Bounds from Hardness Amplification

There is no previous result working directly on the lower bounds of hard-core set constructions. However, one can obtain such bounds from lower bounds for the task of hardness amplification [Vio06, LTW05]. This is because the hard-core set lemma can be used for hardness amplification, as shown in [Im95], and a closer inspection shows that a black-box construction of hard-core set in fact yields hardness amplification in a similar black-box model.

In particular, one can have the following. First, using a recent result of Viola [Vio06], we can derive a lower bound of min(_10ε¹ ,_{5 log n}ⁿ ) on the query complexity of any strongly black-box construction of hard-core set. Note that this bound is always smaller than our bound. Second, we can use the result in [LTW05] to derive an advice lower bound of Ω(log^(1−δ)_ε ²) for any weakly black-box construction of hard-core set. Note that this bound is exponentially worse than ours. Finally, we can use another result of Viola [Vio06] to show that for weakly black-box construction of hard-core set, if the function Dec satisfies the additional condition that it only needs a short (logarithmic in the circuit size of Dec) advice, then it cannot belong to AC⁰[p]. Note that this additional condition is not required in our result and our proof is much simpler.¹

6.1.3 Our Techniques

To have our query lower bound, we show that if a strongly black-box con-struction does not make enough number of queries, then there exist a family

1On the other hand, under this additional condition, Viola achieved something stronger:

such Dec can be used as oracle gates by an AC⁰circuit to compute the majority function exactly (instead of approximately). We can also achieve this, but we omit it here as our proof is similar to Viola’s.

100 CHAPTER 6. HARDCORE SET CONSTRUCTIONS G = {g₁, . . . , g_k} of functions and a function f violating the guarantee of the construction. We prove the existence of G and f by a probabilistic argument.

We choose f randomly and then choose g₁, . . . , g_k independently as k noisy versions of f , with each g_i(x) being f (x) added by a noise of rate (1 − 2ε)/2.

We can show that f is unlikely to have an ε-hard-core set for G, because it is very unlikely to have a subset on which every g_i has a large deviation from f , when k is large enough. On the other hand, we can show that if the function Dec does not make enough number of queries to functions in G, there is a good chance that it is not close to f . This implies the existence of G and f for which Dec fails to work. Thus, we conclude that the query complexity must be high.

To have our advice lower bound, we show the existence of a family G = {g₁, . . . , g_k} of functions such that one can find a large collection Γ of functions with the property that every function in Γ has no hard-core set but no two functions in Γ are close. The candidates for Γ are functions G_I, with I = {i₁, . . . , i_t}, defined as G_I(x) = Maj(gi1(x), . . . , g_it(x)), where Maj denotes the majority function. We will let t = b1/εc, so that every G_I has a good correlation with some g_i for i ∈ I, which implies that G_I has no ε-hard-core set for G. On the other hand, for any G_I and G_J with small I ∩ J , they are likely to be far away because for any input x, P

i∈I∩Jgi(x) is likely to be small, so there is a good chance that the values of G_I(x) and G_J(x) are dominated by P

i∈I\Jg_i(x) andP

j∈J \Ig_j(x), respectively, and hence there is a good chance that G_I(x) 6= G_J(x). This implies that with high probability, each G_I is far away from many other G_J’s, and by the well-known Tur´an’s theorem, there must be many G_I’s which are far away from each other, and they form the set Γ. This gives an advice lower bound of log |Γ|.

To show that a weakly black-box construction can be used to approximate the majority function, we again use the observation that for any G, the function G_I, with |I| = t ≤ 1/ε, has no ε-hard-core set for G. When t ≤ n, we can define the functions g_i, for 1 ≤ i ≤ t, as g_i(x) = x_i (the i-th bit of x). Then for some advice α, Dec^G,α(x) = G_I(x) = Maj(x1, . . . , x_t) for at

6.2. PRELIMINARIES 101 least δ fraction of x, and by an average argument there must exist some fixed

¯

xt+1, . . . , ¯xnsuch that Dec^G,α(x1, . . . , xt, ¯xt+1, . . . , ¯xn) = Maj(x¹, . . . , xt) for at least δ fraction of x₁, . . . , x_t. By hard-wiring α and ¯x_t+1, . . . , ¯x_n into the circuit for Dec, we get a circuit which is δ-close to the majority function on t bits.

6.1.4 Organization of this chapter

First, in Section 6.2 we give some preliminaries and define our two models for black-box constructions of hard-core set. In Section 6.3, we prove a query lower bound for such a strongly black-box construction. Then we show a lower bound on the advice length needed in such a weakly black-box construction in Section 6.4. Finally, in Section 6.5 we show that no such weakly black-box construction can be realized in AC⁰[p].

6.2 Preliminaries

Let F_n denote the set of all Boolean functions f : {0, 1}ⁿ → {0, 1}. Let AC⁰[p](s) denote the class of Boolean functions computed by constant-depth circuits of size s equipped with mod_p gates (which output 0 exactly when the input bits sum to 0 modulo p), and let AC⁰[p] = AC⁰[p](poly(n)). Given a multi-set (or simply a set) S, we let |S| denote the number of elements in it, counting multiplicity. Given a set G = {g₁, · · · , g_k} ⊆ F_n, together with a multi-set I = {i₁, · · · , i_q} ⊆ [k] of indices, let g_I denote the function such that gI(x) = (gi1(x), · · · , giq(x)) for x ∈ {0, 1}ⁿ. We say that two functions f and g in F_n are δ-close if Pr_x∈Un[f (x) 6= g(x)] ≤ δ.

We will also need the following result, known as Tur´an’s Theorem, which can be found in standard textbooks (e.g. [AS00]).

Fact 6 (Tur´an’s Theorem) Given a graph G = (V, E), let d_v denote the degree of a vertex v and α(G) the size of the maximum independent set.

Then α(G) ≥P

v∈V 1 dv+1.

102 CHAPTER 6. HARDCORE SET CONSTRUCTIONS