Download presentation

Presentation is loading. Please wait.

Published byMustafa Decoursey Modified over 2 years ago

1
Derandomization & Cryptography Boaz Barak, Weizmann Shien Jin Ong, MIT Salil Vadhan, Harvard

2
Question Suppose the sequence 666 appears in the digits of both in the 100 th place and in the 1000000 th place. Suppose an archeologist finds a mathematical proof by Archimedes that 666 appears in . Is it possible to recover the place in Archimedes knew about?

3
Our Results Under reasonable assumptions we obtain: Non-interactive WI proof system for NP (in the plain model) First non-interactive proof with secrecy property Non-interactive Commitment Scheme Under incomparable assumptions to [BM]

4
Our Assumptions Assumption A: 9 L s.t. L 2 Dtime(2 cn ) for some c L Ntime(2 n ) / 2 n for some >0 A natural strengthening of EXP * NP NcNc NN NN Thm 1: Assumption A + TDP ) non-interactive WI Thm 2: Assumption A + OWF ) non-interactive commit. In paper: prove Thm 2 under weaker, uniform, assumption. (Uses [GST03])

5
Derandomization: a brief overview* A paradigm that attempts to transform: Probabilistic algorithms => deterministic algorithms. (P BPP EXP NEXP). Probabilistic protocols => deterministic protocols. (NP AM EXP NEXP). We don ’ t know how to separate BPP and NEXP. Can derandomize BPP and AM under natural complexity theoretic assumptions. * Thanks to Ronen Shaltiel for these slides

6
Hardness versus Randomness Initiated by [BM,Yao,Shamir]. Assumption: hard functions exist. Conclusion: Derandomization. A lot of works: [BM82,Y82,HILL,NW88,BFNW93, I95,IW97,IW98,KvM99,STV99,ISW99,MV99, ISW00,SU01,U02,TV02,GST03]

7
Hardness versus Randomness Assumption: hard functions exist. Conclusion: Derandomization.

8
Hardness versus Randomness Assumption: hard functions exist. Exists pseudo-random generator Conclusion: Derandomization.

9
Pseudo-random generators A pseudo-random generator (PRG) is an algorithm that stretches a short string of truly random bits into a long string of pseudo-random bits. pseudo-random bits PRG seed Pseudo-random bits are indistinguishable from truly random bits for feasible algorithms. Consider also generators with O(log n) length seed. ??????????????

10
Pseudo-random generators with O(log n) length seed. Polynomial-sized algorithm can identify pseudo- random strings as follows: Given a long string, enumerate all seeds and check that PRG(seed)=long string. Can distinguish between random strings and pseudo- random strings. Assuming distinguisher can enumerate all seeds. The Nisan-Wigderson setup: distinguisher can not enumerate all seeds. Example: Seed length = 5logn and generator fools circuits of size n 3. PRG can also run in time n 5 Sufficient for derandomization!!

11
State of the art in this direction Thm [NW88, …,IW97]: If 9 L s.t. L 2 Dtime(2 cn ) for some c L Size(2 n ) for some >0 Then BPP=P.

12
Arthur-Merlin Games [BM] Completeness: If the statement is true then Arthur accepts. Soundness: If the statement is false then Pr[Arthur accepts]< ½. Merlin Arthur “xL”“xL” toss coins message I accept

13
Arthur-Merlin Games [BM] Completeness: If the statement is true then Arthur accepts. Soundness: If the statement is false then Pr[Arthur accepts]< ½. The class AM: All languages L which have an Arthur-Merlin protocol. Contains many interesting problems not known to be in NP. (e.g. graph nonisomorphism)

14
The big question: Does AM=NP? In other words: Can every Arthur-Merlin protocol be replaced with one in which Arthur is deterministic? Note that such a protocol is an NP proof.

15
Pseudo-random generators for nondeterministic circuits Nondeterministic algorithm can identify pseudo-random strings as follows: Given a long string, guess a short seed and check that PRG(seed)=long string. Assuming the circuit can run the PRG!! In NW setup circuit cannot run the PRG!!. For example: The PRG runs in time n 5 and fools (nondeterministic) circuits of size n 3.

16
State of the art in this direction Thm [AK,MV,KvM,SU]: If 9 L s.t. L 2 Dtime(2 cn ) for some c L Nsize(2 n ) for some >0 (i.e., if Assumption A holds) Then AM=NP.

17
PRG ’ s for nondeterministic circuits derandomize AM We can model the AM protocol as a nondeterministic circuit which gets the random coins as input. Merlin Arthur “xL”“xL” random message message I accept Hardwire input

18
PRG ’ s for nondeterministic circuits derandomize AM We can model the AM protocol as a nondeterministic circuit which gets the random coins as input. Merlin Arthur “xL”“xL” random input Nondeterministic guess I accept input Nondeterministic guess Hardwire input

19
PRG ’ s for nondeterministic circuits derandomize AM We can model the AM protocol as a nondeterministic circuit which gets the random coins as input. We can use pseudo-random bits instead of truly random bits. Merlin Arthur “xL”“xL” pseudo-random input Nondeterministic guess I accept Nondeterministic guess input Hardwire input

20
PRG ’ s for nondeterministic circuits derandomize AM We have AM protocol w/ deterministic (not probabilistic) Arthur: He sends all pseudo-random strings and Merlin replies on each one. Protocol is sound : otherwise we have a nondeterministic distinguisher. Merlin Arthur “xL”“xL” pseudo-random input Nondeterministic guess I accept Our main observation: If original protocol was WI then new “ protocol ” is also WI!

21
Proof of Thm 1: Thm [DN]: 9 TDP ) 9 AM protocol that is WI for NP Combining this w/ [SU] and observation we get Thm 1: TDP + Assumption A ) 9 Noninteractive WI for NP

22
Proving Thm 2 Use same technique to derandomize Naor ’ s commitment scheme (which is also of “ AM ” type).

23
That ’ s it …

Similar presentations

OK

Using Nondeterminism to Amplify Hardness Emanuele Viola Joint work with: Alex Healy and Salil Vadhan Harvard University.

Using Nondeterminism to Amplify Hardness Emanuele Viola Joint work with: Alex Healy and Salil Vadhan Harvard University.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on social networking sites in india Ppt on model view controller design Ppt on perimeter and area of circle Ppt on transportation in human body Ppt on genders for grade 1 Ppt on teaching learning materials Ppt on median and altitude of a triangle Ppt on history of olympics Ppt on nature and human paintings Ppt on ram and rom