Abstract
Lattice basis reduction algorithms, such as LLL, play a very important role in cryptography, which usually aim to find a lattice basis with good “orthogonality”. However, not every lattice has an orthogonal basis, which means that we can only find some nearly orthogonal bases for these lattices. In this paper, we show that every integral lattice must have two types of special bases related to the orthogonality. First, any integral lattice with rank more than 1 has a class of bases such that the angle between any two basis vectors lies in \([\frac{\pi }{3},\frac{2\pi }{3}]\). Second, any integral lattice with rank more than 2 has a class of bases such that any basis can be divided into two sets and the vectors in every set are pairwise orthogonal. To obtain such results, we introduce the technique called unimodular congruence transformation for the Gram matrix.
This research is supported by the NNSF of China (No. 11201458, No. 11471314, No. 61121062, and No. 61572490), in part by 973 Project (No. 2011CB302401) and in part by the National Center for Mathematics and Interdisciplinary Sciences, CAS.
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We very thank the anonymous referees for their valuable suggestions on how to improve the presentation of this paper.
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Liu, R., Pan, Y. (2016). Two Types of Special Bases for Integral Lattices. In: Kim, Hw., Choi, D. (eds) Information Security Applications. WISA 2015. Lecture Notes in Computer Science(), vol 9503. Springer, Cham. https://doi.org/10.1007/978-3-319-31875-2_8
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DOI: https://doi.org/10.1007/978-3-319-31875-2_8
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