A New Class of Algorithms for Finding Short Vectors in Lattices Lifted from Co-dimension $k$ Codes
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| Format: | Preprint |
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2024
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| _version_ | 1866913205158150144 |
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| author | Lin, Robert Shor, Peter W. |
| author_facet | Lin, Robert Shor, Peter W. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Class of Algorithms for Finding Short Vectors in Lattices Lifted from Co-dimension $k$ Codes Lin, Robert Shor, Peter W. Cryptography and Security Number Theory 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. |
| title | A New Class of Algorithms for Finding Short Vectors in Lattices Lifted from Co-dimension $k$ Codes |
| topic | Cryptography and Security Number Theory |
| url | https://arxiv.org/abs/2401.12383 |