Pseudorandom generators for group products Michal Koucký Institute of Mathematics, Prague Prajakta Nimbhorkar Pavel Pudlák IMSC, Chenai IM, Prague IMSC, Chenai IM, Prague

2 Branching programs j i p ij = Pr[ reaching j from i ] models randomized space bounded computation models randomized space bounded computation space s → width w ≈ 2 O( s ) w t

3 Goal: Estimate probabilities p ij (up-to additive error ε) in small space. Possible solution: Find a small set F  {0,1} t so that p ij ’s are well approximated by taking a random path according to a random sample from F. Want: A single set F working for all branching programs of length n, width n, and all i and j. → a random set F of size 2 O(log n + log 1/ε) will do.

4 Goal: Find an explicit set F  {0,1} n, i.e., F : {0,1} l → {0,1} n computable in small space, where l ≈ O(log n + log 1/ε). Our result: Explicit F : {0,1} l → {0,1} n, where l = O( (2 O(w log w) + log 1/ε) ∙ log n ) that works for all permutation branching programs of width w and length n. permutation b.p. … in each layer the 0-edges form a permutation and 1-edges form a permutation. permutation b.p. … in each layer the 0-edges form a permutation and 1-edges form a permutation.

5 r 1 r 2 r n Equivalent formulation for group products [MZ]: A fixed group G and elements g 1, g 2, …, g n  G approximate thedistribution R on G given by where r 1, r 2, … r n  R {0,1} We have: F : {0,1} l → {0,1} n so that r 1, r 2, …, r n given by the output of F approximate R well for any choice of g 1, g 2, …, g n  G. l = O( (|G| O(1) + log 1/ε) ∙ log n ) We have: F : {0,1} l → {0,1} n so that r 1, r 2, …, r n given by the output of F approximate R well for any choice of g 1, g 2, …, g n  G. l = O( (|G| O(1) + log 1/ε) ∙ log n ) For G=({0,1},+) → ε-biased spaces. For G=({0,1},+) → ε-biased spaces. g 1 ∙ g 2 ∙ ∙ ∙ g n

6 Known results: width n and length n width n and length n [Nisan92] l = O( log 2 n ) [Nisan92] l = O( log 2 n ) [INW94] l = O( log 2 n ) [INW94] l = O( log 2 n ) width w and length n (permutation/regular) width w and length n (permutation/regular) [BV10] l = O( (w 4 log log n + log 1/ε ) log n ) [BV10] l = O( (w 4 log log n + log 1/ε ) log n ) [BRRY10] l = O( (log w + log log n + log 1/ε ) log n) [BRRY10] l = O( (log w + log log n + log 1/ε ) log n) ours l = O( (2 O(w log w) + log 1/ε) log n ) ours l = O( (2 O(w log w) + log 1/ε) log n ) other combinatorial structures other combinatorial structures [LRTV10, MZ09, GMRZ11] l = O( log n + log O(1) 1/ε) cyclic groups [LRTV10, MZ09, GMRZ11] l = O( log n + log O(1) 1/ε) cyclic groups …

7 Techniques: Convolution * R 1, R 2 probability distributions on G R 1 * R 2 probability distribution on G s.t. for any g  G R 1 * R 2 (g) = ∑ h  G R 1 (h) ∙ R 2 (h -1 g) R 1 * R 2 (g) = ∑ h  G R 1 (h) ∙ R 2 (h -1 g) Examples: Examples: r 1 r n/2 r n/2+1 r n g 1 ∙ ∙ ∙ g n/2 * g n/2+1 ∙ ∙ ∙ g n r 1 r 2 r n g 1 * g 2 * * g n

8 Recursive convolution (~INW): D 1 D 2 D 1 D 2 a 1 … a n/2 and a n/2+1 … a n obtained using F n/2 : {0,1} l → {0,1} n/2 1. F n (s,s’) = F n/2 (s) ◦ F n/2 (s’) → D 1 * D 2 leads to F n : {0,1} O( n ) → {0,1} n 2. F n (s,d) = F n/2 (s) ◦ F n/2 ( s(d) )→ D 1 *γ D 2 leads to F n : {0,1} O( k log n ) → {0,1} n s(d) … d-th neighbor of s in a k-regular expander on 2 l vertices a 1 a n/2 a n/2+1 a n g 1 ∙ ∙ ∙ g n/2 * g n/2+1 ∙ ∙ ∙ g n

9 D 1 *γ D 2  D 1 * D 2 – D 1 *γ D 2  < γ D 1 *γ D 2  D 1 * D 2 – D 1 *γ D 2  < γ Thm: If R 1, R 2, … R N are distributions obtained from group products, F is a formula built from R 1, R 2, … R N using *, and F’ is obtained from F by replacing * with *γ then  D F – D F’  < γ 2 c|G| 11 * *γ * *γ R 1 * R 1 *γ * R 4 *γ R 4 R 2 R 3 R 2 R 3 R 2 R 3 R 2 R 3 FF’

10 Proof ideas: D 1, D 2, R 1, R 2 distr. on G D 1 = R 1 + ε 1 D 2 = R 2 + ε 2 D 1 = R 1 + ε 1 D 2 = R 2 + ε 2 where ∑ h  G ε 1 (h) = 0 ∑ h  G ε 2 (h) = 0 D 1 * D 2 = R 1 * R 2 + ε 1 * R 2 + R 1 * ε 2 + ε 1 * ε 2 D 1 * D 2 = R 1 * R 2 + ε 1 * R 2 + R 1 * ε 2 + ε 1 * ε 2 D 1 *γ D 2 = … + ε γ D 1 *γ D 2 = … + ε γ where  ε γ  < γ 1. If R 2 is uniform then ε 1 * R 2 = If R 2 is close to uniform then ε 1 * R 2 is close to If the support of R 2 is the whole group G then  ε 1 * R 2  < (1-δ)  ε 1 .

11 Open problems Improve dependence on the width of the branching program/group size, and on the error ε. Improve dependence on the width of the branching program/group size, and on the error ε. Remove restrictions on the branching programs Remove restrictions on the branching programs