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99
Lattice Reduction: a Toolbox for the Cryptanalyst
 Journal of Cryptology
, 1994
"... In recent years, methods based on lattice reduction have been used repeatedly for the cryptanalytic attack of various systems. Even if they do not rest on highly sophisticated theories, these methods may look a bit intricate to the practically oriented cryptographers, both from the mathematical ..."
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Cited by 72 (10 self)
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In recent years, methods based on lattice reduction have been used repeatedly for the cryptanalytic attack of various systems. Even if they do not rest on highly sophisticated theories, these methods may look a bit intricate to the practically oriented cryptographers, both from the mathematical and the algorithmic point of view. The aim of the present paper is to explain what can be achieved by lattice reduction algorithms, even without understanding of the actual mechanisms involved. Two examples are given, one of them being the attack devised by the second named author against Knuth's truncated linear congruential generator, which has been announced a few years ago and appears here for the first time in journal version.
Special cubic fourfolds
 Compositio Math
, 2000
"... This paper is concerned with smooth cubic hypersurfaces of dimension four (cubic fourfolds) and the surfaces contained in them. A cubic fourfold is special if it contains a surface which is not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreduc ..."
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Cited by 47 (4 self)
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This paper is concerned with smooth cubic hypersurfaces of dimension four (cubic fourfolds) and the surfaces contained in them. A cubic fourfold is special if it contains a surface which is not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families Cd, where each Cd is a divisor in the moduli space C of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. For any family Cd of special cubic fourfolds possessing associated K3 surfaces, we discuss how Cd is related to the moduli space Nd of degree d K3 surfaces. In particular, we prove that the moduli space of cubic fourfolds contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. In many cases, we construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface, which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the cubic fourfold is isomorphic to the Hilbert scheme of length two subschemes of an associated K3 surface.
Efficient Solution of Rational Conics
 Math. Comp
, 1998
"... this paper (section 2), and to Denis Simon for the reference [10]. ..."
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Cited by 26 (0 self)
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this paper (section 2), and to Denis Simon for the reference [10].
On the Proximity Factors of Lattice ReductionAided Decoding
"... Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of ..."
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Cited by 22 (7 self)
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Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of its performance limit. To this aim, the proximity factors are defined to measure the worstcase gap to maximumlikelihood (ML) decoding in terms of the signaltonoise ratio (SNR) for given error rate. The proximity factors are derived analytically and found to be bounded above by a function of the dimension of the lattice alone. As a direct consequence, it follows that lattice reductionaided decoding can always achieve full receive diversity of MIMO fading channels. The study is then extended to the dualbasis reduction. It is found that in some cases reducing the dual can result in smaller proximity factors than reducing the primal basis. The theoretic bounds on the proximity factors are further compared with numerical results.
Solving Quadratic Equations Using Reduced Unimodular Quadratic Forms
 Math. of Comp
, 2005
"... Abstract. Let Q be an n × n symmetric matrix with integral entries and with det Q � = 0, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to ..."
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Cited by 18 (1 self)
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Abstract. Let Q be an n × n symmetric matrix with integral entries and with det Q � = 0, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a quadratic equation q(x, y, z) =0issolvable over Q, a solution can be deduced from another quadratic equation of determinant ±1. The combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over Q, and this gives a polynomial time algorithm (as soon as the factorization of the determinant of Q is known). There are various methods in the literature for solving homogeneous quadratic equations q(x, y, z) =0overQ. Mathematicians seem to be unanimous in saying that the first step consists of reducing to the diagonal case, that is, to Legendre equations of the type ax 2 + by 2 + cz 2 = 0. As we will see in Section 4.2, this is a good idea in theory, but disastrous in practice: the determinant of the new equation
On the distribution of quadratic residues and nonresidues modulo a prime number
 Mathematics of Computation
, 1992
"... you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact inform ..."
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Cited by 17 (2 self)
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you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at.
On Gel'fandGraev characters of reductive groups with disconnected centre
, 1997
"... this paper is, following the theme of [DLM], to present explicit results on the character theory of the group of rational points G F , particularly in the case when the centre of G is not connected. We refer to [DLM] for an explanation as to how the issues treated here relate to the determination ..."
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Cited by 15 (2 self)
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this paper is, following the theme of [DLM], to present explicit results on the character theory of the group of rational points G F , particularly in the case when the centre of G is not connected. We refer to [DLM] for an explanation as to how the issues treated here relate to the determination of character values. Here we prove two results
Cubic modular equations and new Ramanujantype series for 1/π
 Pacific J. Math
"... In this paper, we derive new Ramanujantype series for 1/π which belong to “Ramanujan’s theory of elliptic functions to alternative base 3 ” developed recently by B.C. Berndt, S. Bhargava, and F.G. Garvan. 1. Introduction. Let (a)0 = 1 and, for a positive integer m, (a)m: = a(a + 1)(a +2)···(a+m−1), ..."
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Cited by 14 (2 self)
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In this paper, we derive new Ramanujantype series for 1/π which belong to “Ramanujan’s theory of elliptic functions to alternative base 3 ” developed recently by B.C. Berndt, S. Bhargava, and F.G. Garvan. 1. Introduction. Let (a)0 = 1 and, for a positive integer m, (a)m: = a(a + 1)(a +2)···(a+m−1), and ∞ ∑ (a)m(b)m z
Mathematical method and proof
"... Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that ..."
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Cited by 11 (6 self)
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Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not wellequipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.