Previous Results

For the case of strongly black-box hardness amplification, Viola [Vio04]

proved that no amplification procedures from worst-case hardness to mild hardness is computable in PH. Lu et al. [LTW05] proved a more general

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60CHAPTER 3. WEAKLY BLACK-BOX HARDNESS AMPLIFICATION result, showing the impossibility of amplifying hardness from (1 − δ)/2 to (1 − δ^k)/2 in PH for any super-polynomial k . Furthermore, they showed that such a hardness amplification must be highly non-uniform in nature, in the sense that one must start from a function f which is hard against a very non-uniform complexity class even if one only wants to obtain a function f which is hard against a uniform complexity class [LTW05] (presented in¯ Chapter 2).

Since the strongly black-box approach has its limitation, one may look for a weaker type of hardness amplification. Bogdanov and Trevisan [BT03]

showed that even if one drops the constraint on the encoding procedure, one still cannot amplify from worst-case hardness to mild hardness for functions in NP unless PH collapses, when the decoding procedure is required to be computable non-adaptively in P.

The other possibility is to consider weakly black-box hardness amplifica-tion, in which the constraint on the decoding procedure is dropped, while the encoding procedure is still required to be done in a black-box way. Viola [Vio05] proved that if a weakly black-box procedure amplifying from worst-case hardness to mild hardness can be realized in PH, then one can obtain from it a mildly hard function computable in PH. Although this can be seen as a negative result, it does not rule out the possibility of such a weakly black-box hardness amplification. In fact, it appears difficult to establish impossibility results for such a hardness amplification. This is because if an average-case hard function indeed exists, an amplification procedure may simply ignore the initial hard function and compute the average-case hard function from scratch. This raises the question: can one prove any meaning-ful impossibility result for weakly black-box hardness amplification?

3.1.2 Our Results

We derive two negative results for weakly black-box hardness amplification.

First, we prove that if a weakly black-box hardness amplification realized in TIME(t) can amplify hardness by an ω(t) factor, from o(ε/t) to ε, then it

3.1. INTRODUCTION 61 must embed in it a function computable in TIME(t) with hardness about ε.

Note that a function in TIME(t) cannot be hard against a class containing TIME(t). Therefore, we obtain an unconditional impossibility result: it is impossible to use a procedure in TIME(t) to transform a function which is o(ε/t)-hard against the class C = SIZE(2^n/3) into a function which is ε-hard against a class ¯C ⊇ TIME(t). This rules out the possibility of using a low-complexity procedure to do such a hardness amplification for high-low-complexity functions.¹ Note that when t = 2^o(n), this gives an impossibility result for amplifying from worst-case hardness to mild hardness in sub-exponential time. We also extend this impossibility result to the case with C being any uniform complexity class equipped with an advice of length at most 2^n/3. This says that such a weakly hardness amplification, just as in the strongly black-box case [LTW05], must also be highly non-uniform in nature: it is impossible to have such a weakly hardness amplification if one start from an initial function which is hard against any complexity class with only 2^n/3 bits of non-uniformity (even of arbitrarily high uniform complexity). Second, we prove that if a weakly black-box hardness amplification realized in NP (Σ_kP, respectively) can amplify hardness beyond a polynomial factor, from ε²/n^ω(1) to ε, then one can obtain from it a function computable in NP (ΣkP, respectively) with hardness about ε. This improves the result in [Vio05], as the hard function obtained there seems to need at least the complexity of Σ_k+1P, one level higher than ours in PH. Again, this enables us to derive an unconditional impossibility result: it is impossible to use a procedure in NP (Σ_kP, respectively) for such a hardness amplification, if the new function’s hardness is measured against a class containing NP/poly (ΣkP, respectively),

1It is possible to use a low complexity (oracle) procedure to amplify hardness within certain range for functions in high complexity classes. For example, the derandomized XOR lemma [IW97] (the XOR lemma [Yao82, NW94], respectively) allows us to use a polynomial-time (oracle) procedure to amplify from mild hardness to average-case hardness (hardness close to average-case hardness, respectively) for functions in high complexity classes, such as E. Our result says that this becomes impossible if one wants to amplify hardness beyond certain factor.

62CHAPTER 3. WEAKLY BLACK-BOX HARDNESS AMPLIFICATION when the initial function is hard against a uniform complexity class equipped an advice of length 2^n/3. Note that this excludes the possibility of having such a hardness amplification from worst-case hardness to mild hardness for functions in NP. Following our result, we widen the gap between worst-case and mild hardness within NP.

3.1.3 Organization of this chapter

First, some preliminaries are given in Section 3.2. Then in Section 3.3 and in Section 3.4, we show the results of weakly black-box hardness amplification sub-exponential time and Σ_kP respectively.

3.2 Preliminaries

First, we generalize the hardness from the circuit model to arbitrary ones.

Definition 11 We say that a function f : {0, 1}ⁿ → {0, 1} is (ε, C)-hard, for a complexity class C, if for any C ∈ C, Pr_x∈Un[C(x) 6= f (x)] > ε. We will call f ε-hard when the complexity class C is clear.

The parameter ε in the definition above is allowed to be a function of n, so a better notation should be ε(n), but for simplicity we drop the parameter n. In previous works, people usually consider hardness against circuits, i.e., with C = SIZE(s) for some s. Since we will consider hardness against other complexity classes, we introduce this slightly more general definition. Next, we define the notions of weakly black-box hardness amplification [RTV04].

Definition 12 Let C and ¯C be complexity classes. We say that an oracle algorithm Amp^(·) : {0, 1}^¯n → {0, 1} realizes a weakly black-box (n, ε, ¯ε, C, ¯C) hardness amplification, if given any (ε, C)-hard function f : {0, 1}ⁿ → {0, 1}, the function Amp^f : {0, 1}^n¯ → {0, 1} is (¯ε, ¯C)-hard.

Here, the reduction from the initial function f to the harder function is done in a black-box way, as the harder function Amp^f only uses f as an oracle.

3.3. IMPOSSIBILITY OF HARDNESS AMPLIFICATIONIN IN TIME(T )63