1 Nisan-Wigderson pseudorandom generator

Transcription

1 CSG399: Gems of Theoretical Computer Science. Lecture 3. Jan. 6, Instructor: Emanuele Viola Scribe: Dimitrios Kanoulas Nisan-Wigderson pseudorandom generator and design constuction Nisan-Wigderson pseudorandom generator Today we prove the Nisan-Wigderson pseudorandom generator theorem; we also present a design construction. If x {0, } s and S [s], we denote by x S the S bits of x indexed by S. Theorem (Nisan-Wigderson). Let f : {0, } l {0, } satisfy COR(f, size w) and w set n := w 3 (e.g., think w = l ω() ). Let S,..., S n be a design over universe of size s, where: S i = l, S i S j α and α = log n = log w. 3 Then the Nisan-Wigderson generator G : {0, } s {0, } n defined as fools circuits of size n with error n. G(x) = f(x S ) f(x S n ) Proof. Assume for the sake of contradiction that there exists a circuit c which breaks the generator: E x {0,} se[c(g(s))] E u e[c(u)] n, where u is uniformly distributed over {0, } n. By Yao s next-bit predictor Lemma we have: c : c c + O(), i n: Ex {0,} se[c (f(x S ) f(x S i )) + f(x S i )] n 2. rom c we will construct another circuit c that correlates with f, giving a contradiction. x [s] Si x Si igure : Input x We break the input x in two parts: x ([s] S i ) and x S i (igure ), and write: E x [s] Si [ Ex Si e[c (f(x S ) f(x S i )) + f(x S i )] ] n 2.

2 We now fix x [s] S i to a value that maximizes the above expectation, and for j < i we let f j (x S i S j ) be the function f(x S j ) where the bits of x outside of S i are fixed accordingly. or this fixing and notation we: Ex Si e[c (f (x S S i ) f i (x S i S i )) + f(x S i )] ( ) Note. Each f j O(α2 α ). is a function of α bits of x S i, and thus computable by a circuit of size Now we are going to exhibit the circuit c that correlates with f(x S i ). Given an input y {0, } l, we let x {0, } s be such that y = x S i, and the other bits of x are fixed as before, and c outputs: c(y) := c (f (x S S i ) f i (x S i S i )). Correlation: By our definition of c and ( ), we have E y e[ c(y) + f(y)] n 2 = (w 3 ) 2 = w 2 3 w. Size of c: We need to show that the size of circuit c is less than w. n = w /3 and α = log n, we have: n 2 Recalling that Size( c) c + size required to compute f i (x S S i ) f i (x S i S i ) c + O() + (i )O(α2 α ) n + O() + no(n log n) O(n 2 log n), O(w 2 3 log w 3 ) w. Thus, we have constructed a circuit of size at most w which has correlation at least /w with f. This contradicts our hypothesis and proves the theorem. We mention the following corollary, whose in particular part is Problem. Corollary 2. or every fixed ɛ > 0, there exists a constant c: if f : {0, } l {0, } is computable in time 2 O(l) and COR(f, size 2 ɛl ), then generator: G : {0, } c log n 2 ɛl {0, } n that fools circuits of size n with error /n and it is computable in time poly(n). In particular, P = BPP. Impagliazzo and Wigderson have strengthened the above result to hold under the weaker hypothesis that COR(f, size 2 ɛl ) <, as opposed to COR(f, size 2 ɛl ). In particular, 2 ɛl this leads to the following striking disjunction: Either SAT : {0, } l {0, } has circuits of size 2 l/00000 for arbitrarily large l, or P=BPP. Back from heaven, our goal now is to get an unconditional result. Specifically, we will construct a generator G : {0, } logo() n {0, } n that fools small circuits of fixed depth. We start with the design construction (which of course can be used in other contexts too). 2

3 Design Construction Now we will see how we can construct designs. Designs that are good enough for Corollary 2 can be found by an exhaustive brute-force procedure, in poly(n) time. We can construct more explicit designs using finite fields. We recall some basic facts about finite fields. act. ields of size 2 l can be constructed and operations over them can be performed in time poly(l). A polynomial of degree d over a field, i.e. P :, P (x) = d i=0 xi c i, has at most d roots over. Theorem 3. h power of 2, constant c, design S,..., S n such that n = 2 h, S i S j h, i j, S i = h c, the universe has size (set size) 2 = h 2c, and S i is computable in time poly(h). We are packing exponentially many sets whose pairwise intersection is a small power of the set size, over a universe roughly as large as the set size Proof. Pick a field of size h c, note x has size x = log h = c log log n. Our universe is. Given a string i {0, } log n = {0, } h, construct S i as follows: cloglogn cloglogn... igure 2: string i View i as the coefficients of a polynomial p i over. The degree of p i is The set S i will be the graph of the polynomial p i : S i := {(a, p i (a)) : a }, log n c log log n log n = h. cf. igure 3. Note S i = = h c, and we have n = {0, } h sets. To bound the intersection size, observe that S i S j is the number of roots of the polynomial (p i p j ). Since p i p j has degree less than h, we bound from above the intersection size by h, using act. inally, using again act, S i is computable in poly(h) time. 3

4 igure 3: Graph of polynomial roots of different polynomials igure 4: Bound on the intersection size Constant-depth circuits The depth of a circuit denotes the maximum distance from an input to an output gate (and recall circuits in these lectures have unbounded fan-in; our complexity measure is the number of wires). Remark 4. The depth in Nisan-Wigderson s proof increases by an absolute constant. Therefore, given f : {0, } l {0, } such that COR(f, size w and depth d), the Nisanw Wigderson construction gives a generator G : {0, } s {0, } n that fools circuits of size n and depth d O() with error /n. The complete proof of the next theorem will be given in next lectures. Theorem 5. The parity function on l bits, Parity(x,..., x l ) := x +...+x l mod 2, satisfies: COR(f, size w = 2 lɛ/d and depth d) /w, where ɛ is an absolute constant. 4

5 Now, Problem 2 asks you to put together the Nisan-Wigderson construction, the design construction from the previous section, and the above theorem, to obtain the following. Corollary 6 (Nisan). d, explicit generator G : {0, } logo(d) n {0, } n, that fools circuits of size n and depth d, with error /n. or an application of this corollary, see Problem 3. 5