Havas, G., Majewski, B., and Matthews, K. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics 7 (1998), 125-136.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Dawande [3] To close, we mention a few other key references. Further applications of HNF to the generation of cutting planes are given in Hung Rom [12] and Bockmayr Eisenbrand [8] Fast algorithms for HNF computation can be found in Storjohann Labahn [14] and Havas, Majewski Matthews [11]. Acknowledgements: Thanks are due to the anonymous referee, and also to the Editor, for some helpful comments. ....

G. Havas, B.S. Majewski & K.R. Matthews (1998) Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7, 125-136.

....and section 4.2) Shifts also appear naturally in the context of computing normal forms of matrices over ZZ . Schrijver has shown that integer weights on the rows of a matrix A in ZZ leads to a lattice whose reduced basis gives the Hermite form of A [28, p74] a similar approach is found in [15, 6] where the powers of a positive integer give an appropriate shift. For matrix polynomials we develop a more complete study. Computing shifted forms. The primary aim in this paper is to give a new algorithm for computing a shifted Popov form of an arbitrary rank polynomial matrix. In the case ....

.... In fact we show that there exist multipliers verifying deg u 1 (z) dn 1; 1 deg u k (z) d 1 ; 30) where = deg Gcd (a 1 (z) an (z) Notice that our bound includes the classical one for n = 2 (cf. 14] Also, a straight forward generalization of the integer bound of [15] to the polynomial case would lead to the weaker estimate deg u k (z) dn 1 for all k. In order to show (30) we choose a = 0 and b = d and we take as the u k (z) the entries of the last column of a ( 0; b) minimal multiplier. Degree bounds for the u k (z) are then determined by using ....

G. Havas, B.S. Majewski and K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Mathematics, 7:2, (1998) 125-135.

.... In fact we show that there exist multipliers verifying deg u 1 (z) d n 1; 1 deg u k (z) d 1 ; 30) where = deg Gcd (a 1 (z) a n (z) Notice that our bound includes the classical one for n = 2 (cf. 14] Also, a straight forward generalization of the integer bound of [15] to the polynomial case would lead to the weaker estimate deg u k (z) d n 1 for all k. In order to show (30) we choose a = 0 and d and we take as the u k (z) the entries of the last column of a ( 0; b) minimal multiplier. Degree bounds for the u k (z) are then determined by using ....

G. Havas, B.S. Majewski and K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Mathematics, 7:2, (1998) 125-135.

....et al. 11] developed a polynomial time algorithm based on basis reduction for finding a decomposition into irreducible factors of a non zero polynomial in one variable with rational coefficients. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski and Matthews [8]. Here, the aim is to find a short multiplier vector x such that ax = d, where d = gcd(a 1 ; a 2 ; a n ) In our algorithm we use a basis that is similar to the bases used by Lagarias and Odlyzko [10] and by Schnorr and Euchner [15] The important differences between the approach ....

G. Havas, B.S. Majewski, K.R. Matthews (1996). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

....find small integer solutions to a polynomial in a single variable modulo N , and to a polynomial in two variables over the integers. This has applications to some RSA based cryptographic schemes. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski and Matthews [7]. Here, the aim is to find a short multiplier vector x such that ax = a 0 , where a 0 = gcd(a 1 ; a 2 ; a n ) 3 Structure of initial and reduced basis Here we consider a lattice that contains all vectors of interest to our problem (1) such that ax = a 0 ; 0 x u Without loss of ....

G. Havas, B.S. Majewski, K.R. Matthews (1996). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

....Storjohann for sending this) 4) Another practical algorithm is given in (Havas and Majewski, 1997) where a certain pivoting strategy for the standard algorithm is discussed which tries to reduce the coecient growth during the computation. This is also available via GAP (GAP, 1999) 5) In (Havas et al. 1998) the standard algorithm is combined with an LLL lattice reduction algorithm. This is particularly interesting for nding transforming matrices to the Hermite and Smith normal form with small entries. Also all matrix entries stay small during the computation. We have implemented this algorithm in ....

Havas, G., Majewski, B. S., and Matthews, K. R. (1998). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7:125-135.

....reduction (LLL algorithms in the sense of [LLL82] see also [Coh93] and [GLS88] To the best of the authors knowledge there are two algorithms solving this problem which were carefully analysed. These are the algorithm of BUCHMANN and POHST ( BP89] and the one of HAVAS, MAJEWSKI and MATTHEWS ([HMM98]) The analysation of BUCHMANN and POHST yields O( m n) 6 log 2 (B= 1 (L) binary operations (B : max i=1; m kv i k) VAN DER KALLEN s analysis of the other algorithm yields O( m n) 4 log 2 (mB) arithmetic operations ( vdK00] where all used numbers have bit length O(m log 2 ....

....value smaller than K = 100. All computations were done using exact rational arithmetic. We utilized SHOUP s NTL library 2.0 1 and his implementation of an LLL algorithm for not necessarily linearly independent vectors. Although the implemented algorithm differs from the algorithms [BP89] and [HMM98] we are sure that they have a very similar complexity. At least they are all slowed down if the number of generators is large. 4.2. Successive minima. If we want to compute the successive minima of a lattice we can do similar to algorithm 4.1. There are only two differences. We need a complete ....

GEORGE HAVAS, BOHDAN S. MAJEWSKI, KEITH R. MATTHEWS. Extended GCD and Hermite Normal Form Algorithms via Lattice Basis Reduction. Experimental Mathematics 7 (1998), 125--136.

....rational coecients. Lenstra et al. 1982) developed a polynomial time algorithm based on basis reduction for nding a decomposition into irreducible factors of a non zero polynomial in one variable with rational coecients. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski, and Matthews (1996). Here, the aim is to nd a short multiplier vector x such that ax = d, where d = gcd(a 1 ; a 2 ; a n ) In our algorithm we use a basis that is similar to the basis used by Lagarias and Odlyzko (1985) The important di erences between the approach described above and our approach are the ....

Havas, G., B. S. Majewski, K. R. Matthews. 1996. Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

....have been proposed to compute the Hermite form. They can be classified into three major classes: ffl Direct methods, applying elementary row operations over the ring. To ensure a polynomial running time the potential for exponential growth of intermediate entries (see [6] has to be avoided ([3, 9, 12]) See also [8] ffl Modulo determinant methods, performing arithmetic modulo a multiple of a determinant. The modular arithmetic avoids exponential growth of intermediate expressions ( 4, 5, 7, 10, 17] ffl Coefficient methods for polynomialmatrices, translating the Hermite problem to a ....

....bounds on the degrees of intermediate polynomials; an amortized analysis establishes the worst case running time bound claimed above. Notice that we do not use fast matrix or polynomial 1 arithmetic. Basis reduction is applied to the problem of computing the Hermite form of an integer matrix in [9]. Polynomial degree being non Archimedean is one of the reasons for the good degree bounds we get. Most direct methods triangularize the input matrix from the left top to the right bottom. The algorithm we describe here takes the novel approach of recovering H row by row starting with the last ....

G. Havas, B. S. Majewski, and K. R. Matthews. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7:125---136, 1998.

.... In this paper, we present an algorithm which takes as its starting point the recent LLL based algorithm of Havas, Majewski and Matthews and which often nds a shorter vector (a 1 ; am ) 1 Introduction Let s 1 ; s m be integers and s = gcd (s 1 ; s m ) In a recent paper [Havas, Majewski, Matthews 1998], the author and his collaborators used variants of the LLL algorithm to nd multiplier vectors (a 1 ; am ) of small Euclidean length jjXjj = a 2 1 a 2 m ) 1=2 such that s = a 1 s 1 : am s m . In each case a unimodular m m matrix P is produced such that P [s 1 ; ....

....0; 1) 1; 1; 3; 2; 0) 15 3 (0; 0; 1; 0) 0; 4; 1; 0; 1) 18 2 (0; 1; 0; 0) 2; 0; 1; 3; 2) 18 1 ( 1; 0; 0; 0) 1; 1; 4; 2; 0) 22 0 (0; 0; 0; 0) 3; 0; 2; 1; 0) 14 The shortest multiplier is p = p 5 p 4 = 0; 1; 2; 2; 2] with jjpjj 2 = 13. Property G holds here. Example 3. Example 7. 2 of [Havas, Majewski, Matthews 1998]) Take s 1 ; s 10 to be 763836, 1066557, 113192, 1785102, 1470060, 3077752, 114793, 3126753, 1997137, 2603018. The unimodular matrix P = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 0 3 1 0 0 0 1 1 2 0 1 2 2 1 1 3 1 1 1 2 0 0 1 3 3 1 2 1 0 0 3 2 3 2 3 1 0 0 1 2 2 2 0 1 3 3 2 1 0 2 2 2 5 2 1 2 1 1 0 0 ....

G. Havas, B.S. Majewski, K.R. Mat{ thews, Extended gcd and Hermite normal form algorithms via lattice reduction, Experimental Mathematics 7 No 2 (1998) 125-136.

.... of an algebraic number eld (see [1] There are also applications to group theory, since abelian groups are Z modules (see for example [9] The problem of computing the image of (that is, the HNF basis of the image of ) has been studied extensively, for example in [4] 10] 14] 15] [11]) 7] 6] 12] and [8] The rst ve of these algorithms su er, to one degree or another, from an explosion in the size of integers used in intermediate stages, a phenomenon known as entry explosion which a ects many algorithms over Z. The last four of these algorithms use modular arithmetic, ....

Havas, G., Majewski, B., and Matthews, K. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics 7 (1998), 125-136.

.... Delta u2(z) an (z) Delta un(z) d(z) which satisfy deg u1(z) fl n Gamma ffi Gamma 1; n X k=2 u k 6=0 (1 deg uk (z) fl 1 Gamma ffi: 17) Notice that these bounds include the classical one for n = 2 (cf. 10] Also, a straight forward generalization of the integer bound of [11] to the polynomial case would lead to the weaker estimate deg uk (z) fl n Gamma 1 for all k. There are a number of interesting problems that still remain to be solved. We have shown that it is possible to solve the shifted Popov form problem via some fraction free algorithm by noting that it is ....

G. Havas, B.S. Majewski and K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Mathematics, 7:2, (1998) 125-135.

....find small integer solutions to a polynomial in a single variable modulo N , and to a polynomial in two variables over the integers. This has applications to some RSA based cryptographic schemes. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski and Matthews [7]. Here, the aim is to find a short multiplier vector x such that ax = a 0 , where a 0 = gcd(a 1 ; a 2 ; a n ) 3 Structure of initial and reduced basis Here we consider a lattice that contains all vectors of interest to our problem (1) 9 a vector x 2 Z n such that ax = a 0 ; 0 x ....

G. Havas, B.S. Majewski, K.R. Matthews (1996). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

.... In this paper, we present an algorithm which takes as its starting point the recent LLL based algorithm of Havas, Majewski and Matthews and which often finds a shorter vector (a 1 ; am ) 1 Introduction Let s 1 ; s m be integers and s = gcd (s 1 ; s m ) In a recent paper [Havas, Majewski, Matthews 1998], the author and his collaborators used variants of the LLL algorithm to find multiplier vectors (a 1 ; am ) of small Euclidean length jjXjj = a 2 1 Delta Delta Delta a 2 m ) 1=2 such that s = a 1 s 1 : am s m . In each case a unimodular m Theta m matrix P is ....

....0; 0) Gamma2; 0; 1; 3; Gamma2) 18 1 ( Gamma1; 0; 0; 0) Gamma1; Gamma1; 4; Gamma2; 0) 22 0 (0; 0; 0; 0) Gamma3; 0; 2; Gamma1; 0) 14 The shortest multiplier is p = p 5 p 4 = 0; Gamma1; Gamma2; Gamma2; 2] with jjpjj 2 = 13. Property G holds here. Example 3. Example 7. 2 of [Havas, Majewski, Matthews 1998]) Take s 1 ; s 10 to be 763836, 1066557, 113192, 1785102, 1470060, 3077752, 114793, 3126753, 1997137, 2603018. The unimodular matrix P = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Gamma2 0 Gamma3 1 0 0 0 Gamma1 Gamma1 2 0 Gamma1 2 2 Gamma1 Gamma1 3 Gamma1 1 1 Gamma2 0 0 Gamma1 3 Gamma3 ....

G. Havas, B.S. Majewski, K.R. Mat-- thews, Extended gcd and Hermite normal form algorithms via lattice reduction, Experimental Mathematics 7 No 2 (1998) 125--136.

....et al. 11] developed a polynomial time algorithm based on basis reduction for finding a decomposition into irreducible factors of a non zero polynomial in one variable with rational coefficients. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski and Matthews [8]. Here, the aim is to find a short multiplier vector x such that ax = d, where d = gcd(a 1 ; a 2 ; a n ) In our algorithm we use a basis that is similar to the bases used by Lagarias and Odlyzko [10] and by Schnorr and Euchner [15] The important differences between the approach ....

G. Havas, B.S. Majewski, K.R. Matthews (1996). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

....)X 3 77pu 5 X 2 Z (a 77pu 6 )XZ 2 (b 77pu 7 )Z 3 passes through P i = X i ; Y i ; Z i ) i = 1; 2; 3; 4, and has minimal discriminant. Here we use the techniques described in Steps 7 and 9 and Appendix B of [33] including the HavasMajewski Matthews Hermite normal form algorithm [7]. 4. Finally, check whether the P i are dependent. In case of dependency, compute the dependency relation with smallest coefficients. Experiment A differs from Experiment B only in that M = 1. For the corresponding Experiment C, we chose p = 5167, a p = 2462, b p = 1260, and P 0 = 2; 946) The ....

G. Havas, B. Majewski, and K. Matthews, Extended GCD and Hermite normal form algorithms via lattice basis reduction, Experimental Math. 7 (1998), 125-136.

....to force the lifted points to be dependent. Implementation Note. The existence of a solution to (5) satisfying (6) is guaranteed by condition (ii) in Step 2 and an elementary algebraic lemma. An efficient method to construct small solutions has been described by Matthews [13] as an extension of [6]. See Appendix B for details. Theoretical Note. It is this step which militates against taking r to have the maximum value of 9, because we would like the coefficients of the curve Cu to be of moderate size. A rough back of the envelope calculation suggests that there should exist solutions to ....

....Algorithm. Let B be an m by n matrix with integer coordinates, and let b 2 Z m . The following algorithm determines if the inhomogeneous equation Bx = b has any solutions, and determines a small solution if solutions exist. 1) Apply the (Havas Majewski Matthews) Hermite Normal form algorithm [6] to the matrix G = B t 0 b t 1 : Call the result HNF(G) and let U be the associated unimodular transformation matrix. In other words, HNF(G) UG. 2) Compute H = HNF(B t ) Gamma C 0 Delta , where C consists of the nonzero rows of H. 3) If the equation Bx = b has a ....

G. Havas, B.S. Majewski, K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Math. 7 (1998), 125--136.

....It seems likely that this problem is intractable for almost all norms or metrics of interest. Yet, finding some solution vector x is computable in polynomial time, and a number of algorithms exist to do this. Among all known polynomial time algorithms, the method of Havas, Majewski, and Matthews [7, 8], based on the LLL lattice basis reduction algorithm, gives the best values for kxk in practice. However, a running time proportional to n 4 limits its practicality. The faster sorting GCD method of Majewski and Havas [13] has a running time proportional to n 2 , and gives solution vectors ....

George Havas, Bohdan S. Majewski, and Keith R. Matthews. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7(2):125--136, 1998.

....et al. 11] developed a polynomial time algorithm based on basis reduction for finding a decomposition into irreducible factors of a non zero polynomial in one variable with rational coefficients. In extended g.c.d. computations, basis reduction is used by for instance Havas, Majewski and Matthews [8]. Here, the aim is to find a short multiplier vector x such that ax = d, where d = gcd(a 1 ; a 2 ; a n ) In our algorithm we use a basis that is similar to the bases used by Lagarias and Odlyzko [10] and by Schnorr and Euchner [15] The important differences between the approach ....

G. Havas, B.S. Majewski, K.R. Matthews (1996). Extended gcd and Hermite normal form algorithms via lattice basis reduction. Working paper, Department of Mathematics, The University of Queensland, Australia.

....Hermite normal form algorithm By Keith Matthews October 8, 1998 Abstract A LLL based Hermite normal form algorithm of Havas, Majewski, Matthews is used to find a short integer solution of AX = B, in the case when a nontrivial solution of AX = 0 exists. 1 Introduction In a recent paper [3], LLL based methods were given for obtaining short integer solutions of the linear equation a 1 x 1 Delta Delta Delta a n x n =d=gcd (a 1 ; a n ) Consider the more general problem of solving AX = B, where A; X;B are integer matrices of size m Theta n; n Theta 1; m Theta 1, ....

....to A t B t , is given in M. Pohst s book [4, pages 23 24] There are other methods designed to avoid coefficient explosion such as that of Chou Collins [1] In this note we present another method, based on the recent LLL based Hermite normal form algorithm of Havas, Majewski, Matthews [3] to the matrix G = A t 0 B t 1 . Let H = HNF(A t ) C 0 , where C consists of nonzero rows. Then a solution of AX = B exists, iff and only if HNF(G) 2 4 C 0 0 1 0 0 3 5 and the corresponding unimodular tranformation matrix P , which will have small entries, has the form ....

George Havas, B.S. Majewski, K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice reduction, Experimental Mathematics 7 No2 (1998) 125--136.

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Havas, G., Majewski, B., and Matthews, K. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics 7 (1998), 125-136.

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G. Havas, B.S. Majewski and K.R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Mathematics, 7:2, (1998) 125-135.

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G. Havas, B.S. Majewski and K.R. Matthews. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7:125-136, 1998.

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George Havas, Bohdan S. Majewski, and Keith R. Matthews. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics, 7:125|136, 1998.

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