Average Case Complexity
March 26, 2011
Lecture 4 : Average-Case Completeness
Professor: Valentine Kabanets
1
Scribe: Julian Sahasrabudhe
Introduction
We prove the completeness of a distribution problem with respect to the class (NP,PComp).
Recall that BH

Average Case Complexity
Jan 11, 2011
Lecture 2: Why Average Case Complexity?
Professor: Valentine Kabanets
1
Scribe: Mert Salam
g
Introduction
In this class we attempt to formalize what it means for a function to be eciently computable
on average. The not

Average Case Complexity
February 8, 2011
Impagliazzos Hardcore Lemma
Professor: Valentine Kabanets
1
Scribe: Hoda Akbari
Average-Case Hard Boolean Functions w.r.t. Circuits
In this lecture, we are going to introduce a notion of hardness in the sense of ci

Average Case Complexity
March 16, 2011
Impagliazzos Five Worlds
Professor: Valentine Kabanets
Scribe: Shawn Andrews
The Five Worlds of Impagliazzo
(based on A Personal View of Average Case Complexity by R. Impagliazzo, 1995)
1
Algorithmica:
P = NP
(or N P

Average Case Complexity
March 26, 2011
Lecture 5 : Average-Case Completeness
Professor: Valentine Kabanets
1
Scribe: Julian Sahasrabudhe
Decision vs. Search
Recall that in the worst-case setting that under the assumption P = N P , the problem of nding
a N

Average Case Complexity
Jan 27, 2011
Lecture 7: Samplable Distributions Continued
Professor: Valentine Kabanets
1
Scribe: Ruiwen Chen
The construction of reduction
This section will prove the following theorem.
Theorem 1 (Impagliazzo and Levin). Any (L, D

Average Case Complexity
March 17, 2011
Lecture 12: Random k-SAT
Professor: Valentine Kabanets
1
Scribe: Xiongnan (Newman) WU
Denition and Threshold Phenomenon
Up to now, we have only considered general problems in NP with some P-Samplable distributions.
I

Average Case Complexity
Jan 25, 2011
Lecture 6: Samplable Distributions
Professor: Valentine Kabanets
1
Scribe: Ruiwen Chen
Introduction
So far, we have showed that (BH, U BH ) is complete for (NP, PComp):
(BH, U BH ) AvgP(HeurBPP) (NP, PComp) AvgP(HeurBP

Average Case Complexity
March 8, 2011
Lecture 14: Random k-SAT : Second Moment Method
Professor: Valentine Kabanets
1
Scribe: 301136642
SAT Thresholds
Suppose k,cn
That is, is a formula on n variables with c n clauses. We are interested in sk = supc cfw_c

Average Case Complexity
February 10, 2011
Hardness Amplication: Yaos XOR Lemma
Professor: Valentine Kabanets
1
Scribe: Hoda Akbari
Hardness Amplication
Starting with a function that is somewhat hard on average, our aim is to get an even harder
function.
L

Average Case Complexity
March 10, 2011
Lecture 14: The random k-SAT and k-XORSAT solution space
Professor: Valentine Kabanets
1
Scribe: 301136642
Solution Space Geometry
From last time we have the picture:
Curiously note that the shattering phase picture

Average Case Complexity
March 16, 2011
Security Amplication
Professor: Valentine Kabanets
Scribe: Shawn Andrews
Security Amplication
We want to make a weak OWF a strong OWF.
Denition 1. A poly-time f : cfw_0, 1
(n)
n
cfw_0, 1 is a strongly OWF if
c > 0,

Average Case Complexity
March 17, 2011
Lecture 13: Random k-SAT - Continued
Professor: Valentine Kabanets
1
Scribe: Xiongnan (Newman) WU
Threshold for k-SAT
Theorem 1 (Achlioptas, Peres 2004). For any k 3, the satisability threshold Ck (n) for
random k-SA