Floating-Point LLL Revisited
Identifieur interne : 002471 ( Istex/Corpus ); précédent : 002470; suivant : 002472Floating-Point LLL Revisited
Auteurs : Phong Q. Nguên ; Damien StehléSource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L3) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L3 outputs a so-called L3-reduced basis in polynomial time O(d 5 n log3 B), using arithmetic operations on integers of bit-length O(d log B). This worst-case complexity is problematic for lattices arising in cryptanalysis where d or/and log B are often large. As a result, the original L3 is almost never used in practice. Instead, one applies floating-point variants of L3, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L3) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L3 is not even guaranteed to terminate, and the output basis may not be L3-reduced at all. In this article, we introduce the L2 algorithm, a new and natural floating-point variant of L3 which provably outputs L3-reduced bases in polynomial time O(d 4 n (d + log B) log B). This is the first L3 algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log B, like the well-known Euclidean and Gaussian algorithms, which it generalizes.
Url:
DOI: 10.1007/11426639_13
Links to Exploration step
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<front><div type="abstract" xml:lang="en">Abstract: The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L3) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L3 outputs a so-called L3-reduced basis in polynomial time O(d 5 n log3 B), using arithmetic operations on integers of bit-length O(d log B). This worst-case complexity is problematic for lattices arising in cryptanalysis where d or/and log B are often large. As a result, the original L3 is almost never used in practice. Instead, one applies floating-point variants of L3, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L3) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L3 is not even guaranteed to terminate, and the output basis may not be L3-reduced at all. In this article, we introduce the L2 algorithm, a new and natural floating-point variant of L3 which provably outputs L3-reduced bases in polynomial time O(d 4 n (d + log B) log B). This is the first L3 algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log B, like the well-known Euclidean and Gaussian algorithms, which it generalizes.</div>
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<abstract>Abstract: The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L3) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L3 outputs a so-called L3-reduced basis in polynomial time O(d 5 n log3 B), using arithmetic operations on integers of bit-length O(d log B). This worst-case complexity is problematic for lattices arising in cryptanalysis where d or/and log B are often large. As a result, the original L3 is almost never used in practice. Instead, one applies floating-point variants of L3, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L3) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L3 is not even guaranteed to terminate, and the output basis may not be L3-reduced at all. In this article, we introduce the L2 algorithm, a new and natural floating-point variant of L3 which provably outputs L3-reduced bases in polynomial time O(d 4 n (d + log B) log B). This is the first L3 algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log B, like the well-known Euclidean and Gaussian algorithms, which it generalizes.</abstract>
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<profileDesc><abstract xml:lang="en"><head>Abstract</head>
<p>The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L<hi rend="superscript">3</hi>
) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer <hi rend="italic">d</hi>
-dimensional lattice basis with vectors of norm less than <hi rend="italic">B</hi>
in an <hi rend="italic">n</hi>
-dimensional space, L<hi rend="superscript">3</hi>
outputs a so-called L<hi rend="superscript">3</hi>
-reduced basis in polynomial time <hi rend="italic">O</hi>
(<hi rend="italic">d</hi>
<hi rend="superscript">5</hi>
<hi rend="italic">n</hi>
log<hi rend="superscript">3</hi>
<hi rend="italic">B</hi>
), using arithmetic operations on integers of bit-length <hi rend="italic">O</hi>
(<hi rend="italic">d</hi>
log <hi rend="italic">B</hi>
). This worst-case complexity is problematic for lattices arising in cryptanalysis where <hi rend="italic">d</hi>
or/and log <hi rend="italic">B</hi>
are often large. As a result, the original L<hi rend="superscript">3</hi>
is almost never used in practice. Instead, one applies floating-point variants of L<hi rend="superscript">3</hi>
, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L<hi rend="superscript">3</hi>
) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L<hi rend="superscript">3</hi>
is not even guaranteed to terminate, and the output basis may not be L<hi rend="superscript">3</hi>
-reduced at all. In this article, we introduce the L<hi rend="superscript">2</hi>
algorithm, a new and natural floating-point variant of L<hi rend="superscript">3</hi>
which provably outputs L<hi rend="superscript">3</hi>
-reduced bases in polynomial time <hi rend="italic">O</hi>
(<hi rend="italic">d</hi>
<hi rend="superscript">4</hi>
<hi rend="italic">n</hi>
(<hi rend="italic">d</hi>
+ log <hi rend="italic">B</hi>
) log <hi rend="italic">B</hi>
). This is the first L<hi rend="superscript">3</hi>
algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log <hi rend="italic">B</hi>
, like the well-known Euclidean and Gaussian algorithms, which it generalizes.</p>
</abstract>
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<Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading>
<Para>The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L<Superscript>3</Superscript>
) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer <Emphasis Type="Italic">d</Emphasis>
-dimensional lattice basis with vectors of norm less than <Emphasis Type="Italic">B</Emphasis>
in an <Emphasis Type="Italic">n</Emphasis>
-dimensional space, L<Superscript>3</Superscript>
outputs a so-called L<Superscript>3</Superscript>
-reduced basis in polynomial time <Emphasis Type="Italic">O</Emphasis>
(<Emphasis Type="Italic">d</Emphasis>
<Superscript>5</Superscript>
<Emphasis Type="Italic">n</Emphasis>
log<Superscript>3</Superscript>
<Emphasis Type="Italic">B</Emphasis>
), using arithmetic operations on integers of bit-length <Emphasis Type="Italic">O</Emphasis>
(<Emphasis Type="Italic">d</Emphasis>
log <Emphasis Type="Italic">B</Emphasis>
). This worst-case complexity is problematic for lattices arising in cryptanalysis where <Emphasis Type="Italic">d</Emphasis>
or/and log <Emphasis Type="Italic">B</Emphasis>
are often large. As a result, the original L<Superscript>3</Superscript>
is almost never used in practice. Instead, one applies floating-point variants of L<Superscript>3</Superscript>
, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L<Superscript>3</Superscript>
) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L<Superscript>3</Superscript>
is not even guaranteed to terminate, and the output basis may not be L<Superscript>3</Superscript>
-reduced at all. In this article, we introduce the L<Superscript>2</Superscript>
algorithm, a new and natural floating-point variant of L<Superscript>3</Superscript>
which provably outputs L<Superscript>3</Superscript>
-reduced bases in polynomial time <Emphasis Type="Italic">O</Emphasis>
(<Emphasis Type="Italic">d</Emphasis>
<Superscript>4</Superscript>
<Emphasis Type="Italic">n</Emphasis>
(<Emphasis Type="Italic">d</Emphasis>
+ log <Emphasis Type="Italic">B</Emphasis>
) log <Emphasis Type="Italic">B</Emphasis>
). This is the first L<Superscript>3</Superscript>
algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log <Emphasis Type="Italic">B</Emphasis>
, like the well-known Euclidean and Gaussian algorithms, which it generalizes.</Para>
</Abstract>
<KeywordGroup Language="En"><Heading>Keywords</Heading>
<Keyword>LLL</Keyword>
<Keyword>L<Superscript>3</Superscript>
</Keyword>
<Keyword>Lattice Reduction</Keyword>
<Keyword>Public-Key Cryptanalysis</Keyword>
</KeywordGroup>
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<name type="personal"><namePart type="given">Phong</namePart>
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<affiliation>CNRS/École normale supérieure, DI, 45 rue d’Ulm, 75005, Paris, France</affiliation>
<affiliation>E-mail: Phong.Nguyen@di.ens.fr</affiliation>
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<abstract lang="en">Abstract: The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL or L3) is a very popular tool in public-key cryptanalysis and in many other fields. Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L3 outputs a so-called L3-reduced basis in polynomial time O(d 5 n log3 B), using arithmetic operations on integers of bit-length O(d log B). This worst-case complexity is problematic for lattices arising in cryptanalysis where d or/and log B are often large. As a result, the original L3 is almost never used in practice. Instead, one applies floating-point variants of L3, where the long-integer arithmetic required by Gram-Schmidt orthogonalisation (central in L3) is replaced by floating-point arithmetic. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point L3 is not even guaranteed to terminate, and the output basis may not be L3-reduced at all. In this article, we introduce the L2 algorithm, a new and natural floating-point variant of L3 which provably outputs L3-reduced bases in polynomial time O(d 4 n (d + log B) log B). This is the first L3 algorithm whose running time (without fast integer arithmetic) provably grows only quadratically with respect to log B, like the well-known Euclidean and Gaussian algorithms, which it generalizes.</abstract>
<subject lang="en"><genre>Keywords</genre>
<topic>LLL</topic>
<topic>L3</topic>
<topic>Lattice Reduction</topic>
<topic>Public-Key Cryptanalysis</topic>
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