1. Lucas-Lehmer Primality Tester Presentation 2: Architecture Proposal February 1, 2006 Team: W-4 Nathan Stohs W4-1 Brian Johnson W4-2 Joe Hurley W4-3 Marques Johnson W4-4 Design Manager: Prateek Goenka Overall Objective: The Testing of Prime numbers

2. Status Finished –Project Chosen –C simulations –Behavioral Verilog To Do –Floor Plan –Schematic Design –Layout –Simulaitons

3. Applications 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 www.mersenne.org

4. Application 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. Our goal is to build a chip that uses less power

5. C-Code #include unsigned int mod_square(int value, int mod, int p); int main(int argv, char** argc) { if (argv != 2) { printf("Usage: %s p\n", argc[0]); exit(0); } int p = atoi(argc[1]); int value = 4; int i; int mod = (2 << (p-1)) - 1; for (i=1; i<=(p-2); i++) { int oldval = value; value = mod_square(value, mod, p); printf("round %d: (%d * %d - 2) mod %d = %d\n", i, oldval, oldval, mod, value); } if (value == 0) printf("2^%d-1 is prime\n", p); else printf("2^%d-1 is not prime\n", p); return 0; } unsigned int mod_square(int value, int mod, int p) { int acc, i; for(acc=-2, i=0; i<(sizeof(int)*8-1); i++) { int a = (value >> i) & 1; int temp; if (a) { if (i-p > 0) temp = value << (i-p); else temp = value >> (p-i); acc = acc + temp + ((value << i) & ((1 << p) - 1)); } if (acc >= mod) acc = acc - mod; } return acc; }

6. Simulation Results Fully simulates Mersenne Primes up to 30 Using the algorithm below Numbers above 2 30 make the code overflow because of the squaring in the algorithm 2 P -1 is prime if and only if S p-2 is zero in this sequence: S 0 = 4, S N = (S N-1 2 - 2) mod (2 P -1) S 0 = 4 S 1 = (4 * 4 - 2) mod 127 = 14 S 2 = (14 * 14 - 2) mod 127 = 67 S 3 = (67 * 67 - 2) mod 127 = 42 S 4 = (42 * 42 - 2) mod 127 = 111 S 5 = (111 * 111 - 2) mod 127 = 0 So for 2 7 -1: Result says it is prime

7. Design Decisions What type of adder will we use? Should we broaden the scope of our project? Do we get into mod multiply and mod add for (2 p +1)?

8. Significance of (2 p +1) As previously stated (2 p -1) finds Mersenne Primes (2 p +1) however, can be used to find Fermat numbers This form is also used for encryption

9. Adder Options Parallel-prefix Adders –Serial-Prefix, which is the smallest but slowest –Sklansky parallel-prefix, which is the fastest but lager End-around Carry-Adders –Some take 2 propagations in series, which is slow –Faster ones require two adders and a multiplexer, which is big

10. Modulo Multiply (2 n -1) Has to do with partial products The algorithm calls for a Mod every time that there is a chance Example: Mod 127 23 X 56 18 120 150 1000 Leave Mod

11. Verilog Code module mod_square( value, mod, p, newvalue ); input [31:0] value, mod, newvalue; input [4:0] p; reg [31:0] acc=-2; reg [31:0] temp; reg a; acc=-2; initial begin for(i=0; i<31; i = i + 1) begin a = (value >> i) & 1; if (a) begin if (i-p > 0) temp = value << (i-p); else temp = value >> (p-i); acc = acc + temp + ((value << i) & ((1 << p) - 1)); end if (acc >= mod) acc = acc - mod; end

12. Reg:P Mod Calc (1<<p)-1 Reg:2 P -1 Mod Serial Multiplier Counter 0:P-2 Reg =-2 Comparator Counter = P-2 Check 0 Reg =4 Block Diagram for Prime Numbers P Out

13. Simulation Results Fully simulates Mersenne Primes up to 30 Using the algorithm below Numbers above 2 30 make the code overflow because of the squaring in the algorithm 2 P -1 is prime if and only if S p-2 is zero in this sequence: S 0 = 4, S N = (S N-1 2 - 2) mod (2 P -1) S 0 = 4 S 1 = (4 * 4 - 2) mod 127 = 14 S 2 = (14 * 14 - 2) mod 127 = 67 S 3 = (67 * 67 - 2) mod 127 = 42 S 4 = (42 * 42 - 2) mod 127 = 111 S 5 = (111 * 111 - 2) mod 127 = 0 So for 2 7 -1: Result says it is prime

14. Mod Serial Multiplier Block Diagram multiplicandmultiplier Zero? Mod add Mod Register p 2 p -1

15. What’s Next Continue researching adders Choose which adder to use Decide if we will broaden the scope of our project Start doing layout

16. Problems Prime numbers do not get interesting until P is in the thousands Up to P=32 we could use Registers If we want to take this to the next level we need to use SRAM but will that make this too big of a project?

17. Questions?