Watermarking of SAT Using
Combinatorial Isolation Lemmas
Rupak Majumdar
EECS Dept. University of California, Berkeley, CA
Jennifer L. Wong
CS Dept. University of California, Los Angeles, CA
DAC, June, 2001
[email protected]
Watermarking
• Embedding of Information for ID or
Proof of Authorship
• Technique Adds Extra Constraints to the
Problem
• Effective Intellectual Property Protection
Technique
Boolean Satisfiability
• Instance: A set of
variables V and a
collection C of
clauses over V.
• Solution: A truth
assignment for V
such that at least one
variable in each
clause evaluates
to true.
V = {v1, v2, v3}
C = {{v1, v2}, {v1’}, {v1’, v3},
{v1’, v2’, v3’}, {v3}}
Solution: v1 = False
v2 = True
v3 = True
Boolean Satisfiability
• First NP-complete Problem
• Applications
–
–
–
–
–
Deterministic Test Pattern Generation
Delay Fault Testing
Logic Verification/Synthesis
FPGA Routing
AI, Operations Research, Combinatorial
Optimization
• Backtrack Search, Local Search, Algebraic
Manipulation, Recursive Learning, …
• Watermarking Techniques
– Constraint-Based (Kahng ’98, Qu ‘99)
Fairness & Watermarking
• All possible watermarking signatures
of a given length result in a similar
solutions space
• Difficulty of finding a solution after
WM (given length) -> Runtime
• Quantify by # solutions
1111
0010
1101
Solution Space
Fairness & Watermarking
V = {v1, v2, v3, v4}
C = { (v1 + v3 + v4’) (v2 + v4) (v1’+ v2+ v3’+ v4) (v2’+v3+v4’) }
Embed a Signature of 4 bits
Signature Solutions Signature Solutions Signature Solutions Signature Solutions
0001
3
0011
3
0111
4
1111
4
0010
5
0101
4
1011
5
0100
2
0110
6
1101
4
1000
5
1001
5
1110
4
1010
5
1100
4
Ave Solution Distance 0.85 & Ave Variance 0.21
Fair Technique
Credibility & Watermarking
# solutions after WM
# solutions of with quality  threshold
• Effort required to find a particular solution
Credibility & Watermarking
V = {v1, v2, v3, v4}
C = { (v1 + v3 + v4’) (v2 + v4) (v1’+ v2+ v3’+ v4) (v2’+v3+v4’) }
Embed Signatures which are Multiples of Length 4
Signature Length (bits) 4
8 12
Ave # solutions
4.1 2.1 1.3
Min/Max # Solutions
2/5 1/4 0/3
Ave . Discrepency
0.5 0.6 0.72
16 20
24
28 32
0.5 0.2 0.3 0.1 0
0/1 0/1 0/1 0/1 0/0
0.5 0.32 0.42 0.18 0
Watermarking Flow
Signature
Initial
Specification
Additional
Constraints
Overconstrained
Specification
Off-the-Shelf
Problem Solver
Watermarked
Solution
Isolation Lemma (Valiant & Vazirani)
If f is any CNF formula in x1, x2, …, xn and w1, …wk
{0,1}n, then one can construct in linear time a formula
fk’ whose satisfying assignments v satisfy f and the
equations v•w1 = v•w2 = …= v•wk = 0. Furthermore, one
can construct a polynomial-size CNF formula fk in
variables x1, x2, …, xn,y1, …, ym for some m s.t. there is a
bijection between solutions of fk and fk’ defined by
equality on the values of x1, x2, …, xn.
• “NP is as easy as detecting unique solutions”
• Isolates a solution by randomized reduction
Unique and Fair Solutions to SAT
• CNF Formula over k Variables
• Add Multiple Watermarks of length k
– Adding Additional Constraints to
the Formula
Let a  b  c denote the CNF formula
(a’ V b’ V c’) (a’ V b V c) (a V b V c’) (a V b’ V c)
Unique and Fair Solutions to SAT
•
•
•
•
f is a CNF formula
vi is the ith variable in the set V
wj is the jth watermark of k-bits
xy is the yth created variable
f *= f  ( vi1  vi2  …  vij  1)
Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { {v
(v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }
Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { {v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }
v1 v2 v3 v4
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
1
1
1
1
0
0
1
1
1
1
1
1
1
0
1
1
1
1
0
1
Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { {v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }
Watermark = 0101
Positions of 1’s = {2, 4}
f
*=
f  ( v2  v4  1)
Unique and Fair Solutions to SAT
f
*=
f  ( v2  v4  1)
( v2  v4  1) = (x1  v2  v4 )  (x1  1)
Create x1
V* = {v1, v2, v3, v4, x1}
(x1  v2  v4)
{x1’,v2’, v4’} {x1’,v2, v4}
{x1,v2’, v4} {x1,v2, v4’}
(x1  1)
{x2’}
Unique and Fair Solutions to SAT
f **= { {v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’}
{x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’}}
V1
V2 V3
V4
X1  V2  V4
X1  1
0
0
1
1
0
1
1
1
0
1
1
1
0
0
0
0
1
1
1
1
0
1
0
1
Unique and Fair Solutions to SAT
V* = {v1, v2, v3, v4, x1, x2}
• f *= { {v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’}
{x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’}}
Watermark = 0011
Positions of 1’s = {3, 4}
f
**=
f
*
 ( v3  v4  1)
Unique and Fair Solutions to SAT
f **= f *  ( v3  v4  1)
( v3  v4  1) = (x2  v3  v4 )  (x2  1)
Create x2
V** = {v1, v2, v3, v4, x1, x2}
x2  v3  v4
x2  1
{x2’,v3’, v4’} {x2’,v3, v4}
{x2,v3’, v4} {x2,v3, v4’}
{x2’}
Unique and Fair Solutions to SAT
f **= {
{v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’}
{x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’} {x2’,v3’, v4’}
{x2’,v3, v4} {x2,v3’, v4} {x2,v3, v4’} {x2’} }
V1 V2 V3 V4
X1 
V2  V4
0
0
0
0
0
0
1
1
1
1
1
1
0
1
0
X1  1
1
1
1
X 2
V3  V4
0
0
0
X2  1
1
1
1
Experimentation Enviroment
• SAT Instances
– DIMACS Instances
– Created Instance with Known # Solutions
• Public Domain SAT Solvers
– WalkSAT (Selman & Kautz ’94)
– GSAT (Selman & Kautz ’92)
– NTAB (Crawford & Auton ‘93)
– Rel_SAT (Bayardo & Schrag ‘97)
Credibility
Created Instances
0.7
Normalized Number of
Solutions
0.6
0.5
0.4
0.3
0.2
0.1
0
v
0
1000
v
1 0 s - 0 s -2 0 0 0
10
-2 0 v
1 0 0 s s -2 0 0 v
0
1 0 0 0 0 0 s -2 5 v
00v
10
s -1 0
0
0
v
0
1
s -5 0
Ins
1 0 0 0 0 s -1 5 0 v
ta n
10v
2500
50sce
-1 0 v
s
100s
1x
2x
3x
4x
5x
10x
a
dded W
e
b
m
E
of
Length
te r m a r
k
Credibility (trade-off strength & runtime)
DIMACS
400
Normalized Runtime
(seconds)
350
300
250
200
150
100
50
0
100x
50x
25x
Le
ng
10x
th
o
Wa f Em
5x
ter
b
e
ma dd
ed
rk
2x
1x
i
u i8c
ii1 f225 2
6
pa
-0 9
i
r8- a1
7
f10 i16b
1
-c
1
jnh
0
0
pa f600 1
r
pa par3 16r16
2 -2 1
pa ha
ces
n
n
3
r
a
c
t
3
o
-c
g1
Ins
25 1 2- i4
.17
Fairness (Runtime)
DIMACS
Fairness (Runtime)
Created Instances
Conclusion
• Ultimate Fairness and Credibility
• Arbitrary Problem Application
• Connection between Watermarking &
Sound Mathematics & Theoretical
Computer Science