Compute Cluster Nodes for Factoring Integers

数字字段筛选（NFS）是已知最快的整数分解算法。NFS的多项式选择和筛选步骤易于并行化，可以在许多独立的机器上执行，节点间通信很少。普渡大学的秃鹰系统拥有数万台机器，可以很好地执行这部分算法。但是，NFS的线性代数步骤最好在具有大内存的多核机器的几个核心上执行。尽管普渡大学的秃鹰系统对所有教员开放，但其多核机器的使用仅限于那些通过拨款帮助支付机器费用的教员。多核服务可以通过TeraGrid获得，但这项服务是有限的，需要每年申请一次。该奖金将用于购买新的多核机器的一部分，然后PI和他的学生将在其生命周期内访问它。该机器将用于运行NFS的线性代数步骤，以分解重要数字，例如Cunningham项目中的数字。在Condor系统进行筛分后，多核机器将发现坎宁安数的分解，这在整个数学中具有重要的应用。这些应用包括十进制分数的周期长度，新的代数恒等式的发现，完全数，算术函数的求值，确定环域的结构，椭圆曲线密码系统的构造，以及线性反馈移位寄存器的设计。此外，每次分解所花费的运行时间将为分解整数的难度提供另一个基准，并告诉密码学家为某些密码选择多大的参数。这些密码和椭圆曲线密码系统用于保护商业和政府的私人通信以及许多其他目的。PI的学生将使用多核机器，获得超级计算机编程的经验，并学习分解整数的最佳技术。当他们毕业时，这些学生将把这些知识运用到研究、教育、政府和商业领域。

The Number Field Sieve (NFS) is the fastest known integer factoring algorithm. The polynomial selection and sieving steps of NFS are easily parallelizable and may be performed on many independent machines with little internode communication. Purdue's Condor system has tens of thousands of machines available and is well-positioned to perform this part of the algorithm. However, the linear algebra step of NFS is best performed on several cores of a multicore machine with large memory.Although Purdue's Condor system is available to all faculty, use of its multicore machines is restricted to those faculty who helped to pay for the machine through a grant. Multicore service is available through TeraGrid, but this service is limited and requires annual applications.The award will be used to purchase part of a new multicore machine and then the PI and his students will have access to it during its lifetime. The machine will be used to run the linear algebra step of NFS to factor important numbers, such as those in the Cunningham Project.The factorizations of Cunningham numbers, which will be discovered by the multicore machine after the Condor system performs the sieving, have significant applications throughout mathematics. These applications include period lengths of decimal fractions, discovery of new algebraic identities, perfect numbers, evaluation of arithmetic functions, determining the structure of cyclotomic fields, construction of elliptic curve cryptosystems, and design of linear feedback shift registers. Furthermore, the running time taken by each factorization will provide another benchmark for the difficulty of factoring integers, and tell cryptographers how large to choose parameters for certain ciphers. These ciphers and the elliptic curve cryptosystems are used to secure private communication by business and government and for many other purposes. The PI's students will use the multicore machine and gain experience programming supercomputers as well as learn the best techniques for factoring integers. When they graduate, these students will take this knowledge to careers in research, education, government and business.