1. The Power of Super-Log Number of Players Arkadev Chattopadhyay (TIFR, Mumbai) Joint with: Michael Saks ( Rutgers )

2. A Conjecture f:{0,1} n ! {0,1} 0111111 1101101 0111100.............. 1111110 X1X1 X2X2 X3X3 XkXk (f ± g) ( X 1, X 2, , X k ) = ) MAJ ± MAJ  ACC 0 110 1000 f(g(C 1 ), g(C 2 ),…, g(C n )) n Question: Complexity of (MAJ ± MAJ)? Observation: a la Beigel-Tarui’91 ) MAJ  ACC 0 Proposed by Babai-Kimmel-Lokam’95 g:{0,1} k ! {0,1}.

3. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A Some Upper Bounds SYM ± AND {GIP, Disj,…} Popular Names SYM ± g {GIP, MAJ ± MAJ, Disj…} Deterministic  (n/ 2 k + k ¢ log n ). O(k.(log n) 2 ), k ¸ log n + 2. Grolmusz’91, Pudlak Babai-Gal-Kimmel-Lokam’02 k ¸ 3 Ada-C-Fawzi-Nguyen’12 g: compressible and symmetric SYM ± ANY Simultaneous Almost- Simultaneous O(k.(log n) 2 ), k ¸ log n + 4.

4. Block Composition f:{0,1} n ! {0,1} 011111 110111 011110............ 111110 X1X1 X2X2 X3X3 XkXk (f n ± g r ) ( X 1, X 2, , X k ) = ) MAJ  ACC 0 f(g(A 1 ), g(A 2 ),…, g(A n )) n = 2 r = 3 Conjecture: Fact: Babai-Gal-Kimmel-Lokam’02 g:{0,1} k £ r ! {0,1}. Still Open! A1A1 A2A2 Even for interactive protocols

5. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A Our Result Theorem: SYM n ± ANY r has a 2-round k-party deterministic protocol of cost when, Remark 1: First protocol for r > 1. Remark 2: Corollary: MAJ ± MAJ r has efficient protocol when r is poly-log and k is a sufficiently large poly-log. r = O(log log n)

6. Main Ingredients Computing k-1 degree polynomials is easy for k- players. (Goldman-Hastad’90’s) Degree reduction by basis change. (New Idea)

7. Low degree Polynomials x3 x5 x7x3 x5 x7 Alice Bob x 6 x 10 x 11 x2 x8 x9x2 x8 x9 Charlie Alex x 1 x 4 x 12 Bob, Charlie Alice Alex Alice, Charlie Bob, Alex deg( P ) = 3 k = 4 > deg( P ) Simultaneous k -party deterministic protocol Cost = O( k ¢ log| F |)

8. A Polynomial Fantasy f:{0,1} n ! {0,1} (SYM ± g) ( C 1, C 2, , C n ) = Fantasy: P high (Ci) = 0 for all i !! g:{0,1} k ! {0,1} µ F p. Prime p > n g(X) ´ P(X 1, ,X k ) deg(P) · k P ´ P high (X) + P low (X) deg < k deg = k easy k-player protocol of cost = k.log(p) Bad

9. Shifted Basis 0110 0011 0111 1000 Example: Fact: B u is a basis for every u 2 {0,1} k u = 0 k gives standard basis u -shifted A Def: u is good for A if for all column C of A, u and C agree on some co-ordinate. 1 0 0 0 good 0 1 0 1 bad no agreement

10. Good is Really Good (SYM ± g) ( C 1, C 2, , C n ) = Fact: P high ( C ) = 0 for all C if u is good for A. easy k-player protocol of cost log(p) Bad u Apply u -shift u u Zeroed out!

11. Good Shifts Are Aplenty Observation: If k À log n + 1, Player k spots many good shifts. Protocol: Player k announces a good shift u. All players compute their portions using u. Simultaneous! Cost = k - 1 Cost = k ¢ log(p) = O( k ¢ log n ) Extends to r = O(log log n).

12. Future Direction Can we go to r = O(log n)? Is ? Thank You!