Abstract
The security of lattice-based cryptosystems is generally based on the hardness of the Shortest Vector Problem (SVP). There are two common categories of lattice algorithms to solve SVP: search algorithms and reduction algorithms. The original enumeration algorithm (ENUM) is one of the former algorithms which run in exponential time due to the exhaustive search. Further, ENUM is used as a subroutine for the BKZ algorithm, which is one of the most practical reduction algorithms. It is a critical issue to reduce the computational complexity of ENUM. In this paper, first, we improve the mechanism in the so-called reordering method proposed by Wang in ACISP 2018. We call this improvement Primal Projective Reordering (PPR) method which permutates the projected vectors by decreasing norms; therefore it performs better to reduce the number of search nodes in ENUM. Then, we propose a Dual Projective Reordering (DPR) method permutating the projected vectors in its dual lattice. In addition, we propose a condition to decide whether the reordering method should be adopted or not. Preliminary experimental results show that our proposed reordering methods can successfully reduce the number of ENUM search nodes comparing to the predecessor, e.g., PPR reduces around 9.6% on average in 30-dimensional random lattices, and DPR reduces around 32.8% on average in 45-dimensional random lattices. Moreover, our simulation shows that the higher the lattice dimension, the more the proposed reordering method can reduce ENUM search nodes.
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This work was supported by JSPS KAKENHI Grant Number JP20K23322, JP21K11751 and JP19K11960, Japan.
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Yamamura, K., Wang, Y., Fujisaki, E. (2022). Improved Lattice Enumeration Algorithms by Primal and Dual Reordering Methods. In: Park, J.H., Seo, SH. (eds) Information Security and Cryptology – ICISC 2021. ICISC 2021. Lecture Notes in Computer Science, vol 13218. Springer, Cham. https://doi.org/10.1007/978-3-031-08896-4_8
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