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 (PRGs) New approach to construct PRGs Open problems

Introducing Pseudorandom generators(PRGs) Definition of pseudorandom generator (PRG):

Introducing Pseudorandom generators(PRGs) Definition of pseudorandom generator (PRG): ℱ = 𝑔: −1,1 * ⟶ −1,1 family of functions : tests

Introducing Pseudorandom generators(PRGs) 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

Introducing Pseudorandom generators(PRGs) 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 *

Introducing Pseudorandom generators(PRGs) 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

Introducing Pseudorandom generators(PRGs) 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

Introducing Pseudorandom generators(PRGs) Goal: Construct random variable 𝑌 .

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ?

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? An algorithm to sample random variable 𝑌 ∈ −1,1 *

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction.

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? An algorithm to sample random variable 𝑌 ∈ −1,1 * Use few coin flips in the construction. Algorithm should be “explicit”/ ”easy to compute”

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? 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 *

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? 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 𝑉 4 where 𝑉 4 is uniform over −1,1 4

Introducing Pseudorandom generators(PRGs) Question. What do we mean by “construct” 𝑌 ? 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 𝑉 4 where 𝑉 4 is uniform over −1,1 4 𝑡 is called seed length

Example * characters Example 1: Tests: 𝔾 7 ℱ = 𝑔 𝑦 = ∏ 𝑦 : ∶ 𝑇 ⊆ 𝑜 :∈;

Example * characters Example 1: Tests: 𝔾 7 ℱ = 𝑔 𝑦 = ∏ 𝑦 : ∶ 𝑇 ⊆ 𝑜 :∈; 𝑌 ∶ 𝜁 -bias random variable

Example * characters Example 1: Tests: 𝔾 7 ℱ = 𝑔 𝑦 = ∏ 𝑦 : ∶ 𝑇 ⊆ 𝑜 :∈; 𝑌 ∶ 𝜁 -bias random variable • PRGs with optimal seed length 𝑃 log 𝑜/𝜁 are known.

Example * characters Example 1: Tests: 𝔾 7 ℱ = 𝑔 𝑦 = ∏ 𝑦 : ∶ 𝑇 ⊆ 𝑜 :∈; 𝑌 ∶ 𝜁 -bias random variable • PRGs with optimal seed length 𝑃 log 𝑜/𝜁 are known. • Initiated by [Naor-Naor’90], found many applications

Fractional PRGs 𝑔: −1,1 * → −1,1 1 -1 -1 1 1 1 -1 1

Fractional PRGs 𝑔: −1,1 * → −1,1 multi−linear extension 𝑔: ℝ * → ℝ 1 -1 -1 1 1 1 -1 1

Fractional PRGs 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 1 -1 1

Fractional PRGs Equivalent definition of PRG: 𝑌 ∈ −1,1 * ε -fools ℱ if 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁, ∀𝑔 ∈ ℱ 1 -1 -1 1 1 1 -1 1

Fractional PRGs 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 1 -1 1

Fractional PRGs PRG: random variable 𝑌 ∈ −1,1 * where 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁

Fractional PRGs 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

Fractional PRGs 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 1 -1 1

Fractional PRGs 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 1 -1 1 Trivial f-PRG: 𝑌 ≡ 0 ; we will rule it out later.

Fractional PRGs 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 1 -1 1 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?

Fractional PRGs How to convert 𝑌 ∈ −1,1 * to 𝑌 L ∈ −1,1 * ?

Fractional PRGs How to convert 𝑌 ∈ −1,1 * to 𝑌 L ∈ −1,1 * ? do a random walk that converges to −1,1 * Main idea:

Fractional PRGs How to convert 𝑌 ∈ −1,1 * to 𝑌 L ∈ −1,1 * ? do a random walk that converges to −1,1 * Main idea: the steps of the random walk are from 𝑌

Fractional PRGs How to convert 𝑌 ∈ −1,1 * to 𝑌 L ∈ −1,1 * ? do a random walk that converges to −1,1 * Main idea: the steps of the random walk are from 𝑌 Recall: f-PRG is 𝑌 = (𝑌 M , ⋯, 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

Constructing PRGs from f-PRGs Ma Main theorem: Suppose: ℱ : class of 𝑜 -variate Boolean functions, closed under restrictions

Constructing PRGs from f-PRGs Ma Main theorem: Suppose: ℱ : class of 𝑜 -variate Boolean functions, closed under restrictions 𝑌 ∈ −1,1 * : 𝔽𝑔 𝑌 − 𝑔(0) ≤ 𝜁 ∀𝑔 ∈ ℱ

Constructing PRGs from f-PRGs 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

Constructing PRGs from f-PRGs 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 𝑌 M ,… , 𝑌 T such that 𝑌 M ,… , 𝑌 T are independent copies of 𝑌 ,

Constructing PRGs from f-PRGs 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 𝑌 M ,… , 𝑌 T such that 𝑌 M ,… , 𝑌 T are independent copies of 𝑌 , 𝑌′ ∈ −1,1 * : 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ

Constructing PRGs from f-PRGs 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 𝑌 M ,… , 𝑌 T such that 𝑌 M ,… , 𝑌 T are independent copies of 𝑌 , 𝑌′ ∈ −1,1 * : 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ M * 𝑢 = 𝑃 V log W

Constructing PRGs from f-PRGs 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 𝑌 M ,… , 𝑌 T such that 𝑌 M ,… , 𝑌 T are independent copies of 𝑌 , 𝑌′ ∈ −1,1 * : 𝔽𝑔 𝑌′ − 𝑔(0) ≤ 𝜁𝑢 ∀𝑔 ∈ ℱ M * 𝑢 = 𝑃 V log W • If 𝑌 has seed length 𝑡 then 𝑌′ has seed length 𝑢𝑡