Abstract
In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l ∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Anashin and A. Khrennikov, Applied Algebraic Dynamics (Springer, 2009).
V. Anashin, A. Khrennikov and E. Yurova, “T-functions revisited: New criteria for bijectivity/transitivity,” Designs Codes Crypt. 71, 383–407 (2014).
Y. Bugeaud, Approximation by Algebraic Numbers, Cambridge Tracts Math. (Cambridge Univ. Press, 2004).
M. J. Coster, A. Joux, B. A. Lamacchia, A.M. Odlyzko, C. P. Schnorr and J. Stern, “Improved low-density subset sum algorithms,” Comput. Compl. 2, 111–128 (1992).
N. Gama and Ph. Q. Nguyen, “Predicting lattice reduction,” N. P. Smart (Ed), EUROCRYPT, LectureNotes in Comp. Sci. 4965, 31–51 (Springer 2008).
H. Inoue and K. Naito, “The shortest vector problems in p-adic lattices and simultaneous approximation problems of p-adic numbers,” to appear in proc. of ICM Satellite Conf. 2014 (The 4th Asian Conf. on Nonlinear Anal. and Optim.).
J. C. Lagarias, “The computational complexity of simultaneous Diophantine approximation problems,” SIAM J. Computing 14, 196–209 (1985).
J. C. Lagarias and A. M. Odlyzko, “Solving low-density subset sum problems,” Journal ACM 32, 229–246 (1985).
D. Micciancio and S. Goldwasser, “Complexity of lattice problems, a cryptographic perspective,” Springer Int. Series in Engineering and Computer Science 671 (Springer, 2002).
J. van de Pol and N. P. Smart, “Estimating key sizes for high dimensional lattice-based systems,” IMA Int. Conf., Lecture Notes in Computer Science 8308, 290–303 (Springer, 2013).
A. Shamir, “A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem,” 23rd Ann. Symp. on Foundations of Computer Science 1982, 145–152 (1982).
V. G. Sprindžuk, Mahler’s Problem in Metric Number Theory, Izdat. Nauka i Tehnika (Minsk, 1967) [in Russian]. English transl. by B. Volkmann, Translations of Mathematical Monographs 25, (Amer. Math. Society, Providence, R.I., 1969).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci. Publ., 1995).
B. M. M. de Weger, “Approximation lattices of p-adic numbers,” J. Numb. Theo. 24, 70–88 (1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Inoue, H., Kamada, S. & Naito, K. Simultaneous approximation problems of p-adic numbers and p-adic knapsack cryptosystems - Alice in p-adic numberland. P-Adic Num Ultrametr Anal Appl 8, 312–324 (2016). https://doi.org/10.1134/S207004661604004X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S207004661604004X