On a Reciprocal Tarry-Escott Problem, the Distribution of Roots of Polynomials Modulo a Composite, and Sieve Methods

ponen, Bjorn M.（Croot, Ernest S.）摘要题目：关于一个互反的Tarry-Escott问题，多项式模复合的根的分布和筛法。申请人的研究计划有三个目标：他希望完成他关于Prouhet-Tarry-Escott问题的结果和方法的工作，并将其写在一篇或多篇论文中；他计划继续下去，并在一篇论文中写下他与一名密码学本科生的研究成果；并且，他希望继续发展一种新的筛法来确定整数薄集中素数的个数。作者在Prouhet-Tarry-Escott问题上的工作给出了一种构造多变量丢芬图方程组解的新方法，该方法可能会导致该领域许多未解问题的解决。密码学方面的工作主要围绕着证明攻击公钥密码系统的某些算法，即找到以整数q为模的多项式的低高度根（如Coppersmith的方法），不容易改进。最后，提议者计划继续发展一种筛法，用于在给定的整数集合中计算素数的数量，这允许人们使用关于该集合的附加分析信息（除了组合筛使用的通常数据），希望该方法将导致解决素数理论中一个或另一个已知的，困难的，未解决的问题。自古希腊时代以来，数学家们一直在试图理解质数是如何间隔的；也就是说，当一个人考虑越来越大的质数时，连续质数之间的距离是如何变化的？筛法是作为回答这类问题的理论工具而发展起来的；然而，有许多关于这种间隔的自然问题，他们目前无法回答。提案人的研究目标之一是完成开发一种新的筛选方法，他希望利用这种方法在这些尚未解决的问题上取得进展。提议者在密码学上的工作是由一个本科生研究某种方法（Coppersmith的算法）来攻击RSA密码系统，这是一个通过互联网发送安全数据的过程。申请人（和学生）计划继续研究数论中的一个相关问题，该问题的解决方案将表明这种攻击方法无法得到很大改进。最后，提议者计划继续研究prouet - tarry - escott问题。这个问题是数论中一个长期未解决的问题，倡议者正在开发新的方法来解决这个问题和其他类似的问题。

DMS-0301282Poonen, Bjorn M.(Croot, Ernest S.)AbstractTitle: On a Reciprocal Tarry-Escott Problem, the Distribution of Roots Of Polynomials Modulo a Composite, and Sieve MethodsThe Proposer's research project has three goals: He wishes to finish working on his results and methods on the Prouhet-Tarry-Escott problem, which he will write up in one or more papers;he plans to continue and to write up in a paper his work with an undergraduate on cryptology; and, he wishes to continue to develop a new sieve method for determining the number of primes in thin sets of integers. The proposer's work on the Prouhet-Tarry-Escott problem gives a new method for constructing solutions to certain systems of diophantine equations with many variables, and the method may lead to a solution of one of more unsolved problems in this area. The work on cryptology centers around showing that certain algorithms for attacking public-key cryptosystems, that find low-height roots of polynomials modulo an integer q (such as Coppersmith's method), cannot be easily improved.Finally, the proposer plans to continue developing a sieve methodfor counting the number of primes in a given set of integers, which allows one to use additional analytic information (besides the usual data used by thecombinatorial sieve) about the set, in the hopes that the method willlead to the solution of one or another known, difficult, unsolved problemsin prime number theory.Since the time of the ancient Greeks, mathematicians have been trying tounderstand how the prime numbers are spaced; that is, how does the distancebetween consecutive prime numbers vary as one considers larger and largerprimes? Sieve methods were developed as a theoretical tool for answeringthis type of question; however, there are many natural questions about such spacings that they currently cannot answer. One of the proposer'sresearch goals is to finish developing a new sieve method which he hopes to use to make progress on some of these unsolved problems. The proposer'swork on cryptology was motivated by research with an undergraduate ona certain method (Coppersmith's algorithm) for attacking the RSA cryptosystem, which is a procedure for sending secure data via the internet.The proposer (and student) plans to continue his workon a related problem in number theory, the solution of which would show thatthis method of attack cannot be much improved. Lastly, the proposer plansto continue his work on the Prouhet-Tarry-Escott problem. This problem is a longstanding unsolved question in number theory, and proposer isdeveloping new methods to make progress on it and other, similar problems.