1. Shuchi Chawla, Carnegie Mellon University Guessing Secrets Efficiently Shuchi Chawla 1/23/2002

2. Shuchi Chawla, Carnegie Mellon University A game similar to 20 questions  Alice has k secret n-bit strings  Bob wants to learn them  Bob can ask binary questions about one secret  Alice uses any secret of her choice to reply  What should he ask?  How should he figure out the secrets?  How many questions are needed?  Can he learn all?

3. Shuchi Chawla, Carnegie Mellon University How it all began…  Problem first defined by Chung, Graham & Leighton, EJC’01  Efficiently solve for the 2 secrets case  Alon et al, SODA’02  Chung et al, SODA’02  Micciancio & Segerlind, Unpublished  K secrets case  Largely open

4. Shuchi Chawla, Carnegie Mellon University What can Bob hope to learn?  Consider the two secret case  He may hope to learn one of the secrets  What if Alice has 3 in mind and always goes with the majority of the two  All the three pairs are always consistent with the answers!

5. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph

6. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part

7. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other

8. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other

9. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other

10. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other

11. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other

12. Shuchi Chawla, Carnegie Mellon University Formalizing the problem  Alice’s secret is an edge in the graph  Bob specifies a partition of the graph  Alice chooses one part  Bob deletes all edges inside the other Bob cant do any better!

13. Shuchi Chawla, Carnegie Mellon University A star or triangle  Alice always picks the bigger set  Bob cant delete any more edges!  Bob cannot learn more if all pairs of edges intersect - star or triangle The model generalizes easily to k secrets k-uniform hypergraphs

14. Shuchi Chawla, Carnegie Mellon University Guessing “Efficiently”  Few, short questions, fast inversion algorithm  [Chung et al’01]:  At least 3n-5 queries required – information theoretic argument  Algorithm using 4n+3 queries – each 2 n bit long, O(2 n ) time  [MS02] improve this to O(n)-bit questions, O(n 2 ) time  [Chung et al’01] achieve similar bounds using inner product questions  All the above are Adaptive strategies

15. Shuchi Chawla, Carnegie Mellon University Oblivious strategies  Specify all questions up-front  Desirable – ease of implementation & inversion  [Alon et al]  Basic idea: view the strings as codes

16. Shuchi Chawla, Carnegie Mellon University  Sequence of functions f i : [N] -> {0,1}  Each secret x  [N]   f 1 (x),f 2 (x),…,f m (x)   Call this “code” C  A code is (2,2)-separating iff For all a,b,c,d  [N]  i for which C(a)=C(b)  C(c)=C(d)  Such a code would solve the guessing problem

17. Shuchi Chawla, Carnegie Mellon University Constructing a good code  If a binary linear code C is  –biased, for some  < 1/14, then it is (2,2)-separating  [ABN+92] For any  > 0 there exists an explicit family of constant rate binary linear  -biased codes.  Inversion can be done using list decoding in time O(n 3 )

18. Shuchi Chawla, Carnegie Mellon University Obtaining the solution  Say the secret is (x 1,x 2 )  Bob uses code C  Alice answers a = (a 1,a 2,…,a m )  For each i, C(x 1 ) i =a i or C(x 2 ) i =a i  C is  -biased  for at least (1/2-  )n i, C(x 1 ) i =C(x 2 ) i  Either C(x 1 ) or C(x 2 ) is within distance (1/4+  )n of a

19. Shuchi Chawla, Carnegie Mellon University Decoding a  wlog C(x 1 ) is within distance (1/4+  )n of a 1.Decode C upto a distance of (1/4+  )n 2.For each x returned in the previous step, remove the answers that correspond to x, and then perform an “erasure list decoding” of C obtaining y 3.Return all pairs (x,y)  Paper proves we get either a star or triangle

20. Shuchi Chawla, Carnegie Mellon University K-secrets, k>2  Looking for intersecting k-uniform hypergraphs  Representation is difficult -what does an intersecting hypergraph look like?  [Alon et al] show connection to 2k-universal families of binary strings

21. Shuchi Chawla, Carnegie Mellon University 2k-universal families of binary strings  Find a small subset S  {0,1} N such that:  For every set {i 1,…,i 2k } and string (a 1,…,a 2k ),  x  S such that x i y =a y for each y. 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 S = N=6 k=1

22. Shuchi Chawla, Carnegie Mellon University 2k-universal families of binary strings  Find a small subset S  {0,1} N such that:  For every set {i 1,…,i 2k } and string (a 1,…,a 2k ),  x  S such that x i y =a y for each y. 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 S = Our functions Secrets N=6 k=1

23. Shuchi Chawla, Carnegie Mellon University Filling in details..  There is an explicit strategy for Bob that uses ~O(k2 6k n) questions [ABN+92]  [Alon et al’02] also give a poly(n) time algorithm for inversion that finds a set of size poly(n) secrets that contains at least one secret of Alice.