Simple Affine Extractors using Dimension Expansion .
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Transcript of Simple Affine Extractors using Dimension Expansion .
Simple Affine Extractors using Dimension Expansion.
Matt DeVos and Ariel Gabizon
Vague Definition: A pseudorandom object(e.g. graph, function) has some nice property a random object would have with high probability.
For example: A graph that has no large cliquesor large independent sets. The field of pseudorandomness aims to
explicitly construct pseudorandom objects.
Pseudorandomness
Efficient
Det. Alg.
Explicitly constructing pseudorandom objects
bad objects
Universe of exp(n) objects
good object
Why do we want to explicitly construct pseudorandom objects?
-Insight into the computational power(lessnes) of randomness
-Useful tools in derandomizing algorithms (good example-expanders!)
Still, is constructing pseudorandom objects more meaningful than making money, or trying to become famous?
Thm: Pseudorandomness is meaningless Theoretical Computer Science is meaningless
NP machine
PNP by explicitly constructing pseudorandom objects
functions with poly-size circuits
functions on n bits
function in NP without poly-size
circuits
The nice property can usually be phrased as avoiding a not too large set of bad events.
Example: A function of high circuit complexity avoids the event `being computed by circuit C’ for all small circuits C.
Circuits are hard to understand – let’s first work with bad events that are easier to understand.
The bad event in this paper – a function that is biased on an affine subspace.
Affine Extractors
Finite field F, with |F|=q (q=pl for prime p)Vector Space Fn
An affine extractor is a coloring of Fn such that any large enough affine subspace is colored in a balanced way
For simplicit
y assume only 2 colors
Fn
Just to make sure..
An affine subspace XµFn of dim. k Defined by vectors a(1),…,a(k),b2Fn where a(1),
…,a(k) are independent
X={ (j=1 to k) tj¢a(j) + b|t1,…,tk2F}
Now, more formally.. An affine extractor for dim k, field size q
and error ² is a function D:Fn{0,1} such that for any affine
subspace XµFn of dim k |PrxX(D(x) =1 ) - ½|·²(We will omit ² from now on, think of it as 1/100)
Intuition: D `extracts’ a random bit for the uniform distribution on X.
1/100
Feeling the parameters..k-dimension of subspaceq- field size
k larger problem easier (need to be unbiased only on larger subspaces)
q smaller problem harder(subspaces have less structure - are closed under scalar multiplication from smaller field)
Random function D:Fn{0,1} is w.h.p an affine extractor when q=2 and k = 5¢logn
Previous results and ours: (explicit)G-Raz: Affine Extractor for all k¸1, when
q>n2.Bourgain: Affine Extractor for k=®¢n, for
any constant ®>0, and q=2. (exponentially small error)
Our result: Affine Extractor for all k¸1 , when q=((n/k)2)
Simple Construction and Proof! However: need char(F)=(n/k) (have weaker
result for arbitrary characteristic)
Warm UpSuppose q>n. How can we get a function
f:FnF that is non-constant on lines?
i.e, for every a0, b2Fn want g(t) , f(a¢t + b) = f(a1¢t + b1,…,an¢t + bn) to be a non-constant function
Answer: Take f(x1,..,xn) = i=1 to n) xii.
g(t) , f(a¢t + b) = i=1 to n) (ai¢t + bi)i
Note: ai0 for some i. Suppose that an0. g(t) is a non-constant polynomial of degree n.as q>n, this is a non-constant function on F.
(from G-Raz)
Quadratic Residue Function:QR:F{0,1} , QR(a) = 1 $9b2F such that b2=a
Thm[Weil]: Let F be a field of odd size q.Let g(t) be a non-constant polynomial over F of
odd degree d. Choose t2F randomly.. QR(g(t)) has bias at most d/q1/2
works for multivariate g too..
Weil’s Theorem
Subspace X of dim k defined by a(1),…,a(k),b
For f:FnF, define f|X (t1,..,tk) = f((j=1 to k) tj¢a(j) + b )
Using Weil: Poly f(X1,..,Xn) of degree d such that: f|X
constant for all X of dim kAffine Extractor for dim k and q»d2
`trick’: Using this view can multiply vectors
x,y2(Fq)n - not just add them!
Vector Space\Field Dualitynq
nq FF
Fix 1-1 Φ:(Fq)n -->Fqn s.t. ∀a,b∈Fqn s,t∈Fq: Φ(at+ bs) = Φ(a)∙t + Φ(b)∙s We identify the source output with an element
of Fqn:∑aj∙tj+b --> Φ[∑ aj∙tj+b] =∑Φ(aj)∙tj+Φ(b)(as tj ∈ Fq ) our source coincides with a multivariate
polynomial with coeff in Fqn
(from now omit Φ and think of aj∈Fqn )
Viewing the source over the `big’ field
Suppose we allow f|X to have coeff. in the `big field’ Fqn
can take f(x) = x.For any subspace X f|X (t1,..,tk) = (j=1 to k) aj¢tj + b is non-
constant.but to use Weil need f|X with coeff. in Fq
Idea- if coeff. of f|X span Fqn. over Fq – we can `project down to Fq’ without becoming zero\constant
A,B linear subspaces in Fqn
Dfn: A¢B,span{a¢b|a2A, b2B} (enough to take products of basis elements)
[Heur-Lieng-Xiang]Suppose n is prime. Then dim(A¢B)¸ min{dim(A)+dim(B)-1,n}
(analogous to the classic Cauchy-Davenport on Zp)
` dimension expansion of products of subspaces’
Thm: Suppose n is prime. Let T: Fqn Fq be any non-trivial Fq-linear map. Let d=n/(k-1). Suppose Char(F)>d. Let f(x)=T(xd).
Then for any affine subspace X of dim k,f|X is a non-constant poly of degree d with
coeff in Fq.Proof idea: When Char(F) is large enough,
coefficients of f|X are `independent products’ of basis elements.
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Open question: Similar results over F2
Relates to following: n is prime.V a linear subspace of dim k
in (F2)n , k>min{100logn,n/100}. t=┌2n/k┐. Vt ={x1+2+4+..+2^{t} | x2V}. Show that Vt spans (F2)n over F2.
Cauchy – DavenportA,B½Zp
A+B , {a+b| a2A, b2B}
C-D: |A+B| ¸ min{|A|+|B|-1,p}
C-D: |A+B| ¸ min{|A|+|B|-1,p}Proof: Induction on |A|.
|A|=1 : |A+B| = |B| (=|A|+|B|-1)
Induction step: Assume first that ; ( AÅB ( A
Using Inclusion-Exclusion + Ind. Hyp |AÅB + A[B| ¸ min{|AÅB| + |A[B| -1,p}
= min{|A| +|B| -1,p}Done as AÅB + A[B ½ A+B
justify assumption ; ( AÅB (A:w.l.g: 02A,B (can replace A by –a +A, for
some a2A. This does not change |A+B|)|A|>1 , so can fix 0≠a2A.If B=Zp we are done.Otherwise, fix first c s.t. c∙a ∉B.Replace B by –(c-1)∙a + B.We have 02B but a∉B. (which justifies
above assumption)