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47
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 55 (7 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.
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 21 (0 self)
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this paper (section 2), and to Denis Simon for the reference [10].
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 15 (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 13 (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 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 10 (3 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.
Valuations and Dedekind's Prague Theorem
 J. Pure Appl. Algebra
"... To any field K we associate an entailment relation in the sense of Scott [12]. In this way we can interpret an abstract propositional theory representing a generic valuation ring of a field, and obtain a simple effective proof of Dedekind's Prague theorem [5,6]. Keywords: Valuations, Entailment rel ..."
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Cited by 9 (1 self)
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To any field K we associate an entailment relation in the sense of Scott [12]. In this way we can interpret an abstract propositional theory representing a generic valuation ring of a field, and obtain a simple effective proof of Dedekind's Prague theorem [5,6]. Keywords: Valuations, Entailment relations. AMS class.: 13A10, 13B25, 54H99 1 Introduction To any field K we associate a relation ` between finite sets of non zero elements of K which satisfy the three conditions of an entailment relation in the sense of Scott [12], and some further simple conditions. In this way, we can give constructive sense of a generic valuation ring of a field. Alternatively, this can be seen as a generalisation of the notion of integral element, and this notion can be used to prove that a given element is integral. As an example, we present a simple effective proof of Dedekind's Prague theorem. 2 Valuations Let K be a field, that is a commutative ring in which any element is 0 or is invertible. We write...
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 9 (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
The mean value of the product of class numbers of paired quadratic fields I
, 1999
"... Abstract. This is the second part of a two part paper. In this part, we evaluate the previously unevaluated local densities at dyadic places which appear in the density theorem stated in the first part. For this purpose we introduce an invariant, the level, attached to a pair of ramified quadratic e ..."
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Cited by 9 (2 self)
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Abstract. This is the second part of a two part paper. In this part, we evaluate the previously unevaluated local densities at dyadic places which appear in the density theorem stated in the first part. For this purpose we introduce an invariant, the level, attached to a pair of ramified quadratic extensions of a dyadic local field. This invariant measures how close the fields are in their arithmetic properties and may be of interest independent of its application here. 1.
Distributions of discriminants of cubic algebras
 Department of Mathematical Sciences, University of Tokyo
, 1985
"... Abstract. Let k be a number field andOthe ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras ofOand derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spac ..."
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Cited by 8 (2 self)
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Abstract. Let k be a number field andOthe ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras ofOand derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spaces. In this paper we consider these objects under imposing finite number of splitting conditions at nonarchimedean places. Especially the explicit formulae of residues at s = 1 and 5/6 under the conditions are given. 1.