Strong AverageCase Circuit Lower Bounds from Nontrivial Derandomization

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Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization Lijie Chen MIT This presentation is

Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization Lijie Chen MIT This presentation is based on Lijie’s slides Hanlin Ren IIIS, Tsinghua

(Oversimplified) History and Motivation •

(Oversimplified) History and Motivation •

(Oversimplified) History and Motivation •

(Oversimplified) History and Motivation •

 A plausible (at least in 1980 s) way: prove circuit lower bounds! Okay,

A plausible (at least in 1980 s) way: prove circuit lower bounds! Okay, today we know circuit lower bounds are hard…

 Picture from https: //en. wikipedia. org/wiki/AC 0

Picture from https: //en. wikipedia. org/wiki/AC 0

 Progress basically stops here, until…

Progress basically stops here, until…

 Proven by the so-called Algorithmic Method!

Proven by the so-called Algorithmic Method!

 Algorithmic Method Proving Lower Bounds is a notoriously hard task Why? You must

Algorithmic Method Proving Lower Bounds is a notoriously hard task Why? You must argue that EVERY ALGORITHM (including those we never thought of) does not work Meanwhile, it seems we are much better at finding algorithms: Max Matching, Max-Flow, Linear Programming, FFT, Fast matrix multiplication, Dynamic Programming…

 Algorithmic Questions

Algorithmic Questions

 Algorithmic Questions

Algorithmic Questions

 Subsequent Developments

Subsequent Developments

(Oversimplified) History and Motivation •

(Oversimplified) History and Motivation •

 (Oversimplified) History PRG “Derandomization”

(Oversimplified) History PRG “Derandomization”

 Use Nisan-Wigderson’ 94?

Use Nisan-Wigderson’ 94?

 This lower bound does not yield PRGs!

This lower bound does not yield PRGs!

Strong Average-Case Lower Bounds From Non-trivial Derandomization

Strong Average-Case Lower Bounds From Non-trivial Derandomization

Today’s Plan 0. (Oversimplified) History and Motivation I’ll focus on this interesting ingredient, rather

Today’s Plan 0. (Oversimplified) History and Motivation I’ll focus on this interesting ingredient, rather than other parts of the proof!

Today’s Plan 0. (Oversimplified) History and Motivation

Today’s Plan 0. (Oversimplified) History and Motivation

 Accept? PRG

Accept? PRG

 Accept? PRG

Accept? PRG

Conditional construction of NPRG: A very brief outline It’s actually far from trivial, and

Conditional construction of NPRG: A very brief outline It’s actually far from trivial, and involves PCP of proximity! [Chen-Williams ’ 19]

Conditional construction of NPRG: A very brief outline

Conditional construction of NPRG: A very brief outline

The bottleneck of [Chen’ 19] If Step I is correct, this step is not

The bottleneck of [Chen’ 19] If Step I is correct, this step is not affected!

Circuit complexity of error correction: Intuition

Circuit complexity of error correction: Intuition

Circuit complexity of error correction: Intuition

Circuit complexity of error correction: Intuition

 Circuit complexity of error correction: Intuition

Circuit complexity of error correction: Intuition

The Key Issue on Improving [Chen’ 19]

The Key Issue on Improving [Chen’ 19]

Today’s Plan 0. (Oversimplified) History and Motivation

Today’s Plan 0. (Oversimplified) History and Motivation

Approximate Sum

Approximate Sum

Why is Approx-Sum better than MAJ? Approx-Sum preserves algorithms! A technical detail: Need to

Why is Approx-Sum better than MAJ? Approx-Sum preserves algorithms! A technical detail: Need to verify a given circuit is indeed an Approx-Sum

Error correction via Approximate Sum Let’s see an application first…

Error correction via Approximate Sum Let’s see an application first…

Error Correction via Approximate Sum Idea: Single-Query Randomized Reduction “randomized encodings” for cryptographers!

Error Correction via Approximate Sum Idea: Single-Query Randomized Reduction “randomized encodings” for cryptographers!

Single-Query Randomized Reduction: Setting

Single-Query Randomized Reduction: Setting

Single-Query Randomized Reduction: Protocol

Single-Query Randomized Reduction: Protocol

What if the server has bugs…

What if the server has bugs…

Error correction with multiple buggy servers…

Error correction with multiple buggy servers…

Error correction with Approx-Sum! Approx-Sum of Buggy Servers! Scaling!

Error correction with Approx-Sum! Approx-Sum of Buggy Servers! Scaling!

Updated Three Steps Involves use of PCP of Proximity! See the paper!

Updated Three Steps Involves use of PCP of Proximity! See the paper!

 Conclusion

Conclusion

Open Questions Non-trivial We showed Lower Bounds for Conjectured by [Vyas and Williams] Strong

Open Questions Non-trivial We showed Lower Bounds for Conjectured by [Vyas and Williams] Strong Weak ? Next Step? ? ?

Open Questions Related to the seed length of our unconditional NPRG…

Open Questions Related to the seed length of our unconditional NPRG…

Open Questions

Open Questions

Thanks! Any Questions?

Thanks! Any Questions?