1. Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols Emanuele Viola, IAS (Work partially done during postdoc at Harvard) Joint work with Avi Wigderson June 2007

2. Computational model M –E.g. M = circuits, GF(2) polynomials, multiparty protocols Lower bound: 9 explicit f : {0,1} n ! {0,1} not in M? [This talk] Correlation bound: 9 explicit f : Cor(f,M) ·   ? Cor(f,M) := max p 2 M | E x [ (-1) f(x) + p(x) ] | 2 [0,1] –Want  =  (|x|) small Motivation: Correlation bound ) pseudorandom generator [Yao,Nisan Wigderson] lower bound for M’ ¾ M [Razborov, Hajnal et al.] Basic questions

3. XOR lemma Generic way to boost correlation bound f © k (x 1,…, x k ) := f(x 1 ) ©  © f(x k ) Hope: Cor(f © k,M) · Cor(f,M)  (k) Theorem [ Yao, Levin, Goldreich Nisan Wigderson, Impagliazzo,… ] XOR lemma for M = circuits

4. Problem with XOR lemma Proofs of the XOR lemma [Y,L,GNW,…] don’t work in any model M for which we have lower bounds! Conjecture[V]: Black-box proof of XOR lemma for M ) M closed under polynomially many majorities poly many majorities Power of M Cannot prove lower bounds [Razborov Rudich] Cannot prove XOR lemma [V]

5. Study two fundamental models 1) GF(2) polynomials 2) multiparty protocols Prove new XOR lemmas Improve or simplify correlation bounds Overview of our results

6. Use norm N(f) 2 [0,1] : (I) Cor(f,M) ¼ N(f) (II) N(f © k ) = N(f) k Proof of the XOR lemma: Cor(f © k,M) ¼ N(f © k ) = N(f) k ¼ Cor(f,M) k Q.e.d. Proof for circuits [L,GNW,…] different: “simulation” Main technique

7. Outline Overview GF(2) polynomials Multiparty protocols Direct product

8. P d := polynomials p : {0,1} n ! {0,1} of degree d E.g., p = x 1 + x 5 + x 7 d = 1 p = x 1 ¢ x 2 + x 3 d = 2 Recall correlation bound: Cor(f,P d ) ·  =  (n,d), 8 p 2 P d : | E x [e( f(x) + p(x) )] | ·  (e(X):=(-1) X ) Note: Fundamental barrier: d = log 2 n,  = 1/n ? GF(2) polynomials

9. Gowers norm Idea: Measure correlation with degree-d polynomials by checking if random (d+1)-th derivative vanishes –Other view: perform random parity check Derivative D y p(x) := p(x+y) + p(x) –E.g. D y (x 1 x 2 + x 3 ) = y 1 x 2 + x 1 y 2 + y 1 y 2 + y 3 –p degree d ) D y p(x) degree d-1 –Iterate: D y,y’ p(x) := D y ( D y’ p(x)) t-th Gowers norm of f : {0,1} n ! {0,1}: (t = d+1) U t (f) := E x,y 1,…,y t [e( D y 1,…,y t f(x) )] (e(X):=(-1) X ) Note: p degree d, U d+1 (p)=1, and U d+1 (f+p)=U d+1 (f)

10. Properties of norm U t (f) := E x,y 1,…,y t [e(D y 1,…,y t f(x))] (e(X):=(-1) X ) (I) U d+1 (f) ¼ Cor(f,P d ) : Lemma [Gowers, Green Tao] : Cor(f,P d ) · U d+1 (f) 1/2 d Lemma [Alon Kaufman Krivelevich Litsyn Ron] (Property testing) : Cor(f,P d ) · 1/2 ) U d+1 (f) · 1-2 -  (d) (II) U(f © k ) = U(f) k –Follows from definition

11. XOR lemma for GF(2) polynomials (I) U d+1 (f) ¼ Cor(f,P d ) : Lemma [G, GT] : Cor(f,P d ) · U d+1 (f) 1/2 d Lemma [AKKLR]: Cor(f,P d ) · 1/2 ) U d+1 (f) · 1-2 -  (d) (II) U(f © k ) = U(f) k Theorem[This work]: Cor(f,P d ) · 1/2 ) Cor(f © k,P d ) · exp(-k/2 O(d) ) Proof: Cor(f © k,P d ) · U d+1 (f © k ) 1/2 d = U d+1 (f) k/2 d · (1-2 -  (d) ) k/2 d Q.e.d.

12. Our correlation bound for Mod 3 Mod 3(x 1,…,x n ) := 1 iff 3 |  i x i Theorem [Bourgain ‘05]: Cor(Mod 3,P d ) · exp(-n/8 d ) Theorem [This work]: Cor(Mod 3,P d ) · exp(-n/4 d ) New proof (formalize over ): Cor((x 1 ) © n mod 3,P d ) · U d+1 ((x 1 ) © n mod 3) 1/2 d = U d+1 (x 1 mod 3) n/2 d = exp(-n/4 d ) Q.e.d.

13. Outline Overview GF(2) polynomials Multiparty protocols Direct product

14. d parties wish to compute f : X 1 £ X 2 … £ X d ! {0,1} Party i knows all inputs except x i (on forehead) Cost of protocol = communication Applications to nearly every area of computer science –Circuit/proof complexity, PRGs, TM’s, branching programs…  d,c := d-party protocols communicating c bits Multiparty protocols [Yao,Chandra Furst Lipton] X1X1 X2X2 X3X3

15. Multiparty norm [Babai Nisan Szegedy,Chung Tetali,Raz] [This work]: coherent view like GF(2) polynomials  © d := each party sends one bit, output is XOR –Note: P d-1 µ  © d µ  d,d [Hastad Goldmann] d-party norm of f, captures correlation with  © d R d (f) := E X 0,X 1 [e(  b 2 { 0,1 } d f(X b ) )] (e(X):=(-1) X ) where X 0 = (x 1 0,…,x d 0 ), X 1 = (x 1 1,…,x d 1 ), X b = (x 1 b 1,…,x d b d ) (Perform random parity check) Note: p 2  © d, R d (p)=1, and R d (f+p)=R d (f)

16. XOR lemma for protocols (I) R d (f) ¼ Cor(f,  d,c ) : Lemma[BNS,CT,R]: Cor(f,  d,c ) · 2 c ¢ Cor(f,  © d ) Lemma[BNS,CT,R]: Cor(f,  © d ) · R d (f) 1/2 d Lemma[This work]: Cor(f,  © d ) ¸ R d (f) (II) R d (f © k ) = R d (f) k Theorem[This work]: Cor(f,  © d ) ·  ) Cor(f © k,  d,c ) · 2 c ¢  k/2 d 2-party case already known [Shaltiel]

17. Outline Overview GF(2) polynomials Multiparty protocols Direct product

18. Want to compute f k (x 1,…,x k ) := (f(x 1 ),…,f(x k )) 2 {0,1} k Suc(f,M) := max p 2 M |Pr[f k ( x 1,…,x k )  p( x 1,…,x k )]| Lemma[This work]: Suc(f k,M) · Cor(f © k,M) –Easy converse of [Goldreich Levin]; works whenever © 2 M –Simplifies result from [Impagliazzo Wigderson ‘97] XOR lemmas for P d,  d,c ) direct products [Parnafes Raz Wigderson] direct product for 2-party –Parameters incomparable

19. Study two fundamental models: –1) GF(2) polynomials –2) multiparty protocols Prove new XOR lemmas, correlation bounds, direct products Via norm N : (I) Cor(f,M) ¼ N(f), (II) N(f © k ) = N(f) k –Powerful technique: new PRGs for GF(2) polynomials [BV] Questions: Are XOR lemmas tight? –[This work] “Ideal” XOR lemma  )  2 for 2-party false Bounds for (log n)-degree polynomials? Conclusion

20. Thank you!