Algebraic and Geometric Methods in Algorithmic Number Theory and Algorithmic Self-Assembly
算法数论和算法自组装中的代数和几何方法
基本信息
- 批准号:0306393
- 负责人:
- 金额:$ 34.5万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2003
- 资助国家:美国
- 起止时间:2003-09-01 至 2007-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Modern cryptography has brought several number theoretic problems to the spotlight, most notably integer factoring and the discrete logarithm problem. The significance of these problems grows as the scope of application for public key cryptography is broadened. Algorithmic self-assembly is an emerging research area where interesting number theoretic and algebraic connections, though unexpected at first, have recently been discovered. It has been suggested that self-assembly will ultimately be useful for circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In accordance with its practical importance, self-assembly has received increased theoretical attention over the last few years. This research addresses complexity theoretic issues in algorithmic number theory and algorithmic self-assembly. The objective is to develop efficient algebraic and geometric methods of computation in algorithmic number theory, cryptography, and algorithmic self-assembly.A primary focus of this research is the discrete logarithm problem over various groups, including the multiplicative groups of finite fields and groups associated with elliptic curves. A novel approach is taken which explores global pairings and local dualities in number fields and elliptic curves. The primary goal is to obtain results that address the computational complexity of these fundamental problems and clarify the foundational security of discrete-log based cryptosystems including elliptic curve cryptosystems. Constructive issues important to curve based cryptography will also be investigated. This research also addresses several fundamental issues concerning reversible self-assembly including, characterization and determination of equilibrium of a self-assembly system, determination of initial concentrations of different types of molecular units for achieving a targeted equilibrium behavior, and rate of convergence to equilibrium. Though these problems seem naturally require analytic tools to study, they turn out to have an interesting algebraic perspective as well. The primary goal is to obtain results that advance the mathematical and algorithmic theory of self-assembly, which is much needed for establishing guiding principles for experimental works in this area.
现代密码学带来了几个数论问题的聚光灯下,最显着的整数分解和离散对数问题。 随着公钥密码学应用范围的扩大,这些问题的重要性也在增加。分子自组装是一个新兴的研究领域,有趣的数论和代数的联系,虽然出乎意料的第一,最近被发现。 有人认为,自组装最终将用于电路制造,纳米机器人,DNA计算和无定形计算。 根据其实际的重要性,自组装在过去几年中得到了越来越多的理论关注。本研究针对演算数论与演算自组装的复杂性理论问题。目标是在算法数论、密码学和算法自组装中发展有效的代数和几何计算方法。本研究的主要焦点是各种群上的离散对数问题,包括有限域的乘法群和与椭圆曲线相关的群。一种新的方法,探讨全球配对和当地的对偶数域和椭圆曲线。 主要目标是获得解决这些基本问题的计算复杂性的结果,并澄清基于离散日志的密码系统,包括椭圆曲线密码系统的基本安全性。建设性的问题,重要的曲线为基础的密码学也将进行调查。本研究还解决了可逆自组装的几个基本问题,包括自组装系统的平衡的表征和确定,确定不同类型的分子单元的初始浓度以实现目标平衡行为,以及收敛到平衡的速率。 虽然这些问题似乎自然需要分析工具来研究,但它们也有一个有趣的代数视角。其主要目标是获得推进自组装的数学和算法理论的结果,这对于建立该领域实验工作的指导原则是非常必要的。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Ming-Deh Huang其他文献
Ming-Deh Huang的其他文献
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{{ truncateString('Ming-Deh Huang', 18)}}的其他基金
CT-ISG: The Foundational Security of Elliptic Curve Cryptography
CT-ISG:椭圆曲线密码学的基础安全性
- 批准号:
0627458 - 财政年份:2006
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
Efficient Randomized Algorithms for Multivariate Algebraic Computations
多元代数计算的高效随机算法
- 批准号:
9820778 - 财政年份:1999
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Computational Number Theory and Computational Algebraic Geometry
计算数论和计算代数几何
- 批准号:
9412383 - 财政年份:1995
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
PYI: Arithmetic and Geometric Methods in Computational Complexity
PYI:计算复杂性中的算术和几何方法
- 批准号:
8957317 - 财政年份:1989
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
The Computational Complexity of Problems Related to Number Theory
数论相关问题的计算复杂性
- 批准号:
8701541 - 财政年份:1987
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
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