Our Results

Previous lower bound results only address hardness in a specific range. How-ever, whether or not one can amplify hardness beyond this range is also a natural and interesting question. For example, it is known that a strongly black-box hardness amplification from hardness 1/poly(n) to average-case hardness can be realized in polynomial time [Yao82, GNW95, Im95, IW97].

Can such a hardness amplification be realized in a lower complexity class, such as AC⁰? Can it start from hardness below 1/poly(n) and still be real-ized in polynomial time? Can it be done in a uniform way (with a uniform decoding function)? In general, how does the quality of a hardness amplifi-cation (the amount of hardness increased) determine its inherent complexity or non-uniformity? All these questions will be addressed in this chapter. We generalize previous results [Vio04, TV02] and consider hardness amplification in a much broader spectrum: from hardness (1 − δ)/2 to hardness (1 − δ^k)/2, for general δ ∈ (0, 1) and k ∈ N.

Following [Vio05], we consider a more restricted model called parallel black-box hardness amplification, in which oracle queries by the encoding function are done in a non-adaptive way. More precisely, we say that a circuit class CKT realizes a parallel black-box hardness amplification if its encoding function Amp can be implemented in the following way. Given any input x, it first generates a circuit T_x ∈ CKT together with t query inputs q_x,1, · · · , q_x,t, then queries f at those t inputs, and finally computes Tx(f (qx,1), · · · , f (qx,t)) as its output. Note that here Tx and qx,1, · · · , qx,t

only depend on x but not f . Although this is a more restricted model, almost all previous constructions of hardness amplification can be done in this way, so it would be nice to know its limitation. Furthermore, through a standard simulation [FSS84, Has86], negative results in this model can in

32CHAPTER 2. STRONGLY BLACK-BOX HARDNESS AMPLIFICATION fact be translated to those in the strongly black-box model.

Our first result addresses both the complexity issue and the non-uniformity issue in the same framework, showing how complexity constraints on the en-coding function result in the inherent non-uniformity of the deen-coding func-tion. Formally, we prove that if such a parallel black-box hardness amplifi-cation, from hardness (1 − δ)/2 to hardness (1 − δ^k)/2, is realized by circuits of depth d and size 2^o(k1/d), then the decoding function Dec must need an advice of length 2^Ω(n). Translating this to the general model, we obtain the same advice lower bound when such a (general) strongly black-box hardness amplification is realized in ATIME(O(1), k^o(1)). This implies that no such hardness amplification is possible if the hardness is measured against circuits of size 2^o(n).

Our lower bound is almost tight as the well known XOR lemma [Yao82, GNW95] gives a way to realize a parallel black-box hardness amplification by circuits of depth O(d) and size 2^O(k1/d), with Dec using an advice of length poly(n/δ^k). Note that Viola’s result in [Vio04] is a special case of ours, be-cause he only addressed explicitly the specific case with (1 − δ)/2 = 2⁻ⁿ and (1 − δ^k)/2 = 1/poly(n) (or equivalently, δ = 1 − 2⁻ⁿ⁺¹ and k = 2^Ω(n)).

Although it seems that his technique can be extended to show lower bounds when (1 − δ)/2 is small enough, but beyond that, say with (1 − δ)/2 = Ω(1), it fails to give a meaningful bound. We can in fact cover this case: our result implies that AC⁰ circuits cannot realize a parallel black-box hardness amplifi-cation, say, from hardness 1/3 to hardness (1−2^−Ω(n))/2. On the other hand, our result when restricted to worst-case to average-case hardness amplifica-tion is incomparable to those of [BT03] and [Vio05].¹ Finally, two interesting facts follow from our result. First, it is impossible to produce in a strongly black-box way a function which is (1 − δ^k)/2–hard against a uniform low

1In [BT03], the complexity lower bound is given on the decoding function instead, under the unproven (though widely believed) assumption that PH does not collapse. In [Vio05], a more general type of hardness amplification than ours is considered, but the possibility of such hardness amplification is not ruled out as we do; instead, it was shown that if the encoding function can be computed in PH, a hard function in PH exists unconditionally.

2.1. INTRODUCTION 33 complexity class, say DTIME(O(1)), even if we start from a function which is (1 − δ)/2–hard against a uniform but arbitrarily high complexity class equipped with an advice of length 2^o(n), say DTIME(2²ⁿ)/2^o(n). On the other hand, it is easy to show that hard functions against DTIME(O(1)) do exist.² This demonstrates one severe weakness of strongly black-box hardness am-plifications. Second, when amplifying hardness from (1 − δ)/2 to (1 − δ^k)/2, the complexity of such amplification is determined mainly by the parameter k; a larger value of k results in a higher complexity requirement, for typical values of δ. Thus, to determine the complexity needed for a hardness am-plification process, one should express the initial and final hardness in the forms of (1 − δ)/2 and (1 − δ^k)/2 respectively. This point was not clear from previous works.

Note that our first result becomes meaningless for d = Ω(log k) as the circuit size becomes 2^o(k1/d) = O(1). Our second result takes care of this:

we show that if a parallel black-box hardness amplification, from hardness (1 − δ)/2 to hardness (1 − δ^k)/2, is realized by nondeterministic circuits of size o(k/ log k), even with arbitrary depth, then the decoding function Dec must need an advice of length 2^Ω(n). For example, to amplify hardness from Ω(1) to (1 − 2^−Ω(n))/2, our second result implies that it can not be realized by nondeterministic circuits of size o(n/ log n) in a parallel black-box way.

Our third result shows that even without any complexity constraint on the encoding or decoding function, amplification between certain range of hardness is still inherently non-uniform. For the special case of amplifying hardness beyond 1/4, the need of non-uniformity can be shown using the Plotkin bound [Plo60] from coding theory. We consider hardness amplifi-cation in a general range and obtain a quantitative bound on the amount of non-uniformity. More precisely, we show that to amplify hardness from (1 − δ)/2 to (1 − ε)/2, the decoding function Dec must need an advice of

2For example, the parity function is (1/2 − 2^−Ω(n))–hard against DTIME(O(1)). How-ever, according to our result, its hardness cannot be shown in such a strongly black-box way.

34CHAPTER 2. STRONGLY BLACK-BOX HARDNESS AMPLIFICATION Ω(log(δ²/ε)) bits. Thus, when ε = δ^k, an advice of length Ω(k log(1/δ)) is necessary, and when ε ≤ cδ²for some constant c, such hardness amplification must be inherently non-uniform. Our result generalizes that of Trevisan and Vadhan [TV02].

Finally, we derive similar lower bounds on strongly black-box construc-tions of PRG from hard funcconstruc-tions.