1. Emanuele Viola Harvard University October 2005
Pseudorandom Bits for Low Complexity Classes: New Results and Applications
Emanuele Viola
Harvard University
October 2005

2. Pseudorandom Generator (PRG) [BM,Y,NW]
Efficiently Computable
Big Stretch s(n) À n ( e.g. s(n) = n(1) )
Fools small circuits: 8 small C
PrX, |X| = s(n)[C(X) = 1] ¼ Pr, || = n [C(PRG(s)) = 1]
PRG

3. Do PRG Exist? PRG ) derandomization: BP ¢ P ( EXP [Y,NW,…]
PRG , circuit lower bounds: EXP  P/poly [NW,BFNW,STV,SU,…]
Open Problem: PRG exist?
This Talk: study restricted PRG
Only fool constant-depth circuits
We know lower bounds for constant-depth circuits

4. PRG that fools constant-depth circuits
As before, but only fools small constant-depth circuit C
PrX, |X| = s(n)[C(X) = 1] ¼ Pr, || = n [C(PRG(s)) = 1]
Depth
x1 :x1 x :xs
PRG

5. Previous Results [N’91] PRG : {0,1}n ! {0,1}s(n)
s(n) = 2n , fools AC0 =
) BP ¢ AC0 µ Time(npolylog n), more in [NW,HVV,V]
[LVW’93] PRG : {0,1}n ! {0,1}s(n)
s(n) = n log n, fools SYM ○ AND =
SYM = arbitrary symmetric gate
E.g., SYM = PARITY, MAJORITY Æ Ç Ç Ç Ç Ç Ç Æ Æ Æ Æ Æ Æ Æ Æ
x1 :x1 x :xs
SYM Æ Æ Æ Æ Æ Æ
x1 :x1 x :xs

6. Our Results x1 :x1 x2 . . . . :xs Theorem[V] :
PRG : {0,1}n ! {0,1}s(n)
with s(n) = n log n
fools AC0 with log2n SYM =
Improves on [LVW93]
Fools richer class than [N91] but worse stretch
BP ¢ (AC0 with few SYM) µ SUB-EXP ( EXP
Currently richest BP ¢ class one can derandomize
SYM
SYM Ç Ç Ç Ç
SYM Æ Æ Æ Æ Æ Æ
x1 :x1 x :xs

7. The Pseudorandom Generator
[NW] style
Input =
Output = … ………
f = © = PARITY
[RW] f  Æ Æ  
x xn

8. Outline Why previous results/techniques do not suffice
For PRG need new average-case lower bound for AC0 with few SYM
Proof sketch of average-case lower bound
Other results: BPP vs. PH

9. Known Lower Bounds x1 :x1 x2 . . . . :xs Recall AC0 with log2n SYM =
[H,BNS,HG,RW,HM,CH]: f 2 P that requires
AC0 circuits with log2n SYM of size nlog n
Often, lower bound ) PRG. But NOT this time!
SYM
SYM Ç Ç Ç Ç
SYM Æ Æ Æ Æ Æ Æ
x1 :x1 x :xs

10. Standard Approach
To construct PRG that fools C (e.g. AC0 with few SYM)
h hard for C
f hard on average for C
PRG that fools C
[NW]
[BFNW,STV,SU,…]
Def. f : {0,1}n ! {0,1} average-case hard for C
if 8 small C 2 C
Prx[C(x)  f(x)] ¸ ½ - n- (1)

11. Standard Approach Fails
To construct PRG that fools C (e.g. AC0 with few SYM)
h hard for C
f hard on average for C
PRG that fools C
Proving correctness
9 C 2 C C = h
9 C 2 C comp. f on average
9 C 2 C breaks PRG
Problem: requires C w TC0. Is TC0 w NEXP? [RR]
Conjecture [V]: Black-box construction ) C w TC0

12. Our vs. Previous Lower Bounds
C = AC0 with few SYM
h hard for C
f hard on average for C
PRG that fools C
[H,BNS,HG,RW,HM,CH]
not average-case hard
Theorem[V]: There is f 2 P s.t.
8 AC0 circuit C of size nlog n with log2n SYM
Prx[C(x)  f(x)] ¸ ½ - n-log n

13. Tools Random restrictions  [FSS,H,…]  : {x1, x2,…, xs} ! {0,1,*}
C| subcircuit on *’s
Multiparty communication complexity [CFL]
Thm[BNS]: Gen. Inner Product (GIP) =
has high
communication complexity  Æ Æ
x xn

14. Proof Sketch © Thm[V]: f = GIP ○ PARITY = is average-case hard for
small AC0 circuits with few SYM
Proof sketch: C small AC0 circuit with few SYM.
W.h.p. over random restriction  :
E1: GIP ○ PARITY| ¼ GIP ) high comm. complexity
E1 ( each bottom PARITY has *
E2: C| computable with low comm. complexity
E1 and E2 ) C|(x)  GIP(x) Q.E.D.  Æ Æ  
x xn

15. Conclusion Theorem[V]: PRG : {0,1}n ! {0,1}s(n)
with s(n) = n log n fools AC0 with log2n SYM
Improves [LVW93], fools richer class than [N91]
Currently richest BP ¢ class one can derandomize
Obtained from average-case hardness result
Conj.: PRG from worst-case hardness ) C w TC0
Open problems: (log2n) SYM?
EXP average-case hard for GF(2) poly of deg. log n ?

16. C| low communication complexity
Lemma [V]: C small AC0 circuit w/ log2n SYM W.h.p. over  2 Rp , C| low comm. complexity
Lemma[HG+HM]: Above holds for 1 SYM

17. More SYM gates Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Consider following protocol
SYM3
SYM2 Ç Ç Ç Ç
SYM1 Æ Æ Æ Æ Æ Æ
x1 :x1 x :xs

18. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Previous lemma ) low communication complexity
SYM3
SYM2 Ç Ç Ç Ç
SYM1 Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

19. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Parties compute value of SYM gate
SYM3
SYM2 Ç Ç Ç Ç 1 Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

20. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Previous lemma ) low communication complexity
SYM3
SYM2 Ç Ç Ç Ç 1 Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

21. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Parties compute value of SYM gate
SYM3 Ç Ç Ç Ç 1 Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

22. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Previous lemma ) low communication complexity
SYM3 Ç Ç Ç Ç 1 Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

23. More SYM gates  Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Parties compute value of SYM gate 1 Ç Ç Ç Ç 1 Æ Æ Æ Æ Æ Æ Æ 
x1 :x1 x :xs

24. More SYM gates Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
Total communication =
communication for 1 SYM X number of SYM
Q.E.D.
Union bound over 2#SYM circuits limits # SYM.
Open Problem: Better analysis?

25. New Results: BPP vs. PH New paper [V] applies PRGs to study BPP vs. PH
Theorem[G,L]: BPTime(n) µ S2Time(n2)
Theorem [V]:
BPTime A (n) µ S2Time A (n1.9)
BPTime (n) µ S3Time (n ¢ polylog n)
Uses [N’92] PRG
Improves [LVW93], fools richer class than [N91]
Currently richest BP ¢ class one can derandomize
Obtained from average-case hardness result
Conj.: PRG from worst-case hardness ) C ¶ TC0
Open problems: (log2n) SYM?
EXP average-case hard for GF(2) poly of deg. log n ?

26. Conclusion Theorem[V]: PRG : {0,1}n ! {0,1}s(n)
with s(n) = n log n fools AC0 with log2n SYM
Improves [LVW93], fools richer class than [N91]
Currently richest BP ¢ class one can derandomize
Obtained from average-case hardness result
Conj.: PRG from worst-case hardness ) C w TC0
Other Results: BPP vs. PH
BPTime (n) µ S3Time (n ¢ polylog n), using [N’92] PRG

27. Multiparty Communication Complexity
``Number on the forehead’’ model [CFL]
k-parties want to compute f(x)
x partitioned in k blocks !
i-th party knows all x but xi
Communication = broadcast
Generalized Inner Product. GIP(x) =
Lemma[BNS]:
Low communication complexity protocol P )
Prx[P(x)  GIP(x)] ¸ ½ - n-log n
Discrepancy, [CT,R]
x x  xk n Æ Æ k k
x xnk

28. C| low communication complexity
Restriction [FSS,…]  map variables to {0,1,*}
Rp = uniform distribution, Pr[(xi) = *] = p
C| subcircuit. New input bits = *
Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
First prove 1 SYM, then log2n SYM

29. 1 SYM gate =  Lemma: C small AC0 circuit with 1 SYM
W.h.p. over  2 Rp , C| low comm. complexity
Proof:
[H]
[HG] SYM ○ ANDk-1 low comm. complexity
8 AND 9 party that can compute it (fan-in < k = # blocks)
Parties broadcast # AND = 1
Communication = k ¢ log(size of circuit) Q.E.D.
SYM
SYM Ç Ç Ç Ç Ç Ç = Æ Æ Æ Æ Æ Æ
k-1
k-1 Æ Æ Æ Æ Æ Æ Æ Æ 
x x  xk

30. Summary of Lemmas Lemma[BNS]:
Low communication complexity protocol P )
Prx[P(x)  GIP(x)] ¸ ½ - n-log n
Lemma: C small AC0 circuit with log2n SYM
W.h.p. over  2 Rp , C| low comm. complexity
Want
Theorem: There is f 2 P s.t.
8 AC0 circuit C of size nlog n with log2n SYM gates Prx[C(x)  f(x)] ¸ ½ - n-log n

31. = Pry[P(y)  GIP(y)] (1 - n-log n) ¸ ( ½ - n-log n)
Proof:
f = GIP ○ PARITY =
C small AC0 circuit with log2n SYM
Random Input x = random  + random y for the *
E1: f | ¼ GIP ) high comm. complexity
E1 ( each bottom PARITY has *
E2: C| low comm. complexity
Prx[C(x)  f (x)]
¸ Pr, y[C|(y)  f|(y) | E1, E2] Pr[E1, E2]
= Pry[P(y)  GIP(y)] (1 - n-log n) ¸ ( ½ - n-log n)
Q.E.D.  Æ Æ  
x xn

32. Conclusion Theorem[This Work]: PRG : {0,1}n ! {0,1}s(n)
with s(n) = n log n fools AC0 with log2n SYM
Improves [LVW93], fools richer class than [N91]
Currently richest BP ¢ class one can derandomize
Obtained from average-case hard function
Conj.: PRG from worst-case hardness ) EXP  TC0
Open problems: (log2n) SYM?
EXP average-case hard for GF(2) poly of deg. log n ?

33. Proof Sketch Tools: Random restrictions  [FSS,H,…]
 : {x1, x2,…, xs} ! {0,1,*} , C| subcircuit on *’s
Communication complexity bound for GIP [BNS]
Theorem[This Work]: GIP ○ PARITY is average-case hard for small AC0 circuits with few SYM
Proof sketch: C small AC0 circuit with few SYM.
W.h.p. over random restriction  :
E1: GIP ○ PARITY| ¼ GIP ) high comm. complexity
E2: C| computable with low comm. complexity
E1 and E2 ) C|(x)  GIP(x) Q.E.D.