Lucas-Lehmer Primality Tester Presentation 1: Proposal Team: Nathan Stohs Joe Hurley Brian Johnson Marques Johnson

Applications, History Lucas-Lehmer is a test used to search for Mersenne Primes. Mersenne Primes are primes of the form 2^p – 1 Given p, will conclusively determine primality after p-2 iterations of the algorithm. Computationally heavy, but numbers tested independently, so easily distributable. Difficultly lies in choosing an implementation

Applications, contd. A hardware implementation of this algorithm is not going to save any lives. Why important then? Mersenne primes (found with this test) are the largest prime numbers we know of today. A pool of over 70,000 computers currently run an implementation of this algorithm, with aggregate performance peaking at 14 Teraflops. This is not intended to be a commercial product.

Algorithm Somewhat basic. Mp is Prime iff Simple iterative structure, with p iterations Includes modulo, squaring, arithmetic Need for fast squaring

Design Are we going to be able to beat a 4 GHz Pentium 4 implementing this algorithm in hand optimized assembly using FFTs for squaring? No. However, it would use much less power Design is limited by the maximum value of p which we want to test, due to squaring. Design scales quite nicely with max p

Flowchart Squaring operation will dominate the design in layout area Approx 10k-15k transistors Due to math constraints, will not require full blown divison Approx 4k-5k transistors Check for 0 residue on last iteration Arithmetic, registers, other controls Approx 3k transistors

Problems Will never be able to test untested numbers, at best an experiment for future work. A lot will depend on how we implement the squaring, leading too.. May be too easy! Backup plan is to use FancyMath™ to aid performance and make the project more “interesting”. Other ideas considered: Implementation of Blowfish Cipher