We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the projections mod $P$ of an initial set of non-lattice vectors onto a single dual codeword. The technical tools we introduce are sorting of the projections followed by single-step pairwise Euclidean reduction of the projections, resulting in monotonic convergence of the positive-valued projections to zero. The length of vectors grows by a geometric factor each iteration. For fixed $P$ and $d$, and large enough user-defined input sets, we show that it is possible to minimize the number of iterations, and thus the overall length expansion factor, to obtain a short lattice vector. Thus we obtain a novel approach for controlling the output length, which resolves an open problem posed by Noah Stephens-Davidowitz (the possibility of an approximation scheme for the shortest-vector problem (SVP) which does not reduce to near-exact SVP). In our approach, one may obtain short vectors even when the lattice dimension is quite large, e.g., 8000. For fixed $P$, the algorithm yields shorter vectors for larger $d$. We additionally present a number of extensions and generalizations of our fundamental co-dimension $1$ method. These include a method for obtaining many different lattice vectors by multiplying the dual codeword by an integer and then modding by $P$; a co-dimension $k$ generalization; a large input set generalization; and finally, a "block" generalization, which involves the replacement of pairwise (Euclidean) reduction by a $k$-party (non-Euclidean) reduction. The $k$-block generalization of our algorithm constitutes a class of polynomial-time algorithms indexed by $k\geq 2$, which yield successively improved approximations for the short vector problem.

翻译：我们提出了一类新算法，用于在由$\mathbb{Z}_P^d$上余维$k$码定义的格中寻找短向量，其中$P$为素数。对于余维$1$情形，我们通过利用初始非格向量集合模$P$投影到单个对偶码字上的堆积性质来求解。引入的技术工具包括对投影进行排序，随后进行单步两两欧几里得约化，使得正值投影单调收敛至零。每次迭代中向量长度以几何因子增长。对于固定的$P$和$d$，以及足够大的用户定义输入集合，我们证明能够最小化迭代次数，从而控制整体长度扩展因子，以获取短格向量。由此我们获得了一种控制输出长度的新方法，解决了Noah Stephens-Davidowitz提出的开放问题（是否存在不归约为近精确最短向量问题（SVP）的近似方案）。在该方法中，即使格维度较大（例如8000），仍可获得短向量。对于固定$P$，算法能随$d$增大而生成更短向量。此外，我们给出了基本余维$1$方法的若干扩展与推广，包括：通过将对偶码字乘以整数后模$P$获取多个不同格向量的方法、余维$k$推广、大输入集合推广，以及"块"推广——将两两（欧几里得）约化替换为$k$方（非欧几里得）约化。算法的$k$块推广构成了一类由$k\geq 2$索引的多项式时间算法，这些算法对短向量问题可提供逐步改进的近似解。