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WhydoesSAT-solverbreakit? MathematicsforComputerScience MIT6.042J/18.062J multipliercircuit Factoringreduces x kbits x y toSAT 2kbits y kbits _ __ _ _ ___ size 5k2 AlbertRMeyer March13,2013

SLIDE 1

Albert R Meyer March 13, 2013

SATfctr.1

Factoring reduces to SAT

Mathematics for Computer Science MIT 6.042J/18.062J

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

x y x⋅y

k bits k bits 2k bits multiplier circuit

size ≤ 5k2 _ _ __ _

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

x y

k bits k bits 2k bits multiplier circuit = n? equality circuit

n ::= pq

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

x y

k bits k bits 2k bits multiplier circuit = n? equality circuit

? ? ? ? ? ? ?

n ::= pq

SLIDE 2

Why does SAT-solver break it? Why does SAT-solver break it?

multiplier circuit equality circuit multiplier circuit equality circuit

k bits k bits 2k bits = n?

x

? ? SAT? ? ?

y

? ?

n ::= pq

Albert R Meyer March 13, 2013

SATfctr.‹#›

k bits k bits 2k bits = n?

x

? ? SAT? ? ?

y

? ?

n ::= pq

Albert R Meyer March 13, 2013

SATfctr.‹#›

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

x y

k bits k bits 2k bits multiplier circuit = n? equality circuit

? ? ? ? ? ? SAT?

n ::= pq

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

x y

k bits k bits 2k bits multiplier circuit = n? equality circuit 1

? ? ? ? ? ?

n ::= pq

SAT?

SLIDE 3

Why does SAT-solver break it?

multiplier circuit equality circuit

Why does SAT-solver break it?

multiplier circuit equality circuit

SAT?

k bits k bits 2k bits = n? 1

x

? ? ?

y

? ?

n ::= pq

Albert R Meyer March 13, 2013

SATfctr.‹#›

SAT?

k bits k bits 2k bits = n? 1

x

? ? ?

y

? ?

after 2k SAT tests…

Albert R Meyer March 13, 2013

SATfctr.‹#›

Albert R Meyer March 13, 2013

Why does SAT-solver break it?

SATfctr.‹#›

p q

k bits k bits 2k bits multiplier circuit = n equality circuit 1

1 1 1

found the factors p,q !

SAT-solvers work on formulas. But there’s a simple trick to find an equi-satisfiable formula about the same size as circuit.

Why does SAT-solver break it?

Albert R Meyer March 13, 2013

SATfctr.14

SLIDE 4

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6.042J / 18.062J Mathematics for Computer Science

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