2000 character limit reached
Effective module lattices and their shortest vectors (2402.10305v1)
Published 15 Feb 2024 in math.NT, cs.IT, and math.IT
Abstract: We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.
- Moments of the number of points in a bounded set for number field lattices. arXiv:2308.15275v2, 2023.
- Explicit hard instances of the shortest vector problem. In Post-Quantum Cryptography: Second International Workshop, PQCrypto 2008 Cincinnati, OH, USA, October 17-19, 2008 Proceedings 2, pages 79–94. Springer, 2008.
- A complete analysis of the BKZ lattice reduction algorithm. Cryptology ePrint Archive, Paper 2020/1237, 2020.
- Miklós Ajtai. Random lattices and a conjectured 0-1 law about their polynomial time computable properties. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 733–742. IEEE, 2002.
- The nearest-colattice algorithm. arXiv:2006.05660, 2020.
- Worst-case to average-case reductions for module lattices. Designs, Codes and Cryptography, 75(3):565–599, 2015.
- Reductions from module lattices to free module lattices, and application to dequantizing module-LLL. Cryptology ePrint Archive, Paper 2023/886, 2023.
- On ideal lattices and learning with errors over rings. In Henri Gilbert, editor, Advances in Cryptology – EUROCRYPT 2010, pages 1–23, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg.
- Pseudorandomness of ring-LWE for any ring and modulus. In STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 461–473. ACM, New York, 2017.
- Seungki Kim. Adelic Rogers integral formula. J. Lond. Math. Soc. (2), 109(1), 2024.
- Nathan Hughes. Mean values over lattices in number fields and effective Diophantine approximation. arXiv:2306.02499, 2023.
- On the equidistribution of Hecke points. Forum Math., 13(2), 2003.
- Philippe Moustrou. On the density of cyclotomic lattices constructed from codes. Int. J. Number Theory, 13(05):1261–1274, 2017.
- Yonatan R Katznelson. Integral matrices of fixed rank. Proc. Amer. Math. Soc., 120(3):667–675, 1994.
- Application of automorphic forms to lattice problems. J. of Math. Crypt., 16(1):156–197, 2022.
- Claude A. Rogers. Existence theorems in the geometry of numbers. Ann. of Math. (2), 48(4):994–1002, 1947.
- Dense packings via lifts of codes to division rings. IEEE Trans. Inform. Theory., 69(5), 2022.
- Hecke operators and equidistribution of Hecke points. Invent. Math., 144(2):327–351, 2001.
- Wolfgang M. Schmidt. On heights of algebraic subspaces and Diophantine approximations. Ann. of Math. (2), 85(3):430–472, 1967.
- Jeffrey Lin Thunder. Asymptotic estimates for rational points of bounded height on flag varieties. Comp. Math., 88(2):155–186, 1993.
- John William Scott Cassels. An introduction to the geometry of numbers. Springer Science & Business Media, 2012.