For the generalized hardcore set lemma in Section 6.1, one may wonder whether we can find a larger binary hardcore set, for example, a set with density δ/D. In this section, we give an upper bound on the size of binary hardcore sets of black-box construction. First, we introduce a black-box construction of a hardcore set.
Definition 6.4.1. We say that an oracle algorithm Dec(·) is a black-box (δ, ε, D) -
con-struction of a hardcore set, if the following holds. Given any function f :
{0, 1}ℓ → [D], where ℓ = Ω(log D), and a family of functions G ={gI|I ⊆ [D] with |I| = 2} satisfying that for each gI ∈ G and H ⊆ f−1(I) with size s, Prx∈H[gI(x)6= f(x)] ≤ (1 − ε)/2, then Prx∈{0,1}ℓ[DecG(x)6= f(x)] ≤ δ. We call s the size complexity of black-box construction.Now, we give an upper bound on the size of binary hardcore sets.
Theorem 6.4.2. Suppose that Ω(1/Dc1)≤ δ for some constant c1
, ε
≤ 1/5 and D ≥ 4.Then, any black-box (δ, ε, D)-construction must have size complexity O(δ2
ℓ/D2).Proof. We use a probabilistic method to show that there exist a function f and a family of
functions G ={gI|I ⊆ [D] with |I| = 2} satisfying that for each gI ∈ G and H ⊆ f−1(I) with size 10δ2ℓ/D(D− 1), Prx∈H[gI(x) 6= f(x)] ≤ (1 − ε)/2, but Prx∈{0,1}ℓ[DecG(x) 6=f (x)]≥ δ/2.
Suppose that Dec is a (δ, ε, D)-black-box construction. We choose a random function f and a random family of functions G = {gI|I ⊆ [D] and |I| = 2} as follows. First, we pick D2 disjoint sets A{1,2}, A{1,3},· · · , A{D−1,D} ⊆ {0, 1}ℓ, each with size 4δ·2ℓ/D(D−1), and then partition{0, 1}ℓ\ S
i1<i2A{i1,i2} into D sets: B1, B2· · · , BD (note that Bicould be empty). We define f as in Figure 6.7, and gI for each I ={i1, i2} ⊆ [D] as in Figure 6.8.
Next, we claim that for each gI ∈ G and H ⊆ f−1(I) with size 10δ2ℓ/D(D − 1), gI predicts f well in H.
Claim 6.4.3. For each gI ∈ G, H ⊆ f−1(I) with size 10δ2ℓ/D(D− 1), and ε ≤ 1/5,
xPr∈H[gI(x)6= f(x)] ≤ (1 − ε)/2.
Proof. Fix any I =
{i1, i2} ⊆ [D], and H ⊆ f−1(I) with size 10δ2ℓ/D(D− 1). For each x ∈ H, let Zx,f,G be the indicator random variable for the event of gI(x)6= f(x). Note• Input: x ∈ {0, 1}ℓ
Proof. Note that for any x
∈ AI for some I ⊆ [D] with |I| = 2, we have thatf,GPr[Bx,f,G= 1] ≥ 1/2.
Hence, by the Chernoff bound of Lemma 2.7.6, we get
f,GPr
X
x∈{0,1}ℓ
Bx,f,G< δ2ℓ 2
≤ Pr
f,G
"
X
x∈∪IAI
Bx,f,G< δ2ℓ 2
#
< e−Ω(δ2ℓ)= o(1),
for ℓ = Ω(log D) and δ > 1/Dc1 for some constant c1.
From Claim 6.4.3, and 6.4.4, we conclude that there exist f and G = {gI|I ⊆ [D] with |I| = 2} such that for each gI ∈ G and H ⊆ f−1(I) with size 10δ2ℓ/D(D− 1), Prx∈H[gI(x)6= f(x)] ≤ (1 − ε)/2, but Prx∈{0,1}ℓ[DecG(x)6= f(x)] ≥ δ/2.
6.5 Open Problems
In section 6.2, we show that our extractor for independent-symbol sources still works for computational independent-symbol sources. We would like to find a better extractor for computational independent-symbol sources or show that our extractor is optimal.
On the other hand, there are several ways to prove the well-known XOR lemma [20], and one is through the hardcore set lemma. In Section 6.3, we show how to prove the generalized XOR lemma using the generalized hardcore set lemma. It would be interesting to consider other proofs for the XOR lemma to prove the generalized XOR lemma.
Chapter 7
Conclusion and Future Works
In this thesis, we consider the problem of deterministically extracting almost perfect random bits from several classes of random sources. First, we consider multiple weakly random sources that are mutually independent. We generalize the well-known leftover hash lemma, and this lemma gives us a way to extract randomness from two independent sources as long as two sources have enough min-entropy. We also extend our construction to extract randomness from t≥ 3 independent sources as long as two of them have enough min-entropy. One nice feature is that the extractor still works even with all but one source exposed. Moreover, we apply our extractor for a cryptographic task in which a group of parties want to agree on a secret key for group communication over an insecure channel, without using ideal local randomness.
We also consider the independent-symbol sources which are the sources lie in between multiple independent sources and bit-fixing sources. Each independent-symbol source consists of a sequence of n independent symbols from {0, 1}d, and the only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which extracts about Ω(log k) bits, for any n, d, k∈ N. For sources with a larger min-entropy, we can extract even more randomness. When k ≥ n1/2+γ, for any constant γ ∈ (0, 1/2), we can extract m = k − O(d log(1/ε)) bits with any error ε≥ 2−Ω(nγ). When k≥ logcn, for some constant c > 0, we can extract m = k− (1/ε)O(1) bits with any error ε≥ k−Ω(1). Our results generalize those of Kamp and Zuckerman [33]
and Gabizon et al. [17] which only work for bit-fixing sources (with d = 1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a
non-explicit deterministic extractor which can extract m = k−O(log(1/ε)) bits whenever k = ω(d + log(n/ε)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k−m = Ω(log(1/ε)). This generalizes a lower bound of Radhakrishnan and Ta-Shma on extracting from general sources.
Then, we go to the other direction to look for a more general class of sources from which seedless extraction is still possible. The sources we consider have the form of a conditional distribution (f (X )|X ), for some function f and some distribution X , and we say that such a source has computational min-entropy k if any circuit of size 2k can only predict f (x) correctly with probability at most 2−k given input x sampled from X . We first show that it is impossible to have a seedless extractor for one single source of this kind.
Then we show that it becomes possible if we are allowed a seed which is weakly random (instead of perfectly random) but contains some statistical min-entropy, or even a seed which is not random at all but contains some computational min-entropy. This can be seen as a step toward extending the study of multi-source extractors from the traditional, statistical setting to a computational setting. We reduce the task of constructing such extractors to a problem in learning theory: learning linear functions under arbitrary distribution with adversarial noise. For this problem, we provide a learning algorithm, which may have interest of its own.
Finally, we consider computational (n, D, k, s)-sources, which, just as (n, D, k)-sources, consist of n mutually independent parts, (f1(X1)|X1),· · · , (fn(Xn)|Xn), each fi(Xi) of length d such that for each i if given input xi sampled from Xi, any circuit of size s can only predict fi(xi) with probability at most 2−ki for some ki ≤ d, and the sum of ki’s is k. Note that we can set the circuit size as a separate parameter to define the computational independent-symbol sources. We generalize the well-known hardcore set lemma to show that our extractor for independent-symbol sources still works for computa-tional independent-symbol sources. In addition, the result of extractors for computacomputa-tional independent-symbol sources implies a generalization of the well-known XOR lemma. Fi-nally, we show an upper bound on the size of a binary hardcore set in any black-box construction.
Since the proofs of Lemma 4.4.3 and 5.4.2 are complicated, in the future, we would like to simplify these proofs. Moreover, we will go on to construct better extractors for
these classes of sources or prove that these extractors are optimal.
In addition, in the proof of the lower bound on entropy loss for independent-symbol sources, we provide a size lower bound on an ”almost” t-wise independent space, and this immediately implies a size lower bound on any approximate t-wise independent space. It may be interesting to find a better size lower bound on an approximate t-wise independent space.
On the other hand, there are several ways to prove the well-known XOR lemma [20], and one is through the hardcore set lemma. In Chapter 6, we generalize the hardcore set lemma to prove the generalized XOR lemma. It may be interesting to consider other proofs for the XOR lemma to prove the generalized XOR lemma.
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Appendix A
An Example of Pair-wise Independent Hash Family
We claim that the best result of [13] is a special case of generalized leftover hash lemma.
Let A1, . . . , Am be n× n matrices over GF [2], such that, ∀S ⊆ [m], S 6= ∅, the rank of AS def= P
i∈SAi is n. Define a family of hash functions H = {fx|fx :{0, 1}n → {0, 1}m} by
fx(y) =hA1x, yi ◦ hA2x, yi ◦ · · · ◦ hAmx, yi where Aix is a matrix-vector multiplication over GF [2].
We show that the family H is pair-wise independent. For any y, y′ ∈ {0, 1}n, y 6= y′, define z = z1◦ · · · ◦ zn = y− y′ 6= 0. Let wt(z) denote the Hamming weight of z. Since z 6= 0, wt(z) = k for some 1 ≤ k ≤ n. W.L.O.G., suppose that z1 = z2 =· · · = zk = 1, and zk+1 = zk+2 = zn = 0. Let Aij denote the transpose of the jth row of Ai, and let x = x1◦ · · · ◦ xn. Then we obtain
fxPr∈H[fx(y) = fx(y′)] = Pr
fx∈H[∀i, hAix, yi = hAix, y′i]
= Pr
fx∈H[∀i, hAix, zi = 0]
= Pr
fx∈H[∀i, hAi1, xiz1+· · · + hAin, xizn= 0]
= Pr
fx∈H[∀i, hAi1, xi + hAi2, xi + · · · + hAik, xi = 0]
= Pr
fx∈H[∀i, hAi1+ Ai2+· · · + Aik, xi = 0]
where + is the addition over GF [2].
Now we evaluate the number of x = x1 ◦ · · · ◦ xn satisfying the following system of equations:
hA11+ A12+· · · + A1k, xi = 0 hA21+ A22+· · · + A2k, xi = 0
...
hAm1 + Am2 +· · · + Amk, xi = 0
Suppose that some of the above m equations are dependent, then there exist b1, . . . , bt
for some 2≤ t ≤ m such that
(Ab11 + Ab21+· · · + Abk1) + (Ab12 + Ab22 +· · · + Abk2) +· · · + (Ab1t + Ab2t +· · · + Abkt) = 0 It means that the sum of the first k rows of Ab1 + Ab2 +· · · + Abt is 0, contradict Ab1 + Ab2 +· · · + Abt having full rank. Hence these m equations are independent. There are 2n−m different values of x to satisfy the above system of m different equations and n variables, hence
fxPr∈RH[fx(y) = fx(y′)] = Pr
fx∈RH[∀i, hAix, y− y′i = 0] = 1 2m. We complete the proof.
Appendix B
An Elementary Proof of Extractors for Independent-Symbol Sources
We give an explicit seedless extractor for independent-symbol sources, which works for any min-entropy k but only extracts about log k bits.
Theorem B.0.1. For any n, k, D∈ N and any prime number M ≥ D, there is an explicit (n, D, k, ε)-extractor Ext0 : [D]n→ [M], with ε ≤ 12 ·√
M · e−k/(8M2log D)
.
Note that for k ≥ Ω(M2log2D), our extractor has ε≤ 2−Ω(k/(M2log D)). Alternatively, for any ε∈ (0, 1), our extractor can extract Ω(log k − log log D − log log(1/ε)) bits. This achieves the same asymptotic bound as the recent result in [32], but here we provide a different and completely elementary proof.
To extract randomness, we will work on the group ZM, for a prime M, and see any symbolXi ∈ [D] of the source as an element in ZM. Throughout this section, operation + or− on elements in ZM is understood as an operation over the group ZM. Our extractor Ext0 : [D]n→ [M] is then defined as
Ext0(X ) = X
t∈[n]
Xt,
which can be seen as taking an n-step walk on the group ZM, using the n symbols from the source in the following way. Each time when we are at some state v ∈ ZM (initially at 0 ∈ ZM) and read a symbol a from the source, we go to the state v + a ∈ ZM. The
extractor of Kamp and Zuckerman [33] for bit-fixing sources can be seen as a special case of ours, with D = 2 andXt ∈ {−1, 1}.
As in [33], we will show that each step of the walk brings the distribution closer to uniform if the symbol read from the source contains some randomness. See a distribution over ZM as an M-dimensional vector in the natural way. Suppose the current distribution isP = (P1, . . . ,PM) and the next symbol in the source has a distribution β = (β1, . . . , βM) shows the progress we can make after each step.
Lemma B.0.2. k¯δk22 ≤ kδk22· (1 − H∞(β)/(4M2log D)).
where the last line follows from the fact that P
jβj2+P
j6=ℓβjβℓ = (P
jβj)2 = 1. Then we need the following two claims.
Claim B.0.3. For any nonzero s∈ ZM, P
of ZM. Thus, there exists an integer t ∈ [1, M − 1] such that i1 = i0+ ts over ZM. By a
which by the Cauchy-Schwartz inequality of Lemma 2.7.2 is at least
X
Using the bounds of the claims in our derivation before, we have
k¯δk22 ≤ kδk22· 1−X
Now let us see how it can be used to prove the theorem.
Proof. (of Theorem B.0.1)
From Lemma B.0.2, we know that after reading the t’th symbol Xt from the source, the L2-distance between the resulting distribution and the uniform one decreases by a factor
1− H∞(Xt)/(4M2log D)≤ e−H∞(Xt)/(4M2log D).
Therefore, we have
kExt0(X ) − Uk22 ≤ Y
t∈[n]
e−H∞(Xt)/(4M2log D) = e−Pt∈[n]H∞(Xt)/(4M2log D).
Since the n symbols of the source are independent of each other, we haveP
t∈[n]H∞(Xt) = H∞(X ) = k, so the bound above becomes e−k/(4M2log D). Then by the Cauchy-Schwartz inequality of Lemma 2.7.2,
kExt0(X ) − Uk1 ≤√
M· kExt0(X ) − Uk2 ≤√
M · e−k/(8M2log D).