4.3 Proof of Main Theorem
4.3.1 Discussion
Our generator uses a seed of length O(n3/2), and as a result, we can only amplify hardness to 1/2 − 1/s0(n) against size s0(n) with s0(n) = 2Ω(n2/3). The main bottleneck is the generator for rectangles. However, to achieve the goal of having s0(n) = 2Ω(n)using our approach, we need to improve both the generator for branching programs and the generator for rectangles. Without improving Nisan’s PRG, even if we could have an optimal ε-PRG for R(m, d), with seed length Θ(log m + log d + log(1/ε)), the resulting generator would still need a seed of length Θ(n log n)+O(n) = Ω(n log n) (see the definition of the generator G and the calculation of its seed length in the previous page), and we would only be able to achieve s0(n) = 2Ω(n/ log n).
Chapter 5
Hardness and
Pseudorandomness in NP
5.1 Introduction
In this chapter, we study the problem of transforming a pseudorandom gen-erator into a hard function and the problem of transforming a hard function into a harder one. It is known that in a high complexity class such as ex-ponential time, one can convert from worst-case hardness to average-case hardness, from average-case hardness to pseudorandomness, and from pseu-dorandomness back to worst-case hardness. However, in lower complexity classes, such as NP, some of the relationships remains unclear. We establish the equivalence between pseudorandomness and average-case hardness, and widen the gap between worst-case hardness and average-case hardness within NP.
By the result of Impagliazzo and Levin ([IL90] in FOCS’ 90), one can build the equivalence between pseudorandomness and average-case hardness within NP. For completeness of this thesis, we give a proof which shows how to transform a pseudorandom generator into a mildly hard function com-putable in NP. We give a strongly black-box construction, with both the transformation procedure and the hardness proof done in a black-box way.
83
84 CHAPTER 5. HARDNESS AND PSEUDORANDOMNESS IN NP 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.
5.1.1 Previous Results
The reduction from average-case hardness to pseudorandomness within NP is done in [NW94]. However, from pseudorandomness to average-case hardness, the method in [NW94] can only transform a PRG back to a worst-case hard function [NW94] within NP. Since the hardness amplification from worse-case hardness to average-case hardness is believed to be impossible (see Chapter 2 and Chapter 3), it is not clear how to obtain the reduction from PRG to average-case hardness within NP. One can use the method developed in [IL90] to achieve this reduction.
Figure 5.1.1 summaries these known relationships between various hard-ness assumptions and pseudorandomhard-ness. Note that the above transforma-tion can be done in a black-box way, in which the decoding procedure is realized in P while the encoding procedure needs the complexity of NP. This raises the following two questions. First, can the complexity of the encod-ing procedure be reduced? Next, since the transformation from worst-case hardness to average-case hardness seems to require high complexity, can we transform a PRG directly into an average-case hard function, using a low-complexity procedure, say in NP (or even in P)?
5.1.2 Our Results
In this chapter, we provides a strongly black-box constructions of average-case hard functions from PRGs although it also can be done via method developed in [IL90]. As a result, we build the equivalence between PRG and average-case hardness within NP as shown in Figure 5.1.1. Combining with
5.1. INTRODUCTION 85
Figure 5.1: The relationship among PRG and various hardness assumptions within NP. Arrows indicate black-box transformations. BB and WBB trans. indicate black-box and weakly black-box transformations respec-tively. Note that the slash symbol means ”the transformation cannot be done in NP”.
results in the previous chapter, we are able to look closer the relationship between pseudorandomness and hardness within NP.
Our main result gives strongly black-box constructions of average-case hard functions from PRGs. The first construction has the encoding pro-cedure realized in NP and the decoding propro-cedure realized in P/poly (or randomized polynomial time). This improves the result of [NW94] which, using an encoding procedure in NP as well, obtains only a worst-case hard function. A natural question then is: can we further reduce the complexity of the encoding procedure, or can we prove a complexity lower bound? We give a partial answer to this by providing another strongly black-box con-struction with the encoding procedure realized in P but at the expense of increasing the complexity of the decoding procedure to NP, which rules out the possibility of proving a complexity lower bound for the encoding proce-dure without restricting the complexity of the decoding proceproce-dure. This still leaves open the question of whether or not one can have both the encoding
86 CHAPTER 5. HARDNESS AND PSEUDORANDOMNESS IN NP and decoding procedures realized in P. Our positive results also imply some impossibility results. By combining with the impossibility results of strongly black-box hardness amplification in [Vio04, LTW05] and the previous im-possibility results of weakly black-box hardness amplification in Chapter 3, respectively, we can obtain corresponding impossibility results of strongly and weakly black-box PRG constructions from hard functions.