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    <title>LLLBases -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</title>
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      <h1>LLLBases -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</h1>
      <div>
        <h2>Description</h2>
<em>LLLBases</em> is a package implementing several variants of LLL bases. Some of these are implemented in the Macaulay2 engine, some by Victor Shoup's NTL package, and some are implemented at top level.        <p></p>
A matrix over ZZ determines a lattice: this is the ZZ-module generated by the columns.  It also determines a generating set. See the book: [H. Cohen, ...] for the definition and the basic algorithms for computing LLL bases.        <p></p>
LLL bases have nice theoretical properties, but the main benefit of LLL bases is that the entries of the resulting matrix often have dramatically small size, even smaller than theory would imply.        <p></p>
This package implements the following functions.      </div>
      <div>
        <h3>Menu</h3>
        <h4>main LLL algorithm</h4>
        <ul>
          <li><span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</span></li>
        </ul>
        <h4>applications and variants</h4>
        <ul>
          <li><span><span class="tt">kernelLLL</span> (missing documentation)
</span></li>
          <li><span><a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</a> -- compute the gcd of integers, and small multipliers</span></li>
          <li><span><span class="tt">hermite</span> (missing documentation)
</span></li>
        </ul>
        <h4>support routines that are occasionally useful</h4>
        <ul>
          <li><span><span class="tt">gramm</span> (missing documentation)
</span></li>
          <li><span><a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a> -- is a basis an LLL basis?</span></li>
        </ul>
      </div>
      <div>
        <div>
          <div>
            <h2>Author</h2>
            <ul>
              <li><a href="http://www.math.cornell.edu/~mike/">Michael E. Stillman</a><span> &lt;<a href="mailto:mike%40math.cornell.edu">mike@math.cornell.edu</a>></span></li>
            </ul>
          </div>
          <div>
            <h2>Version</h2>
            <p>This documentation describes version <b>1.1</b> of LLLBases, released <b>July 7, 2005</b>.</p>
          </div>
          <div>
            <h2>Citation</h2>
            <p>If you have used this package in your research, please cite it as follows:</p>
            <table class="examples">
              <tr>
                <td>
                  <pre><code class="language-bib">@misc{LLLBasesSource,
  title = {{LLLBases: Lenstra-Lenstra-Lovasz bases. Version~1.1}},
  author = {Michael E. Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}
</code></pre>
                </td>
              </tr>
            </table>
          </div>
          <div>
            <h2>Exports</h2>
            <div class="exports">
              <ul>
                <li>Functions and commands                  <ul>
                    <li><span><a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</a> -- compute the gcd of integers, and small multipliers</span></li>
                    <li><span><span class="tt">gramm</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">hermite</span> (missing documentation)
</span></li>
                    <li><span><a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a> -- is a basis an LLL basis?</span></li>
                    <li><span><span class="tt">kernelLLL</span> (missing documentation)
</span></li>
                    <li><span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</span></li>
                  </ul>
                </li>
                <li>Methods                  <ul>
                    <li><span><kbd>gcdLLL(List)</kbd> -- see <span><a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</a> -- compute the gcd of integers, and small multipliers</span></span></li>
                    <li><span><span class="tt">gramm(Matrix)</span> (missing documentation)
</span></li>
                    <li><span><span class="tt">hermite(Matrix)</span> (missing documentation)
</span></li>
                    <li><span><kbd>isLLL(Matrix)</kbd> -- see <span><a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a> -- is a basis an LLL basis?</span></span></li>
                    <li><span><span class="tt">kernelLLL(Matrix)</span> (missing documentation)
</span></li>
                    <li><span><kbd>LLL(Matrix)</kbd> -- see <span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</span></span></li>
                    <li><span><kbd>LLL(MutableMatrix)</kbd> -- see <span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</span></span></li>
                  </ul>
                </li>
                <li>Symbols                  <ul>
                    <li><span><a title="compute BKZ reduced basis instead of an LLL reduced basis" href="___B__K__Z.html">BKZ</a> -- compute BKZ reduced basis instead of an LLL reduced basis</span></li>
                    <li><span><a title="use the original Macaulay2 LLL algorithm" href="___Cohen__Engine.html">CohenEngine</a> -- use the original Macaulay2 LLL algorithm</span></li>
                    <li><span><a title="use the Macaulay2 language LLL algorithm" href="___Cohen__Top__Level.html">CohenTopLevel</a> -- use the Macaulay2 language LLL algorithm</span></li>
                    <li><span><span class="tt">fpLLL</span> (missing documentation)
</span></li>
                    <li><span><a title="use Givens rotations instead of Gram-Schmidt during LLL" href="___Givens.html">Givens</a> -- use Givens rotations instead of Gram-Schmidt during LLL</span></li>
                    <li><span><span class="tt">Hermite</span> (missing documentation)
</span></li>
                    <li><span><a title="use the all-integer LLL strategy from NTL library" href="___N__T__L.html">NTL</a> -- use the all-integer LLL strategy from NTL library</span></li>
                    <li><span><a title="use double precision real numbers" href="___Real__F__P.html">RealFP</a> -- use double precision real numbers</span></li>
                    <li><span><a title="use quadruple precision real numbers" href="___Real__Q__P.html">RealQP</a> -- use quadruple precision real numbers</span></li>
                    <li><span><a title="use a combination of double precision and quad precision real numbers" href="___Real__Q__P1.html">RealQP1</a> -- use a combination of double precision and quad precision real numbers</span></li>
                    <li><span><a title="use arbitrary precision real numbers" href="___Real__R__R.html">RealRR</a> -- use arbitrary precision real numbers</span></li>
                    <li><span><a title="use extended exponent real numbers" href="___Real__X__D.html">RealXD</a> -- use extended exponent real numbers</span></li>
                    <li><span><a title="the LLL threshold, in interval (1/4,1]" href="___Threshold.html">Threshold</a> -- the LLL threshold, in interval (1/4,1]</span></li>
                  </ul>
                </li>
              </ul>
            </div>
          </div>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="lattice reduction (Lenstra-Lenstra-Lovasz bases)" href="index.html">LLLBases</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">LLLBases.m2</span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">LLLBases.m2:818:0</span>.</p>
        </div>
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