Complexity structure analysis on the orbits of solutions of nonlinear partial differential equations by p-adic analysis

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540180
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto University
• Principal Investigator: NAITO Koichiro 熊本大学, 自然科学研究科, 教授 (10164104)
• Co-Investigator(Kenkyū-buntansha): SHIROMOTO Keisuke 熊本大学, 大学院自然科学研究科, 教授 (00343666) ; MISAWA Masashi 熊本大学, 大学院自然科学研究科, 教授 (40242672) ; WADA Takeshi 島根大学, 大学院総合理工学研究科, 教授 (70294139)
• Co-Investigator(Renkei-kenkyūsha): OGAWA Takayoshi 東北大学, 大学院理学研究科, 教授 (20224107)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: P-進数論 / 格子理論 / 暗号理論 / 力学系 / 非線形偏微分方程式

Outline of Final Research Achievements

Analyzing simultaneous rational approximations of irrational p-adic numbers by using multi-dimensional p-adic approximation lattices, we investigate some recurrent properties of discrete orbits given by quasi-periodic dynamical systems, the frequencies of which are weak Liouville p-adic numbers and we show some unpredictability properties of the orbits. For the symbolic dynamical systems given by the coefficient sequences of expansions of p-adic numbers we give some inequality relations between the recurrent dimensions and the topological entropy of these systems.
For the shortest vector problems of p-adic approximation lattices we compare the theoretical solutions given by the simultaneous approximation problems (SAP) and the numerical solutions estimated by the LLL algorithm. By using these results we propose a new lattice based cryptosystem, the private keys of which are the SAP solutions of p-adic lattices.

Free Research Field

基礎解析