b T 1 b 2 ) b T 1 b 1 ) Lagarias [7] showed that the Gaussian algorithm is polynomial. His analysis can be used to show that GAUSS requires O(n 2 ) bit operations for 1 Algorithm. GAUSS(b 1 ; b 2 ) repeat arrange that b 1 is the shorter vector of b 1 and b 2 find k 2 Z such that b 2 kb 1 is of minimal euclidean length b 2 ....

J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142--186, 1980.

....A form which is reduced in the Gaussian sense is not necessarily reduced in the sense of (1) The Gaussian notion of reduction was modi ed by Seeber [13] such that a reduced form satis es (1) with 3 = 3. Gau [5] showed later that 3 = 2. The reduction algorithm of Gau was modi ed by Lagarias [7] to produce so called quasi reduced forms. They satisfy the slightly weaker condition that the rst diagonal element is at most twice the cubic root of the determinant. Lagarias proved that his modi ed ternary form algorithm runs in polynomial time. However, a quasi reduced form is not necessarily ....

....3:03 3:04 3:05 3:06 3:07 3:08 3:09 3:10 3:11 3:12 3:13 3:14 3:15 3:16 3:17 3:18 3:19 3:20 3:21 3:22 3:23 3:24 3:25 3:26 3:27 3:28 3:29 3:30 3:31 3:32 3:33 3:34 3:35 3:36 3:37 3:38 3:39 3:40 We proceed as follows. First we show that the Gaussian ternary form algorithm, in the variant of Lagarias [7], requires O(M(s) log 2 s) bit operations. This is achieved via a re nement of the analysis given by Lagarias. Then we prove that, given a quasi reduced ternary form, it takes at most O(M(s) log s) bit operations to compute an equivalent reduced form. Therefore, a ternary form can be reduced ....

[Article contains additional citation context not shown here]

J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142186, 1980.

.... [vEB81,Ajt97,Mic98,Cai99] Fortunately, even approximate answers to the reduction problem 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 ....

.... (ii) 8i 2 f1; 2; n 1g (1= jb i hb1 ; b i 1 i j j b (i 1) hb1 ; b i 1 i j: The optimal LLL(1) algorithm is a possible generalization of its 2 dimensional version, which is nothing but the famous Gauss algorithm, whose precise analysis is already done both in the worst case [Lag80,Val91,KS96] and in the average case [DFV97] In the sequel, a reduced basis denotes always a LLL(1) reduced basis. When we talk about the algorithm without other precision, we always mean the optimal LLL algorithm. We adopt the following notations for all integer i in f1; 2; n 1g, 8 : u i : b ....

J. C. Lagarias. Worst--case complexity bounds for algorithms in the theory of integral quadratic forms. J. Algorithms, 1:142--186, 1980.

....= 0 return (b 1 ; b 2 ) The integer k in the repeat loop of algorithm GAUSS is the nearest integer to the number (b T 1 b 2 ) b T 1 b 1 ) Figure 1 shows the effect of a normalization step. The length of the second basis vector b 2 has been reduced by subtracting integral multiples of b 1 . Lagarias (1980) showed that the Gaussian algorithm has worst case complexity O(n 3 ) where n is the size of the binary encoding of the input. Rote 1997) showed that the 2dimensional modm shortest vector problem can be reduced to the classical case. See, e.g. Yap 1999) for a thorough treatment of the ....

.... norm. Let C be a reduced basis according to (1) Then the first column of C is a shortest vector of L w.r.t. the 2 norm. Let b (1) and c (1) be the first columns of B and C respectively. It follows that p 2kc (1) k kb (1) k holds, and thus that the basis B is almost reduced . Lagarias (1980, proof of Theorem 4.2) has shown that in this case the algorithm GAUSS requires only a constant number of runs through the repeat loop to reduce B. We thus have the following corollary. Corollary 5. There exists an algorithm that computes in time O(M(n)logn) a reduced basis C of a ....

Lagarias, J. C. (1980), `Worst-case complexity bounds for algorithms in the theory of integral quadratic forms', Journal of Algorithms 1, 142--186.

....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 # ....

J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. J. of Algorithms, 1:142--186, 1980. 21

....at the top or at the bottom of Figure 4 (since jb 2 j jb 1 j and since the projection of b 2 must fall within the two vertical lines about b 1 ) iii) The length of b 1 strictly decreases in each iteration. Hence the algorithm is finite. That it is polynomial time takes a little more argument [82]. 46 (iv) The short vector b 1 produced by the algorithm satisfies jb 1 j (1:075) d(L) 1 2 ) The only known polynomial time algorithm for constructing a reduced basis in an arbitrary dimensional lattice [86] is not much more complicated than the 60 o algorithm. However, the proof of ....

J.C.Lagarias, Worst-case complexity bounds for algorithms in the theory of integral quadratic forms, Journal of Algorithms 1 (1980) 142-186.

....forms, to decide the equivalence of integral binary quadratic forms, to compute the fundamental unit of real quadratic orders, to compute in the class group of quadratic fields etc. The efficiency of the methods for solving the above problems depends heavily on the efficiency of reduction. In [Lag80] Lagarias slightly modified the classical reduction algorithm and proved that his modification reduces a form f in time O(nM(n) where n is the length of the bit string necessary to represent f (see below for a more precise definition) and M(n) is the time for multiplying two n bit numbers. This ....

....form is a polynomial f(X; Y ) aX 2 bXY cY 2 2 ZZ[X; Y ] We also write f = a; b; c) We set size(f) size(a) size(b) size(c) The discriminant of f is Delta(f ) b 2 Gamma 4ac. The form f is reducible in ZZ[X; Y ] if and only if Delta(f ) is a square in ZZ. As explained in [Lag80] the reduction of an irreducible form can be effected by one gcdcomputation of numbers of size O(size(f) Corollary 4.2.4 of [BS96] shows that gcd s can be computed in quadratic time. If Delta(f ) 0 then f is called indefinite. For Delta(f ) 0 the form f is positive definite if a 0, and ....

[Article contains additional citation context not shown here]

J.C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. J. Algorithms, 1:142--186, 1980.

....problem. Lenstra [Len92] outlines an approach to obtaining a probabilistic subexponential algorithm in the general case. Lenstra s paper [Len92] is an important source for information concerning algorithms and open problems concerning algebraic number fields. Ref32 [Gol85] Sha72] Sch82] [Lag80b], Buc90] 33 Solvability of binary quadratic diophantine equations C33 Input a; b; c; d; e; f 2 Z. Output 1 if there exists x; y 2 Z with ax 2 bxy cy 2 dx ey f = 0 and there does not exist a g 2 Z with b 2 Gamma 4ac = g 2 , 0 otherwise. O33a Is C33 NP hard O33b Is C33 ....

Jeffrey C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142--186, 1980.

....[1] Mayer [16, 17, 18, 19] and Hensley [9] This paper provides a detailed analysis of the Gaussian algorithm, both in the average case and in probability. Like its one dimensional counterpart, the algorithm is known to be of worst case logarithmic complexity, a result due to Lagarias [13], with best possible bounds being provided by Vall ee [25] and KaibSchnorr [10] The probabilistic behaviour of the Gaussian algorithm turns out to be appreciably different however. The main results of the paper are as follows. The average case complexity of the Gaussian algorithm (measured in ....

Lagarias, J. C. Worst--case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms 1, 2 (1980), 142--186.

....ideal representation whenever it is possible the ideals are reduced after each multiplication in Algorithm 3.1. So the reduction algorithm is applied very frequently and therefore it is necessary to have a fast reduction method. Our method is a refinement of the following well known algorithm (see [8] or [9] Algorithm 3.4 (Reduction of ideals, classical version) Input: Primitive ideal (A 0 ; B 0 ) Output: Reduced ideal (A; B) in normal presentation) equivalent to the input ideal. 1) A A 0 ; B B 0 ; 2) B round(B=A) Delta A Gamma B; 3) AN (B 2 Gamma D) A; 4) ....

J.C. Lagarias, Worst-Case Complexity Bounds for Algorithms in the Theory of Integral Quadratic Forms, Journal of Algorithms 1 (1980), 142 -- 186.

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J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142186, 1980.

No context found.

J. C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms, 1:142186, 1980.

No context found.

J.C. Lagarias. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. J. Algorithms, 1:142--186, 1980.

No context found.

J.C. Lagarias, Worst-Case Complexity Bounds for Algorithms in the Theory of Integral Quadratic Forms, Journal of Algorithms 1 (1980), 142 -- 186.

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