This motivates us to define our first model of hard-core set constructions as follows. We say that a (non-uniform) oracle algorithm Dec(·) with a decision function D : {0, 1}q → {0, 1} realizes a strongly black-box (δ, ε, k)-construction (of hard-core set) if the following holds. First, Dec will be given a family G = {g1, · · · , gk} of functions as oracle together with a multi-set I = {i1, . . . , iq} as advice, and for any input x, it will query the functions gi1, · · · , giq, all at x, and then output D(gi1(x), · · · , giq(x)). Moreover, it satisfies the property that for any G and for any f which has no ε-hard-core set of density Ω(δ) for G, there exists a multi-set I of size q such that the function DecG,I is δ-close to f (DecG,I(x) 6= f (x) for at most δ fraction of x). We call q the query complexity of Dec, and observe that it relates to the loss of circuit size in the hard-core set lemma, with s0 = O(s/q). Note that the known hard-core set constructions [Im95, KS03] are in fact done in such a strongly black-box way.
Our second model of hard-core set constructions generalizes the first one by removing the constraint on how the algorithm Dec works; the algorithm Dec and its advice now are allowed to be of arbitrary form. We say that a (non-uniform) oracle algorithm Dec(·) realizes a weakly black-box (δ, ε, k)-construction (of hard-core set) if the following holds. For any family G of k functions and for any function f which has no ε-hard-core set of density Ω(δ) for G, there exists an advice string α such that DecG,α is δ-close to f .
1.5 Our results and Organization of this the-sis
Chapter 2 - Strongly Black-Box Hardness Amplification. We show that hardness amplification from hardness (1 − δ)/2 to hardness (1 − δk)/2 cannot be carried out in some black-box way by a circuit of depth d and size 2o(k1/d) or by a nondeterministic circuit of size o(k/ log k) (and arbitrary depth). In particular, for k = 2Ω(n), such hardness amplification cannot be
24 CHAPTER 1. INTRODUCTION done by a strongly black-box model in ATIME(O(1), 2o(n)). Therefore, hard-ness amplification in general requires a high complexity. Furthermore, we show that even without any restriction on the complexity of the amplifica-tion procedure, such a strongly black-box hardness amplificaamplifica-tion must be inherently non-uniform in the following sense. Given as an oracle any algo-rithm which agrees with f0 on (1 − δk)/2 fraction of the input, we still need an additional advice of length Ω(k log(1/δ)) in order to compute f correctly on (1 − δ)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f0, one has to start with a function f which is hard against non-uniform circuits. Finally, we derive similar lower bounds for any strongly black-box construction of pseudorandom generators from hard functions.
Chapter 3 - Weakly Black-Box Hardness Amplification. From worst-case hardness to average-worst-case hardness, we consider a class of hardness ampli-fications called weakly black-box hardness amplification, in which the initial hard function is only used as a black box to construct the harder function.
First, we show that if an amplification procedure in TIME(t) can amplify hardness beyond an O(t) factor, then it must embed in itself a hard function computable in TIME(t). As a result, it is impossible to have such a hardness amplification with hardness measured against TIME(t). Next, we show that, for any k ∈ N, if an amplification procedure in ΣkP can amplify hardness beyond a polynomial factor, then one can obtain from it a hard function in ΣkP. A similar impossibility result can also be derived.
Chapter 4 - Hardness Amplification within NP. We study the prob-lem of hardness amplification in NP. We prove that if there is a balanced function in NP such that any circuit of size s(n) = 2Ω(n)fails to compute it on a 1/poly(n) fraction of inputs, then there is a function in NP such that any circuit of size s0(n) fails to compute it on a 1/2 − 1/s0(n) fraction of inputs, with s0(n) = 2Ω(n2/3). This improves the result of Healy et al. (STOC’04), which only achieves s0(n) = 2Ω(n1/2) for the case with s(n) = 2Ω(n).
1.5. OUR RESULTS AND ORGANIZATION OF THIS THESIS 25 Chapter 5 - Pseudorandomness and Hardness in NP. To build the equivalence between pseudorandomness and average-case hardness, we show how to transform a pseudorandom generator into a mildly hard function computable in NP. We give a strongly black-box construction, with both the transformation procedure and the hardness proof done in a black-box way.
This improves a previous result of Nisan and Wigderson, which can only ob-tain a worst-case hard function from a pseudorandom generator [NW94].
Therefore, we now know that the transformations among mild hardness, average-case hardness, and pseudorandomness all can be done in the com-plexity class NP.
Chapter 6 - Hard-core Set Construction. We study a fundamental result of Impagliazzo (FOCS’95) known as the hard-core set lemma. Consider any function f : {0, 1}n → {0, 1} which is “mildly-hard”, in the sense that any circuit of size s must disagree with f on some δ fraction of inputs. Then the hard-core lemma says that f must have a hard-core set H of density δ on which it is “extremely hard”, in the sense that any circuit of size s0 = O(s/(ε12 log(εδ1))) must disagree with f on at least (1 − ε)/2 fraction of inputs from H.
There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low complexity class. We introduce two models of hard-core set constructions, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models.
First, we show that in any strongly black-box construction, one can only prove the hardness of a hard-core set for smaller circuits of size at most s0 = O(s/(ε12 log 1δ)). Next, we show that any weakly black-box construction must be inherently non-uniform — to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with Ω(1εlog |G|) bits of advice. Finally, we show that weakly black-box constructions in general cannot be realized in a
26 CHAPTER 1. INTRODUCTION low-level complexity class such as AC0[p] — the assumption that f is hard for AC0[p] is not sufficient to guarantee the existence of a hard-core set.