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

DMS-0301282 Poonen，Bjorn M.（Croot，Ernest S.）关于Tarry-Escott问题的根的分布 多项式模复合，和筛方法的提议者的研究项目有三个目标：他希望完成工作，他的结果和方法的Prouhet-Tarry-Escott问题，他将写在一个或多个文件;他计划继续和写在一份文件中，他的工作与本科生密码学;并且，他希望继续发展一种新的筛选方法，用于确定稀疏整数集合中素数的个数。 Prouhet-Tarry-Escott问题的提出者的工作给出了一种新的方法来构造多变量丢番图方程组的解，该方法可能导致解决这一领域中的一个或多个未解决的问题。 密码学的工作集中在证明某些攻击公钥密码系统的算法，即找到多项式模整数q的低高度根。（如Coppermith的方法），不能轻易地改进。最后，提议者计划继续开发一种筛子方法，用于计算给定整数集中素数的数量，它允许人们使用额外的分析信息（除了通常的数据所使用的组合筛）的集合，希望该方法将导致解决一个或另一个已知的，困难的，自古希腊时代以来，数学家们一直试图了解素数是如何间隔的;也就是说，当人们考虑越来越大的素数时，连续素数之间的距离是如何变化的？ 筛方法是作为回答这类问题的理论工具而开发的;然而，关于这种间距，有许多自然问题目前无法回答。 提议者的研究目标之一是完成开发一种新的筛选方法，他希望利用该方法在其中一些未解决的问题上取得进展。 该提议者的密码学工作的动机是与一名大学生研究某种方法（Coppermith算法）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。提议者（和学生）计划继续他的工作在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不会有太大的改进。 最后，提议者计划继续他对Prouhet-Tarry-Escott问题的研究。 这个问题是数论中一个长期未解决的问题，提出者正在发展新的方法来解决它和其他类似的问题。

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.