A. Lenstra, H. Lenstra and L. Lovasz, "Factoring polynomials with rational coefficients ", Mathematische Annalen, 1982  Home/Search   Document Not in Database   Summary   Related Articles   Check

....generator matrix is KZ reduced if and only if its lower triangular representation is KZ reduced. It is known [64] that every lattice has at least one KZ reduced generator matrix. The LLL reduction is named after Lenstra, Lenstra, and Lovsz, who suggested the corresponding reduction criteria in [53]. The LLL reduction is often used in situations where the KZ reduction would be too time consuming. A lower triangular generator matrix (4) is LLL reduced if either , or else each of the following three conditions holds: 8) for (9) and the submatrix (7) is LLL reduced. As before, an arbitrary ....

....(8) for (9) and the submatrix (7) is LLL reduced. As before, an arbitrary generator matrix is LLL reduced if its lower triangular representation is LLL reduced. Any KZ reduced matrix is clearly also LLL reduced. The motivation for the latter reduction is that there exists an efficient algorithm [53] to convert any generator matrix into an LLL reduced one. This algorithm, which operates in polynomial time in and , has become very popular. It was improved upon in [69] and [66] The LLL reduction algorithm has been modified in a number of ways, see [20, pp. 78 104] Hybrids between KZ and LLL ....

A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovsz, "Factoring polynomials with rational coefficients," Math. Annalen, vol. 261, pp. 515--534, 1982.

....Lenstra s reduction technique was only polynomial time for fixed dimension, which was however enough in [89] That inspired Lov asz to develop a polynomial time variant of the algorithm, which computes a so called reduced basis of a lattice. The algorithm reached a final form in the seminal paper [88] where Lenstra, Lenstra and Lov asz applied it to factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [121, 122] Those algorithms have proved invaluable in ....

....algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based on the LLL algorithm [88], which can approximate SVP and SBP. However, it should be emphasized that these algorithms typically perform much better than is theoretically guaranteed, on instances of practical interest. Given as input any basis of a lattice L, LLL provably outputs in polynomial time a basis (b 1 ; b ....

A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513--534, 1982.

....time and approximates the shortest vector within a factor of 2 n=2 . Actually, a reduction algorithm of the same flavor had already been included in Lenstra s work on integer programming (cf. Len83] circulated around 1979) and the lattice reduction algorithm reached a final form in the paper [LLL82] of Lenstra, Lenstra and Lov asz, from which the name LLL algorithm comes. Further refinements of the LLL algorithm were proposed by Schnorr ( Sch87, Sch88] who has improved the above factor into (1 ffl) Babai [Bab86] gave an algorithm that approximates the closest vector by a factor of ....

A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Methematische Ann., 261:513--534, 1982.

....1 ; z m Gamma1 2 Z: Lattice basis reduction can be used to find good multipliers. Such an approach dates back at least to Rosser [22] and Ficken [8] who used it for some small examples. A particularly effective algorithm for lattice basis reduction is due to Lenstra, Lenstra and Lov asz [17]. For a brief description of the LLL algorithm, see Section 2. Of importance in the LLL algorithm is a parameter ff, which is in the range ( 1] The complexity of the algorithm increases with ff, as does the quality guarantee on the basis vectors. One approach to the extended gcd problem, ....

A.K. Lenstra, H.W. Lenstra Jr., and L. Lov'asz. Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515--534.

....is small, e.g. 3. This is the so called low exponent RSA, which is quite popular in the real world. The most powerful known attack against low exponent RSA is due to Coppersmith [5, 6] At Eurocrypt 96, Coppersmith presented two applications [5, 4] of a novel use of the celebrated LLL algorithm [12]. Both applications were searches for small roots of certain polynomial equations: one for univariate modular equations, the other for bivariate integer equations. Instead of using lattice reduction algorithms as shortest vector oracles, Coppersmith applied the LLL algorithm to determine a ....

A. K. Lenstra, H. W. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Math. Ann., 261:515--534, 1982.

....algorithm that computes a so called reduced basis of a lattice. Actually, a reduction algorithm of the same flavor had already been included in Lenstra s work on integer programming (cf. Len83] circulated around 1979) and the lattice reduction algorithm reached a final form in the paper [LLL82] of Lenstra, Lenstra and Lov asz, from which the name LLL algorithm comes. Further refinements of the LLL algorithm were proposed by Schnorr ( Sch87, Sch88] The relevance of those algorithms to cryptography was immediately understood: in April 1982, Shamir ( Sha82] found a polynomial time ....

....be the associated matrix. Denote by B the value of the matrix obtained as an output of the LLL algorithm and by b 1 ; Delta Delta Delta ; b q its column vectors. Finally let 1 be the length of a shortest non zero vector of L (in the usual euclidean sense) The following essentially comes from [LLL82] Fact 1 i) b 1 ; Delta Delta Delta ; b q is a basis of L ii) jb 1 j 2 Theta iii)jb 1 j 2 q(q Gamma1) 2 Theta ( Delta(L) In the above, Delta(L) denotes the determinant of L, that is the (euclidean) volume of the q dimensional parallelepiped enclosed by b 1 ; Delta Delta ....

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A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Methematische Ann., 261:513--534, 1982.

....mathematics, and have been studied extensively. See [25] for example, for a survey of the field. In the area of combinatorial mathematics alone it is possible to phrase many different problems as questions about lattices. Integer programming [20] factoring polynomials with rational coefficients [27], integer relation finding [16] integer factoring [35] and diophantine approximation [36] are just a few of the areas where lattice problems arise. In some cases, such as integer programming existence problems, it is necessary to determine whether a convex body in R contains a lattice point ....

....lattices and lattice reduction theory. Section 1.3 mentions some of the algorithms which currently exist for computing a reduced lattice basis B given a basis B of a particular point lattice. In particular, we detail the operation of the LenstraLenstra Lov asz (LLL) basis reduction algorithm [27], which is currently the best known method for finding short vectors in a lattice. 1.2 Reduced Lattice Bases Any lattice L may be described by many different lattice bases. Let B 1 ; B 2 ; be distinct sets of vectors, all of which form bases of lattice L. We can imagine that there exists ....

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A. K. Lenstra, H. W. Lenstra, and L. Lov'asz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534.

....programming (see [74] which was, among others, based on a novel but non polynomial time lattice reduction technique. That algorithm inspired Lov asz to develop a polynomial time algorithm that computes a so called reduced basis of a lattice. It reached a final form in the seminal paper [73] where Lenstra, Lenstra and Lov asz applied it to factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in ....

....algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based on the LLL algorithm [73], which can approximate SVP and SBP. However, it should be emphasized that these algorithms typically perform much better than is theoretically guaranteed, on instances of practical interest. Given as input any basis of a lattice L, LLL provably outputs in polynomial time a basis (b 1 ; b ....

A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513--534, 1982.

....column for each k orbit. Choosing k orbits for a t (v; k; design means to multiply M by an appropriate 0=1 vector on the right such that a vector with constant entries results. There have been different approaches to finding such 0=1 vectors. We have implemented a variant of the LLL algorithm [12], see [18] which in comparison to Kreher and Radzizowski [8] has the new feature of considering as a variable. This helps find unsuspected values of : After applying the LLL algorithm all solutions are determined by an exhaustive search. The Kramer Mesner matrix is computed by a new version of ....

A.K. Lenstra, H.W. Lenstra Jr., L. Lov'asz, Factoring Polynomials with Rational Coefficients, Math. Ann. 261 (1982), 515-534.

....is no efficient algorithm to find the closest lattice point, or indeed to find the smallest lattice point. There are however polynomial time algorithms which provably find close vectors and small lattice vectors, namely Babai s algorithm [1] and the LLL algorithm of Lenstra, Lenstra and Lovasz [8]. Since such small (close) vectors are rare, the vectors these algorithms recover, tend to be, or be closely related to, the lattice vector we are looking for. In practice these algorithms work remarkably well. One problem with which we are left, is how to ensure our hidden vector is small. The ....

....given 3 bits per nonce for 170 messages using a 160 bit prime with high probability. However, no such CVP1 oracle exists, hence in practice we need to use a method which is not guaranteed to find a closest vector with respect to the 1 norm. In practice such methods are based on the LLL algorithm [8]. To see how successful we can actually be we implemented the following experiment in C LiDIA. For each experiment, we used a newly generated 160 bit prime q and as an alternative to Babai s algorithm, the close vectors were found by embedding the matrix M and the vector u in the square matrix: ....

A.L. Lenstra, H.W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Math Ann., 261, 515--534, 1982.

....that depend explicitly on the discrete nature of the set of feasible solutions. Perhaps most important is the use of lattice basis theory. This theory has become extremely important in the theory of integer programming due to Lenstra s result [Len83] and Lovasz s basis reduction algorithm [LLL82]. Computationally, reduced bases of a lattice have not been used too often in linear integer programming. One reason might be that the computation of a reduced basis of a lattice becomes quite expensive when the number of rows and columns of the matrix are in the range of a couple of hundreds. ....

A. K. Lenstra, H. W. Lenstra Jr., L. Lovasz, Factoring polynomials with rational coefficients, Mathematische Annalen 261, 515--534 (1982).

....problem. Consider the problem of finding feasible solutions of equality constrained integer programs, or more specifically of finding points in the set z: where A Z 4xv and b Z 4. For this problem, Aardal et al. 2] have recently developed a successful two step approach based on basis reduction [3, 6, 11]. In step i they use basis reduction on the associated lattice 0 i of A b dimension N M 1 to construct an alternative representation of the feasible set Z of the form Z x: x q q PA, A Z N M, X 0 , 2) where q, P are integer, Aq = b, P is an integral basis of the null space of A ....

A. K. Lenstra, H.W. Lenstra, and L. Lov&sz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515-534, 1982.

....solution is the shortest whenever (n) 1:0629 Delta n. 12] also contains some evidence showing the limitation of this method. The above mentioned papers suggest as a second stage finding the these shortest vectors using the lattice base reduction algorithm of Lenstra, Lenstra and Lov asz [33] (or some modification of it, cf. 44] This algorithm is not guaranteed to find a shortest vector but one that is at most 2 times the shortest. In order to get that with high probability the shortest vector is much shorter than the other vectors, one has to require (n) n . However, the ....

A. K. Lenstra, H. W. Lenstra and L. Lov'asz, Factoring polynomials with rational coefficients, Math. Ann., vol 261, 1982, 515-534.

....as described in e.g. 3] computes an orthogonal basis B n g for R such that = 0 for 1 j i n using the ordinary inner product. For simplicity, we shall restrict ourselves to n dimensional spaces; all our algorithms are easily transferred to work on lower dimensional subspaces. In [9], statement following proof of (1.28) on p. 523, the authors hint at a method for computing B from B using exact division in the case where B spans a subspace of D . The domain D is an integral domain with the added property that P i x i 6= 0 when x i 6= 0. This additional property ....

....320H. The two SUNs differ in that the earlier Sparc architecture does not support full integer multiplication, only providing instructions to perform 277 a multiplication in several separate steps. The inputs to the long integer executables were lattices from the factoring algorithm of [9]. The input to the double and null executables was a random 100 Theta 100 matrix of single decimal digits. The results of the timed runs can be seen in the tables of figure 3. The times shown are averages of several individual runs on the same input, run on the same workstation under identical ....

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Lenstra, A. K., Lenstra, H. W., and Lov' asz, L. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982).

.... 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 O(n) algorithm for SVP 1 and the polynomial time Turing reduction from approximate CVP to SVP given in [10] the present authors obtained a p n=2 approximation algorithm that runs in 2 ....

A. K. Lenstra, H. W. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982. 9

....version of this problem is to find a non zero lattice vector whose length is at most ff times the length of the shortest non zero lattice vector. The shortest lattice vector problem and its approximate versions have a rich history. In a celebrated result, Lenstra, Lenstra, and Lov asz [9] gave an algorithm (the LLL or L 3 algorithm) that computes a 2 n=2 approximate shortest vector in polynomial time. This was improved in a generalization of the LLL algorithm by Schnorr [12] who obtained a hierarchy of algorithms that provide a uniform trade off between the running time and ....

A. K. Lenstra, H. W. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

....comments in Section 2) and Ajtai s proof is easily one of the most intricate and beautiful NP completeness results known. While Ajtai s proof is a major step forward, the complexity of the shortest vector problem remains far from being well understood. The best known algorithms for this problem [LLL82, Sch85] produce vectors whose length is only guaranteed to be within an exponentially large factor IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA. Email: ravi almaden.ibm.com y Department of Computer Science, University of Houston, Houston, TX 77204, USA. Supported in part by the ....

A. Lenstra, H. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

....the lattice L which we clarify. First, we assumed that 1 sh(L) 2. This can be realized by appropriately scaling L by integers 0; Sigma1; Sigma2; Secondly, we assumed that bl(L) is at most 2 O(n) this can be realized by appropriate applications of the well known LLL algorithm [11]. In fact, the above algorithm can find all ff approximate shortest vectors for any constant ff 1; the exponent of the running time would depend on ff. Also, the algorithm can be extended to work with other norms as well. An adaptation of this algorithm yields a 2 O(n) algorithm to find a ....

A. K. Lenstra, H. W. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

....size of the polynomial and the size of the field (see [Kal92] also [ALRS92, Sud96] in our case, this is polynomial in n. The choice of the field to be GF(2 m ) isn t so important here; bivariate factorization algorithms are available for other fields as well (e.g. the breakthrough result of [LLL82] for the case of rationals) In all cases, the fact that the field size needed is only polynomial in n is crucial; for larger finite fields, the only polynomial time algorithms for bivariate factoring are randomized. The crucial claim is that (v Gamma P a (u) must be present as an irreducible ....

A. Lenstra, H. Lenstra, and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

....Remainder code is not even linear in the usual sense and so linear algebra is not applicable in our case. Thus for solving analogies of simple problems in the algebraic case, we employ the continued fraction method for the Unique Decoding task, and the approximate basis reduction algorithm [21] for the List Decoding task. Our final algorithms achieve decoding capabilities comparable to those in algebraic cases. In particular, the list decoding algorithm recovers from n Gamma o(n) errors, provided p n = p O(1) 1 and k = o(n) 3 Applications. As mentioned earlier the Chinese ....

.... ; 2) 2. Output all roots of the integer polynomial C(x) P i=0 c i x i . The running time of Step 2 above is easily bounded by a polynomial in n; log N and log F . For example, one can use the algorithm for factoring polynomials over the integers due to Lenstra, Lenstra and Lovasz [21] (LLL) Faster algorithms are also known for the simpler task of rootfinding . In particular, one may use classical results on the bound of real roots to isolate an interval which includes all the real roots of the given polynomial. Then one may perform binary search to approximate the real ....

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A. K. Lenstra, H. W. Lenstra and L. Lovasz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

....that its length kvk, which is understood in terms of the L 2 (euclidean) norm, is smallest. For this problem e.g. the algorithm of Hastad et al. 6] for finding integer relationships between real vectors can be employed, which is closely related to the Lovasz Lenstra Lenstra (L 3 ) algorithm [9]. Given linearly independent vectors v 1 ; v s 2 Z n , and k 0, the algorithm in [6] finds a 1 vector x 2 Z n in polynomial time such that v i x T = 0 for all i = 1; s or reports that no such vector of length 2 k exists. The vector computed is not shortest, but ....

A. Lenstra, H. Lenstra, Jr., and L. Lovasz. Factoring Polynomials with Rational Coefficients. Mathematische Annalen, 21:515--534, 1982.

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A. K. Lenstra, H. W. Lenstra, Jr., L. Lov'asz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515--534.

....[4] The main ingredients of our algorithm are as follows. First we choose a lattice that seems particularly useful for our problem. We write down an initial basis that spans the given lattice and apply Lov asz basis reduction algorithm, as described in the paper by Lenstra, Lenstra and Lov asz [11], to the initial basis. We will refer to this algorithm as the basis reduction algorithm . The parameters of the initial basis are chosen such that the reduced basis contains one vector x d satisfying Ax d = d, and n Gamma m linearly independent vectors satisfying Ax = 0. If the vector x d ....

....; 1 j l. The vectors j ; 1 j l and the real numbers jk ; 1 k j l are determined from b j ; 1 j l by the recurrence j = b j Gamma j Gamma1 k=1 jk b k (3) jk = b j ) k = b k ) k : 4) Let jj jj denote the Euclidean length in IR . Lenstra, Lenstra and Lov asz [11] used the following definition of a reduced basis: 5 Definition 2 A basis b 1 ; b 2 ; b l is called reduced if j jk j for 1 k j l (5) j j;j Gamma1 b for 1 j l: 6) The vector b j is the projection of b j on the orthogonal complement of k=1 IRb k , and the ....

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A.K. Lenstra, H.W. Lenstra, Jr., L. Lov'asz (1982). Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515--534.

....would obtain a vector x sucht that ax 6= d. It is obviously a serious drawback that the algorithm terminates with an infeasible solution. To overcome the mentioned deficiencies we have developed an algorithm based on the L basis reduction algorithm as developed by Lenstra, Lenstra and Lov asz [9]. The motivation behind choosing basis reduction as a core of our algorithm is twofold. First, basis reduction allows us to work directly with integers, which avoids the round off problems. Second, basis reduction finds short, nearly orthogonal vectors belonging to the lattice described by the ....

....is an algorithm for deriving orthogonal vectors j ; 1 j n from independent vectors b j ; 1 j n. The vectors b j ; 1 j n and the real numbers jk ; 1 k j n are defined inductively by: j = b j Gamma j Gamma1 jk b jk = b j ) b k ) Lenstra, Lenstra and Lov asz [9] used the following definition of a reduced basis: Definition 2 A basis b 1 ; b 2 ; b n is reduced if j jk j for 1 k j n (5) for 1 j n: 6) The vector b j is the projection of b j on the orthogonal complement of Rb k , and the vectors b j Gamma1 and b j ....

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A.K. Lenstra, H.W. Lenstra, Jr., L. Lov'asz (1982). Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515--534.

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A. Lenstra, H. Lenstra and L. Lovasz, "Factoring polynomials with rational coefficients ", Mathematische Annalen, 1982

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A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz, Factoring polynomials with rational coefficients, Math. Annalen 261 (1982), 515--534.

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A.K. Lenstra, H.W. Lenstra, L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen 261, pages 515--534 (1982).

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515--534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515--534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515-- 534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515--534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515--534, 1982.

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A.K. Lenstra, H.W. Lenstra, L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515-534 (1982).

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Arjen K. Lenstra, Hendrik W. Lenstra, Jr., and L'aszl'o Lov'asz. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513--534, 1982.

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A.K. Lenstra, H.W. Lenstra Jr. and C. Lovasz, Factoring polynomials with rational coefficients, Math Anm. 261 (1982), 515-534.

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A.K. Lenstra, H.W. Lenstra, L. Lowasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 513-534.

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A. K. Lenstra, H. W. Lenstra Jr., L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515-534

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A.K. Lenstra, H.W. Lenstra and L. Lovasz, Factoring polynomials with rational coefficients. Mathematische Annalen 261 (1982) pp. 513-534

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A. K. Lenstra, H. W. Lenstra and L. Lovasz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

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A. Lenstra, H. L. Jr., and L. Lovasz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515--534, 1982.

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A. K. Lenstra, H. W. Lenstra, Jr., and L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513--534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen,

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A. K. Lenstra, H. W. Lenstra, and L. Lovsz, "Factoring polynomials with rational coefficients," Math. Ann., vol. 261, pp. 515--534, 1982.

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A.K. Lenstra, H.W. Lenstra, L. Lov'asz. Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515-534 (1982).

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A. K. Lenstra, H. W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Math. Annalen, vol. 261, pp. 513--534, 1982.

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A. K. Lenstra, H. W. Lenstra, and L. Lovsz. Factoring polynomials with rational coefficients. Math. Annalen, 261:515 -- 534, 1982.

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A. K. Lenstra, H. W. Jr. Lenstra, and L. Lovasz. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515--534, 1982.

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A.K. Lenstra, H.W. Lenstra Jr., and L. Lov'asz, Factoring polynomials with rational coefficients, Mathematische Annalen, 261 (1982), no. 4, 515--534.

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A.K. Lenstra, H.W. Lenstra, L. Lov'asz, Factoring polynomials with rational coefficients, Math. Annalen 261 (1982), 515--534.

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A. K. Lenstra, H. W. Lenstra, and L. Lov asz, "Factoring polynomials with rational coefficients," Math. Annalen, vol. 261, pp. 513--534, 1982.

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