钝角格基

摘要：

格约简是一种算法，它将给定的格基转换为另一个格基，这样找到最短向量和最接近向量等问题就更容易解决了。我们定义了一类基地称为钝角基地，并表明，任何格的基础上，可以转化为钝角的基础。最短向量$\mathbf{s}$可以写为$\mathbf{s}=v_1\mathbf{b}_1+\dots+v_n\mathbf{b}_n$，其中$\mathbf{b}_1,\dots,\mathbf{b}_n$是输入基向量，而$v_1,\dots,v_n$是整数。当输入基是钝角时，所有这些整数都可以被选择为最短向量的正数。钝角基的这种性质使得寻找最短向量的格枚举算法以指数方式更快。我们实现了使基变钝的算法，并对一些小基进行了测试。

英文摘要：

A lattice reduction is an algorithm that transforms the given basis of the lattice to another lattice basis such that problems like finding a shortest vector and closest vector become easier to solve. We define a class of bases called obtuse bases and show that any lattice basis can be transformed to an obtuse basis. A shortest vector $\mathbf{s}$ can be written as $\mathbf{s}=v_1\mathbf{b}_1+\dots+v_n\mathbf{b}_n$ where $\mathbf{b}_1,\dots,\mathbf{b}_n$ are the input basis vectors and $v_1,\dots,v_n$ are integers. When the input basis is obtuse, all these integers can be chosen to be positive for a shortest vector. This property of obtuse bases makes the lattice enumeration algorithm for finding a shortest vector exponentially faster. We have implemented the algorithm for making bases obtuse, and tested it some small bases.

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