D. Coppersmith. Solving linear equations over GF(2) via block Wiedemann algorithm. Math. Comp., 62(205):333--350, Jan. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Lemma 7) The other is when # = 1, which means that the minimal polynomial of T equals its characteristic polynomial, and better lower bounds are given in Theorem 9. Our work was motivated by a connection with block iterative methods for solving large sparse linear systems over finite fields, see [3, 4, 8, 12, 14]. It improves upon the result in the report [15] used in an analysis of the block Wiedemann algorithm. A more di#cult and important question in the analysis of such algorithms is to bound the probability that certain truncated Krylov subspaces generate the whole space. More precisely, let Kry(T ....

D. Coppersmith, Solving linear equations over GF (2): block Lanczos algorithm, Linear Algebra and Its Applications, 192 (1993), pp. 33--60.

....a given line cannot be multiplied too many times (otherwise we would have to allow the coe cient to grow above one machine word) this makes the elimination less e cient. Afterwards, we found it enlightening to use the block Wiedemann algorithm. This algorithm has been proposed by Coppersmith in [12], extending a previous algorithm by Wiedemann [43] Another algorithm, the block Lanczos algorithm [34] is often preferred to the block Wiedemann algorithm. We used the latter because it gave us an opportunity to successfully experiment the accelerating procedure described in [39] the crux of ....

D. Coppersmith. Solving linear equations over GF(2) via block Wiedemann algorithm. Math. Comp., 62(205):333350, Jan. 1994.

....introduces the possibility of a division by zero when the standard Lanczos algorithm is applied. Lanczos methods with lookahead attempt to address this problem, and either reduce or eliminate the possibility of a division by zero; algorithms of this type have been described by Coppersmith [1], Montgomery [9] and Teitelbaum [12] These algorithms are somewhat more complicated than the standard algorithm, and may require additional storage space, additional matrix vector multiplications, or both so that the apparent advantage of using the Lanczos method over Wiedemann s algorithm ....

....that blocking may also be a means to diminish failure probabilities in the small field case. Whether any of these two advantages (fewer matrix times vector products, smaller failure probability) of the block Wiedemann algorithm in its sequential setting carry over to the block Lanczos approach [1, 9] is unknown to us. ....

Coppersmith, D. Solving linear equations over GF(2); block Lanczos algorithm. Linear Algebra Appl. 192 (1993), 33--60.

....introduces the possibility of a division by zero when the standard Lanczos algorithm is applied. Lanczos methods with lookahead attempt to address this problem, and either reduce or eliminate the possibility of a division by zero; algorithms of this type have been described by Coppersmith [1], Montgomery [10] and Teitelbaum [13] These algorithms are somewhat more complicated than the standard algorithm, and may require additional storage space, additional matrix vector multiplications, or both so that the apparent advantage of using the Lanczos method over Wiedemann s algorithm ....

....suggest that blocking is a means to diminish failure probabilities in the small field case. Whether any of these two advantages (fewer matrix times vector products, smaller failure probability) of the block Wiedemann algorithm in its sequential setting carry over to the block Lanczos approach [1, 10] is unknown to us. ....

Coppersmith, D. Solving linear equations over GF(2); block Lanczos algorithm. Linear Algebra Appl. 192 (1993), 33--60.

....7) The other is when = 1, which means that the minimal polynomial of T equals its characteristic polynomial, and better lower bounds are given in Theorem 9. 2 Our work was motivated by a connection with block iterative methods for solving large sparse linear systems over finite fields, see [3, 4, 8, 12, 14]. It improves upon the result in the report [15] used in an analysis of the block Wiedemann algorithm. A more difficult and important question in the analysis of such algorithms is to bound the probability that certain truncated Krylov subspaces generate the whole space. More precisely, let ....

D. Coppersmith, Solving linear equations over GF (2): block Lanczos algorithm, Linear Algebra and Its Applications, 192 (1993), pp. 33--60.

....using sparse matrix techniques for the second stage, the overall runtime of Dixon s method becomes Ln [1=2; # 2] Ln [1=2; # 2] Ln [1=2; # 2] Asymptotically the relation collection stage and the matrix step take the same amount of time. Sparse matrix methods are not further treated here; see [27, 58, 80, 96, 114] for various methods that can be applied. Dixon s method has the advantage that its expected runtime can be rigorously analysed and does not depend on unproved hypotheses or heuristics. In practice, however, it is inferior to the original Morrison Brillhart continued fraction approach and to the ....

D. Coppersmith, Solving linear equations over GF(2) using block Wiedemann algorithm, Math. Comp. 62 (1994) 333-350.

....introduces the possibility of a division by zero when the standard Lanczos algorithm is applied. Lanczos methods with lookahead attempt to address this problem, and either reduce or eliminate the possibility of a division by zero; algorithms of this type have been described by Coppersmith [1], Montgomery [9] and Teitelbaum [12] These algorithms are somewhat more complicated than the standard algorithm, and may require additional storage space, additional matrix vector multiplications, or both so that the apparent advantage of using the Lanczos method over Wiedemann s algorithm ....

....that blocking may also be a means to diminish failure probabilities in the small field case. Whether any of these two advantages (fewer matrix times vector products, smaller failure probability) of the block Wiedemann algorithm in its sequential setting carry over to the block Lanczos approach [1, 9] is unknown to us. ....

Coppersmith, D. Solving linear equations over GF(2); block Lanczos algorithm. Linear Algebra Appl. 192 (1993), 33--60.

....log N) where M(d) is the cost for multiplying two polynomials of degree d. We discuss the implications of this improvement for the overall cost of the block Wiedemann algorithm and how its parameters should be chosen for best e ciency. 1. INTRODUCTION Coppersmith s block Wiedemann algorithm [9] applies to the solving of large, sparse linear systems over nite elds. More precisely, a primary statement of the context of the algorithm would be as follows. We are given a large, sparse square matrix B of size N N de ned over K = Fq , where q can be any prime power. We know that the matrix ....

.... Massey or Extended Euclidean algorithm. Both of these require O(N 2 ) scalar multiplications, but subquadratic variants [1, 13] bring O(M(N) log N) instead, where M(d) is the cost for multiplying polynomials of degree d. M(d) is O(d log d) with fast Fourier transform (FFT) Coppersmith [9] brought the following interesting possibility: instead of vectors x and y, use blocks of vectors, of size N m and N n, respectively, where m and n are chosen integers. One sample x T B k y therefore contains more information because it is made up of several scalars. This enables us to ....

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Coppersmith, D. Solving linear equations over GF(2) via block Wiedemann algorithm. Math. Comp. 62, 205 (Jan. 1994), 333350.

.... of success of the algorithm with respect to its random inputs x and y is not impaired by our enhancement: furthermore, this probability of success is comparable to the probabilities reached with the original Wiedemann algorithm or the Lanczos (standard or block version) algorithm with lookahead [8, 17, 21]. Section 2 addresses the problem of nding a linear generator for a given matrix sequence. Coppersmith s algorithm for this task is presented in 2.3. Existing subquadratic approaches and our new algorithm, are presented in section 3. Section 4 exposes the block Wiedemann algorithm, and shows how ....

Coppersmith, D. Solving linear equations over GF(2): Block Lanczos algorithm. Linear Algebra Appl. 192 (Jan. 1993), 3360.

....To nd such subsets, it looks for smooth n s, factors each n as a product of powers of 1; 2; 3; 5; and then nds linear relations among the exponent vectors mod 2. See [101] 124] 180] 140] 129] 152] 168] 181] 178] 172] 146] and [147] See also [149] 99] 176] 56] [55], 107] 122] 67] and [16] for relevant linear algebra algorithms. The integers n in the continued fraction method are bounded in absolute value by x for some x 2 D 1=2 o(1) one chooses y with (log y) 2 2 (1=2 o(1) log x log log x. It seems that the rst y 2 o(1) values of n always ....

Don Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear Algebra and its Applications 192 (1993), 33-60. MR 94i:65044.

....provided that w T i Aw i 6= 0 when w i 6= 0 during this process. Working over the field F 2 instead of R has the advantage that one can apply a matrix to N different vectors simultaneously, where N is the computer word 8. The block Lanczos method 13 size. Inspired by work of Coppersmith [6], Montgomery [17] implemented the block Lanczos method which exploits this advantage. The block Lanczos algorithm generates a sequence of subspaces W i instead of the vectors fw i g. Applying standard Lanczos over F 2 gives the problem that in approximately half of the cases the requirement w T ....

D. Coppersmith. Solving linear equations over GF(2): Block Lanczos algorithm. Linear Algebra and its Applications, 192:33--60, 1993.

....384 j ( Gamma32) Delta ( Gamma27) Delta ( Gamma2) Delta 243 Delta ( Gamma1) mod 77) becomes 144000 2 j 648 2 (mod 77) which again gives the factorization 77 = 7 Delta 11. Traditionally, one solved the system Be = 0 by a variation of Gaussian elimination. Recently some iterative methods [2, 3, 7, 18] have been found. The iterative methods are superior when the matrix is large, since they require less storage (matrices arising from integer factorization problems are very sparse) For these large, sparse, matrices, the iterative methods are also faster if B is an n Theta n matrix, then ....

Don Coppersmith. Solving linear equations over GF(2): Block Lanczos algorithm. Linear Algebra and its Applications, 192:33--60, October 1993.

....structured Gaussian elimination and the Lanczos and conjugate gradient algorithms [LaMacchiaO] that these methods came into wide use. The main advances in linear algebra for index calculus algorithms in the 1990s came from the parallelization of the Lanczos and Wiedemann algorithms by Coppersmith [Coppersmith2, Coppersmith3]. Currently the most widely used parallel method is Montgomery s version of the Lanczos algorithm [Montgomery] where it is used after structured Gaussian elimination reduces the matrix to manageable size. These parallelization methods essentially speed up the basic algorithms over the field of ....

D. Coppersmith, Solving linear equations over GF (2): block Lanczos algorithm, Linear Algebra Appl. 192 (1993), 33--60.

....3 = 0 et L 1 L 2 L 4 = 0 sont des relations de d ependance pour A. Il nous suffit de savoir que trouver des relations de d ependance dans une matrice k Theta k peut se faire en temps O(k r ) avec r 3 (des algorithmes meilleurs que le pr ec edent sont d ecrits dans [Wie86, LO90, Cop91a, Cop91b] Pour r esumer la m ethode, on a construit des couples (u x ; w x ) v erifiant u 2 x j w x mod N; et on factorise les w x sur une base. On peut montrer que cette m ethode, due a Dixon [Dix81] a un temps de calcul O(L(N) 2 ) o u L(x) exp q log x log log x: Le nombre de nombres ....

Coppersmith (Don). -- Solving linear equations over gf(2) via block Wiedemann algorithm. -- 1991. Submitted to Math. Comp.

....L 1 L 3 = 0 et L 1 L 2 L 4 = 0 sont des relations de d ependance pour A. Il nous suffit de savoir que trouver des relations de d ependance dans une matrice k Theta k peut se faire en temps O(k r ) avec r 3 (des algorithmes meilleurs que le pr ec edent sont d ecrits dans [Wie86, LO90, Cop91a, Cop91b] Pour r esumer la m ethode, on a construit des couples (u x ; w x ) v erifiant u 2 x j w x mod N; et on factorise les w x sur une base. On peut montrer que cette m ethode, due a Dixon [Dix81] a un temps de calcul O(L(N) 2 ) o u L(x) exp q log x log log x: Le nombre de ....

Coppersmith (Don). -- Solving linear equations over GF (2). -- RC n 16997, IBM Research, T. J. Watson Research Center, July 1 1991. Updated version of August 6, 1991.

....introduces the possibility of a division by zero when the standard Lanczos algorithm is applied. Lanczos methods with lookahead attempt to address this problem, and either reduce or eliminate the possibility of a division by zero; algorithms of this type have been described by Coppersmith [1], Montgomery [10] and Teitelbaum [13] These algorithms are somewhat more complicated than the standard algorithm, and may require additional storage space, additional matrix vector multiplications, or both so that the apparent advantage of using the Lanczos method over Wiedemann s algorithm ....

....suggest that blocking is a means to diminish failure probabilities in the small field case. Whether any of these two advantages (fewer matrix times vector products, smaller failure probability) of the block Wiedemann algorithm in its sequential setting carry over to the block Lanczos approach [1, 10] is unknown to us. ....

Coppersmith, D. Solving linear equations over GF(2); block Lanczos algorithm. Linear Algebra Appl. 192 (1993), 33--60.

....that we know the structure of the solution set) is of course the core of elementary linear algebra. Fast probabilistic algorithms which exploit sparsity in A have been developed to solve the problem of finding solutions x if one exists (over fields by Wiedemann 1986, Kaltofen Saunders 1991, Coppersmith 1993, Coppersmith 1994, Kaltofen 1995, Lambert 1996, Villard 1997, and over the integers by Giesbrecht 1997) Furthermore a purported solution is easily checked, so that these algorithms are of Las Vegas type when the hypothesis is made a priori that the system is consistent. However, less attention ....

D. Coppersmith. Solving linear equations over GF(2); block Lanczos algorithm. Linear algebra and its applications 192, pp. 333--350, 1993.

....where such linear systems arise with N over 200; 000 [23, 25, 19] This has motivated several authors to develop fast finite field counterpart to numerical iterative methods. The conjugate gradient method has been used in [23] the Lanczos method in [23, 12] and the block Lanczos method in [8, 29]. But up to now, only the probabilistic analysis of Wiedemann [39] was giving a provably reliable and efficient method to solve Aw = 0 over small fields. This method is based on finding relations in Krylov subspaces using the BerlekampMassey algorithm [28] The same analysis could be applied to ....

D. Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear Algebra and its Applications 192 (1993), 33--60.

....where such linear systems arise with N over 200; 000 [23, 25, 19] This has motivated several authors to develop fast finite field counterpart to numerical iterative methods. The conjugate gradient method has been used in [23] the Lanczos method in [23, 12] and the block Lanczos method in [8, 29]. But up to now, only the probabilistic analysis of Wiedemann [39] was giving a provably reliable and efficient method to solve Aw = 0 over small fields. This method is based on finding relations in Krylov subspaces using the BerlekampMassey algorithm [28] The same analysis could be applied to ....

D. Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear Algebra and its Applications 192 (1993), 33--60.

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D. Coppersmith. Solving linear equations over GF(2) via block Wiedemann algorithm. Math. Comp., 62(205):333--350, Jan. 1994.

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Coppersmith, D. Solving linear equations over GF(2): block Lanczos algorithm. Linear Algebra and its Applications 192 (1993), 33--60.

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C1. D. Coppersmith, Solving linear equations over GF(2); block Lanczos algorithm., Linear Algebra and its Applications 192 (1993), 33--60.

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