Factoringreduces x kbits x y toSAT 2kbits y kbits _ __ _ _ - PowerPoint PPT Presentation

Why does SAT-solver break it? Mathematics for Computer Science MIT 6.042J/18.062J multiplier circuit Factoring reduces x k bits x ⋅ y to SAT 2k bits y k bits _ __ _ _ ___ � size ≤ 5k2 Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr.1 SATfctr .‹#› Why does SAT-solver break it? Why does SAT-solver break it? multiplier circuit multiplier circuit equality circuit equality circuit 0 ? x x k bits k bits ? ? = n? = n? 2k bits 2k bits ? ? y y k bits k bits ? ? n :: = pq n :: = pq Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr .‹#› SATfctr .‹#› 1

Why does SAT-solver break it? Why does SAT-solver break it? multiplier circuit equality circuit multiplier circuit equality circuit 0 0 ? ? x x k bits k bits ? ? ? ? = n? = n? SAT? SAT? 2k bits 2k bits ? ? ? ? y y k bits k bits ? ? ? ? n :: = pq n :: = pq Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr .‹#› SATfctr .‹#› Why does SAT-solver break it? Why does SAT-solver break it? multiplier circuit multiplier circuit equality circuit equality circuit 0 0 0 1 x x k bits k bits ? ? ? ? = n? = n? SAT? SAT? 2k bits 2k bits ? ? ? ? y y k bits k bits ? ? ? ? n :: = pq n :: = pq Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr .‹#› SATfctr .‹#› 2

Why does SAT-solver break it? Why does SAT-solver break it? multiplier circuit equality circuit multiplier circuit equality circuit 0 0 1 1 x x k bits k bits 0 0 ? ? = n? = n? SAT? SAT? 2k bits 2k bits ? ? ? ? y y k bits k bits ? ? ? ? n :: = pq after 2k SAT tests… Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr .‹#› SATfctr .‹#› Why does SAT-solver break it? Why does SAT-solver break it? multiplier circuit equality circuit SAT-solvers work on formulas. 0 1 p k bits But there’s a simple trick to find 0 1 = n an equi-satisfiable formula 2k bits 1 about the same size as circuit. q 0 k bits 1 0 found the factors p,q ! Albert R Meyer March 13, 2013 Albert R Meyer March 13, 2013 SATfctr .‹#› SATfctr.14 3

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