Pseudorandom generators from polarizing random walks Ka Kaave Ho - - PowerPoint PPT Presentation
Pseudorandom generators from polarizing random walks Ka Kaave Ho Hossei eini (UC San Diego) Eshan Chattopadhyay (IAS Cornell) Pooya Hatami (UT Austin Ohio State) Shachar Lovett (UC San Diego) Outline Introduce Pseudorandom generators
Ka Kaave Ho Hossei eini (UC San Diego) Eshan Chattopadhyay (IAS → Cornell) Pooya Hatami (UT Austin → Ohio State) Shachar Lovett (UC San Diego)
Introduce Pseudorandom generators (PRGs) New approach to construct PRGs Open problems
Definition of pseudorandom generator (PRG):
Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests
Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests 𝑉 : Random variable uniform over −1,1 * : truly random object
Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests 𝑉 : Random variable uniform over −1,1 * : truly random object A random variable 𝑌 over −1,1 *
Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests 𝑉 : Random variable uniform over −1,1 * : truly random object A random variable 𝑌 over −1,1 * is 𝜁-pseudorandom for ℱ if 𝔽𝑔 𝑌 − 𝔽𝑔 𝑉 ≤ 𝜁 ∀𝑔 ∈ ℱ
Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests 𝑉 : Random variable uniform over −1,1 * : truly random object A random variable 𝑌 over −1,1 * is 𝜁-pseudorandom for ℱ (𝑌 𝜁-fools ℱ) if 𝔽𝑔 𝑌 − 𝔽𝑔 𝑉 ≤ 𝜁 ∀𝑔 ∈ ℱ
Goal: Construct random variable 𝑌.
An algorithm to sample random variable 𝑌 ∈ −1,1 *
An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction.
An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction. Algorithm should be “explicit”/ ”easy to compute”
An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction. Algorithm should be “explicit”/ ”easy to compute” 𝐻: −1,1 4 ⟶ −1,1 *
An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction. Algorithm should be “explicit”/ ”easy to compute” 𝐻: −1,1 4 ⟶ −1,1 * 𝑌 = 𝐻 𝑉4 where 𝑉4 is uniform over −1,1 4
An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction. Algorithm should be “explicit”/ ”easy to compute” 𝐻: −1,1 4 ⟶ −1,1 * 𝑌 = 𝐻 𝑉4 where 𝑉4 is uniform over −1,1 4 𝑡 is called seed length
Example 1: Tests: 𝔾7
* characters
ℱ = 𝑔 𝑦 = ∏ 𝑦:
:∈;
∶ 𝑇 ⊆ 𝑜
Example 1: Tests: 𝔾7
* characters
ℱ = 𝑔 𝑦 = ∏ 𝑦:
:∈;
∶ 𝑇 ⊆ 𝑜 𝑌 ∶ 𝜁-bias random variable
Example 1: Tests: 𝔾7
* characters
ℱ = 𝑔 𝑦 = ∏ 𝑦:
:∈;
∶ 𝑇 ⊆ 𝑜 𝑌 ∶ 𝜁-bias random variable
are known.
Example 1: Tests: 𝔾7
* characters
ℱ = 𝑔 𝑦 = ∏ 𝑦:
:∈;
∶ 𝑇 ⊆ 𝑜 𝑌 ∶ 𝜁-bias random variable
are known.
𝑔: −1,1 * → −1,1
1 1 1 1 1
𝑔: −1,1 * → −1,1
multi−linear extension
𝑔: ℝ* → ℝ
1 1 1 1 1
𝑔: −1,1 * → −1,1
multi−linear extension
𝑔: ℝ* → ℝ Only consider points in [−1,1]* so 𝑔: [−1,1]*→ [−1,1]
1 1 1 1 1
Equivalent definition of PRG: 𝑌 ∈ −1,1 * ε-fools ℱ if 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁, ∀𝑔 ∈ ℱ
1 1 1 1 1
Equivalent definition of PRG: 𝑌 ∈ −1,1 * ε-fools ℱ if 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁, ∀𝑔 ∈ ℱ because 𝔽𝑔 𝑉* = 𝑔 𝔽𝑉* = 𝑔 0
1 1 1 1 1
PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁
PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Fractional PRG (f-PRG): random variable 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁
PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Fractional PRG (f-PRG): random variable 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁
1 1 1 1 1
PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Fractional PRG (f-PRG): random variable 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Trivial f-PRG: 𝑌 ≡ 0 ; we will rule it out later.
1 1 1 1 1
PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Fractional PRG (f-PRG): random variable 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Trivial f-PRG: 𝑌 ≡ 0 ; we will rule it out later. Question. Are f-PRGs easier to construct than PRGs? Can f-PRGs be used to construct PRGs?
1 1 1 1 1
How to convert 𝑌 ∈ −1,1 * to 𝑌L ∈ −1,1 *?
How to convert 𝑌 ∈ −1,1 * to 𝑌L ∈ −1,1 *? Main idea: do a random walk that converges to −1,1 *
How to convert 𝑌 ∈ −1,1 * to 𝑌L ∈ −1,1 *? Main idea: do a random walk that converges to −1,1 * the steps of the random walk are from 𝑌
How to convert 𝑌 ∈ −1,1 * to 𝑌L ∈ −1,1 *? Main idea: do a random walk that converges to −1,1 * the steps of the random walk are from 𝑌 Recall: f-PRG is 𝑌 = (𝑌M, ⋯, 𝑌*) ∈ [−1,1]* where 𝔽 𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Trivial solution: 𝑌 ≡ 0 Need to enforce non-triviality: require 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … , 𝑜
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … ,𝑜
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … ,𝑜 Then there is 𝑌′ = 𝐻 𝑌M,… , 𝑌T such that 𝑌M,… , 𝑌T are independent copies of 𝑌,
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … ,𝑜 Then there is 𝑌′ = 𝐻 𝑌M,… , 𝑌T such that 𝑌M,… , 𝑌T are independent copies of 𝑌, 𝑌′ ∈ −1,1 *: 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … ,𝑜 Then there is 𝑌′ = 𝐻 𝑌M,… , 𝑌T such that 𝑌M,… , 𝑌T are independent copies of 𝑌, 𝑌′ ∈ −1,1 *: 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ 𝑢 = 𝑃
M Vlog * W
Ma Main theorem: Suppose: ℱ: class of 𝑜-variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 *: 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ 𝔽 𝑌: 7 ≥ 𝑞 for all 𝑗 = 1, … ,𝑜 Then there is 𝑌′ = 𝐻 𝑌M,… , 𝑌T such that 𝑌M,… , 𝑌T are independent copies of 𝑌, 𝑌′ ∈ −1,1 *: 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ 𝑢 = 𝑃
M Vlog * W
Goal: use the f-PRG to define a random walk
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step?
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step?
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step?
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step?
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
Goal: use the f-PRG to define a random walk f-PRG: 𝑌 ∈ [−1,1]* where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 Equivalently: 1st step from 0 Question: what about the 2nd step? We have to assume the class is closed under restriction. Lemma: In second step error is still ≤ 𝜁: because function in scaled cube is in the convex hull of restrictions of 𝑔.
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1]
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
Proof: always we have 1 − |𝑍
:| < 1 − 𝑍 :ZM
1 − 𝑌:
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
Proof: always we have 1 − |𝑍
:| < 1 − 𝑍 :ZM
1 − 𝑌: 𝔽 1 − 𝑍
:
< 𝔽 1 − 𝑍
:ZM 𝔽 1 − 𝑌:
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
Proof: always we have 1 − |𝑍
:| < 1 − 𝑍 :ZM
1 − 𝑌: 𝔽 1 − 𝑍
:
< 𝔽 1 − 𝑍
:ZM 𝔽 1 − 𝑌:
𝔽 1 − 𝑌: = 1, however, 𝔽 1 − 𝑌: < 1 − 𝔽\`
]
a = 1 − 𝑑
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
Proof: always we have 1 − |𝑍
:| < 1 − 𝑍 :ZM
1 − 𝑌: 𝔽 1 − 𝑍
:
< 𝔽 1 − 𝑍
:ZM 𝔽 1 − 𝑌:
𝔽 1 − 𝑌: = 1, however, 𝔽 1 − 𝑌: < 1 − 𝔽\`
]
a = 1 − 𝑑
𝔽 1 − |𝑍
:| < 𝔽 (1 − |𝑍 :ZM|) 1 − 𝑑 < 1 − 𝑑 :
∎
It’s enough to prove it for one dimension: so let 𝑌 be a r.v. on [−1,1] Lemma: Let 𝑍
Y = 0 , 𝑍 T = 𝑍 TZM + 1 − 𝑍 TZM 𝑌T be a random walk with 𝔽𝑌: = 0.
Then after 𝑃
M 𝔽 \ ] log M W
steps, w.h.p 1 − 𝑍
T ≤ 𝜁
Proof: always we have 1 − |𝑍
:| < 1 − 𝑍 :ZM
1 − 𝑌: 𝔽 1 − 𝑍
:
< 𝔽 1 − 𝑍
:ZM 𝔽 1 − 𝑌:
𝔽 1 − 𝑌: = 1, however, 𝔽 1 − 𝑌: < 1 − 𝔽\`
]
a = 1 − 𝑑
𝔽 1 − |𝑍
:| < 𝔽 (1 − |𝑍 :ZM|) 1 − 𝑑 < 1 − 𝑑 :
∎ Round to sign{𝑍
T} once the random walk is close enough to the boundary
𝑔: −1,1 * → {−1,1} Fourier coefficients: 𝑔 f 𝑇 = 𝔽 𝑔 𝑦 ∏ 𝑦:
:∈;
, 𝑇 ⊆ [𝑜]
𝑔: −1,1 * → {−1,1} Fourier coefficients: 𝑔 f 𝑇 = 𝔽 𝑔 𝑦 ∏ 𝑦:
:∈;
, 𝑇 ⊆ [𝑜] 𝑔 has bounded Fourier growth if g |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 c = 𝑜 is a trivial bound.
|𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1
|𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1
𝑍
: :∈;
< 𝜁 , ∀𝑇 ⊆ 𝑜 , 𝑇 ≠ 𝜚
|𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1
𝑍
: :∈;
< 𝜁 , ∀𝑇 ⊆ 𝑜 , 𝑇 ≠ 𝜚
7o 𝑍 , note: 𝑌 ∈ − M 7o , M 7o *
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] ,
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇 𝔽 ∏ 𝑌:
:∈; ;q∅
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝔽 ∏ 𝑍
: :∈; ;q∅
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝔽 ∏ 𝑍
: :∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝜁
;q∅
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝔽 ∏ 𝑍
: :∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝜁
;q∅
≤ ∑ 𝑑h
M 7o h
𝜁
hsM
Proof : 𝑔: −1,1 * → {−1,1} with ∑ |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 Construction: 𝑌 = M
7o 𝑍 , 𝑍 ∈ −1,1 * is 𝜁-bias r.v: |𝔽∏
𝑍
: :∈;
| < 𝜁 , ∀𝑇 ⊆ [𝑜] , 𝔽𝑔 𝑌 − 𝑔 0 = ∑ 𝑔 f 𝑇 ⋅ 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇 𝔽 ∏ 𝑌:
:∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝔽 ∏ 𝑍
: :∈; ;q∅
≤ ∑ 𝑔 f 𝑇
M 7o ;
𝜁
;q∅
≤ ∑ 𝑑h
M 7o h
𝜁
hsM
≤ ∑ 2Zh𝜁
hsM
≤ 𝜁
𝑔: −1,1 * → −1,1 , g |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 seed length = 𝑑7 log
* u
loglog𝑜 + log
M u
𝑔: −1,1 * → −1,1 , g |𝑔 f 𝑇 | ≤ 𝑑h
;: ; ih
∀𝑙 ≥ 1 seed length = 𝑑7 log
* u
log log 𝑜 + log
M u
Classes of functions:
Functions with sensitivity 𝑡:
𝑑 = 𝑃(𝑡)
Gopalan-Servedio-Wigderson’16
Permutation branching programs of width 𝑥:
𝑑 = 𝑃(𝑥7)
Reingold-Steinke-Vadhan’13
Read once branching programs of width 𝑥:
𝑑 = logw 𝑜
Chattopadhyay-Hatami-Reingold-Tal’18
Circuits of depth 𝑒:
𝑑 = logy 𝑡
Tal’17
𝑌M ⋮ 𝑌T
𝑌M ⋮ 𝑌T 𝒉 𝒉 𝒉
𝐻 𝑌M, … ,𝑌T = 𝑌M,M,… , 𝑌T,M ,… , 𝑌M,*,… , 𝑌T,*
𝑌M ⋮ 𝑌T 𝒉 𝒉 𝒉
Thank you!