Protocol for Low Entropy

We have seen that in the case that the entropy is significantly larger than the number of players, or the entropy rate is larger than half, we have very simple network extractor protocols. In this section we describe better results for the case of low entropy sources. We shall start by describing a network extractor protocol that is good enough to get the following theorem:

Theorem 4.3.3 (Polynomial Entropy Synchronous Extractor). There exists a con- stant c > 0 such that for every γ > δ > 0, β > 0 and p large enough, there exists a 1 round (δp, (1 − 2γ)p, 2−kc) synchronous extractor for min-entropy k ≥ nβ _{in the}

full-information model.

Building on the ideas that go into proving the above theorem, we can give a O(log log n/ log log k) round protocol that works even when the entropy k is as small as log10n. This result appears in section 4.3.1.5.

Theorem 4.3.4. If k > log p and n ≤ exp(t) then for t large enough there exists a (t, p − 2t − (1.1 log log n_{log log k} )t, 2−kΩ(1)) synchronous network extractor for min-entropy k > log10n that runs in O(log log n/ log log k) rounds in the full-information model.

Our protocol will be a variation on Protocol 4.3.2. Instead of trying every possible C-tuple of strings from the set A, we shall use a derandomized sample of these tuples.

We shall need the concept of an AND-disperser :

Definition 4.3.5 (AND-disperser). An (l, r, d, δ, γ) AND-disperser is a bipartite graph with left vertex set [l], right vertex set [r], left degree d s.t. for every set V ⊂ [r] with |V | = δr, there exists a set U ⊂ [l] with |U | ≥ γl whose neighborhood is contained in V .

Each vertex on the left identifies a d-tuple of vertices on the right. Thus when l = r_d
, we can easily build an AND-disperser with great performance, just by considering every possible such tuple. We shall construct a much better AND disperser, i.e., one where l, r are much closer to each other.

In our application, we shall need a (l, r, C, δ, γ) AND-disperser with l as small as possible, δ as small as possible and γ as large as possible. We shall prove the following lemma:

Lemma 4.3.6. For every C, δ > 0, there exist constants h, γ > 0 and a polynomial time constructible family of (hr, r, C, δ, γ) AND-dispersers.

Before we see how to prove this lemma, we describe the rest of our construction. Another well studied object that we need is a construction of a bipartite ex- pander.

Definition 4.3.7 (Bipartite Expander). A (l, r, d, β) bipartite expander is a bipartite graph with left vertex set [l], right vertex set [r], left degree d and the property that for any two sets U ⊂ [l], |U | = βl and V ⊂ [r], |V | = βr, there is an edge from U to V .

Pippenger proved the following theorem:

Theorem 4.3.8 (Explicit Bipartite Expander [Pip87, LPS88]). For every β > 0, there exists a constant d(β) < O(1/β2_{) and a family of polynomial time constructible}

(l, l, d(β), β) bipartite expanders.

We shall actually need unbalanced expanders, which can be easily obtained just by deleting vertices from the above graph. We get the following corollary:

Corollary 4.3.9. For every 1 > β > 0 and constant h > 0, there exists a con- stant d(β, h) and a family of polynomial time constructible (r, hr, d(β, h), β) bipartite expanders.

We use these objects to design Protocol 4.3.10, which is the protocol in The- orem 4.3.3. We can show that Protocol 4.3.10 is a network extractor for entropy k.

Protocol 4.3.10. For a synchronous network Player Inputs: Player i has xi ∈ {0, 1}n

Player Outputs: Player i ends up with zi ∈ {0, 1}m

Sub-Routines and Parameters:

• Let 1 > γ > δ > 0 be any constants.

• Let SRExt, n, m, 1, k be an extractor with parameters as in Theorem 3.5.11.

Let IExt be a C source extractor with parameters n, k, m2 = k, 2 as in The-

orem 3.5.9. • Set r = γp.

• Let G1, γ0, h be such that there is a (hr, r, C,γ−δ_γ , γ0)-AND-disperser promised

byLemma 4.3.6. • Set λ = min{γ0_,γ−δ

1−γ}.

• Let G2 denote the (p − r, hr, d, λ) bipartite expander given byCorollary 4.3.9.

We break up the players into two sets, A = [1, r] and the rest of the players in B. We identify every player in A with a vertex in the right vertex set of the graph G1

and identify every player in B with a vertex in the left vertex set of the graph G2.

We identify the left vertex set of G1 with the right vertex set of G2.

Communication in Round 1 : Every player i ∈ A announces his string xi.

Computation :

1. For every vertex g in the left vertex set of G1, every remaining

player j computes the string yj

g = IExt(xjg1, x j g2, . . . , x j gC), where here xj g1, x j g2, . . . , x j

gC are the strings announced by the C neighbors of g.

2. Every player j ∈ B computes the d × k matrix sj _{whose w’th row is y}j jw,

where here jw is the w’th neighbor of j in G2.

Proof of Theorem 4.3.3. Let SRExt be as in Theorem 3.5.11, set up to extract from an (n, k = nγ_{) source and an independent k}0.9 _{× k somewhere random source with}

error 2−kΩ(1). Let IExt, C be as inTheorem 3.5.9, set up to extract k random bits from C independent (n, k) sources with error 2−kΩ(1).

Let X1, . . . , Xp be any independent (n, k) sources. Since there are at most

t = δp faulty players in the set A, at least a γp−δp_r = γ−δ_γ fraction of the strings xi for i ∈ A must be samples from an (n, k) source. Since G1 is a (hr, r, C,γ−δ_γ , γ0)

AND-disperser, we must have that at least a γ0 fraction of the vertices g in the left vertex set of G1 are such that Yg is 2 close to uniform.

Now every non-faulty player j ∈ B who has at least one such g as a neighbor, ends up with a distribution Sj _{that is }

2 close to being a d × k somewhere random

source. Let H denote the set of non-faulty players in B that don’t get such a some- where random source. Then we see that |H| < λ(p − r) = λ(1 − γ)p < (γ − δ)p, since G2 is a (p − r, hr, d, λ, γ0}) expander and by the definition of λ. Thus, all but

(γ − δ)p + t = γp of the players in B compute a somewhere random source. Then, by the properties of the extractor SRExt, each of these players computes a private random string with an additional error of 1. Since both of these errors are 2−k

Ω(1)

, we get that the final error is also 2−kΩ(1).

Next, we complete the proof by showing how to prove Lemma 4.3.6.

Proof of Lemma 4.3.6. We break up [r] into equally sized disjoint sets S1, . . . , Sδr 2C, so

that for every i, |Si| = 2C/δ. Then consider all subsets T ⊂ Si, with |T | = C. The

We define the bipartite graph with left vertex set [hr], right vertex set [r] and left degree C, by connecting every vertex on the left with the corresponding subset of elements of [r]. To see that this graph is an AND-disperser, let V ⊂ [r] be any subset of density δ. Then, by averaging, we must have that V is at least δ/2-dense in at least a δ/2 fraction of the Si’s. But every Si in which V is δ/2 dense has at least

2C δ

δ

2 = C elements of V . For every such Si, there is a vertex in the left vertex set of

the graph whose neighbors all lie in V .

Thus, there must be at least δ_22Cδr = γhr such vertices.

Protocol 4.3.10addresses the issue of getting network extractors with low en- tropy (we can at least handle polynomially small entropy). However, it only guaran- tees that close to p−2t of the p−t non-faulty players end up with useable randomness. We shall soon see that we cannot hope to give a one round protocol which does better than this, for low min-entropy.