optical architecture for (restricted) exponential time hard problems nova fandina ben-gurion...
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Optical Architecture for (Restricted)
Exponential Time Hard Problems
Nova Fandina Ben-Gurion University of the Negev, Israel
Joint work with: Prof. Shlomi Dolev & Prof. Joseph Rosen
Ben-Gurion University of the Negev, Israel
Ben Gurion University of the Negev 19/5/2011
Ben Gurion University of the Negev 2
Outline
Searching for Hard Problem
Succinct Permanent (mod p) is NEXP Time Hard
Succinct Permanent (mod p) Has “Many ” Hard Instances
Holographic Based Optical Architecture
Modern Cryptographic Schemes
Based on unproven complexity assumptions…
what happens if P = NP ?
NEXP hard: don’t have a polynomial time solution
Hard on the average: randomly chosen instance is hard with high probability
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Succinct representation of Graphs[GW83]
Small Circuit Representation output
0 1 2 3 n-1 0 1 2
n-1 log n log n
1
0010
0011
0010 0011
1
1
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Computational problems with succinctly represented inputs
Succ_ problem:𝚷input: succinct representation of the graph Goutput: (G)𝚷 [PY86] If 3SAT 𝚷 then Succ_ 𝚷 is NEXP time hard
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Succinct Permanent
Permanent problem
where the summation is over all permutations σ of {1,…n}
#P - Complete [Val77] Hard on Average as on the Worst Case [Lip91]
Succinct Permanent the output can be too big
11 1
1
n
n n
n nn
a a
A
a a
n
iσ(i)σ i 1
Perm(A) a
|
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Succinct Permanent modulo (small) prime p
input: small boolean circuit representing an integer matrix A with bounded (positive and negative)
entries, prime p (in binary representation)
k- constant c -constant
output: perm A (mod p)
NEXP hard & Big Set of Hard Instances
Outline
8Ben Gurion University of the Negev
Searching for Hard Problem
Succinct Permanent (mod p) is NEXP Time Hard
Succinct Permanent (mod p) Has “Many ” Hard Instances
Holographic Based Optical Architecture
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Zero Succinct Permanent :input: small boolean circuit C representing integer matrix A with bounded entriesoutput: permanent(A)==0
Zero Succinct Permanent (mod p):input: small boolean circuit C representing integer matrix A with bounded entries, small prime poutput: permanent(A)==0 (mod p)
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Roadmap
• Zero Succinct Permanent NEXP time hard
• Zero Succinct Permanent
Zero Succinct Permanent (mod p)
• Zero Succinct Permanent (mod p ) ≤ Succinct Permanent (mod p)
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[PY86]• Succinct 3SAT is NEXP hard
[Val77]• #3SAT Permanen
Zero Succinct Permanent is NEXP hard
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Roadmap
• Zero Succinct Permanent NEXP time hard
• Zero Succinct Permanent
Zero Succinct Permanent (mod p)
• Zero Succinct Permanent (mod p ) ≤ Succinct Permanent (mod p)
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• C represent an integer matrix A with integer values constant |Permanent (A)|
• Chinese Reminder Theorem: Permanent(A) can be computed by computing Permanent(A) modulo each prime p {p
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Define X be a set of first 2|U| primes
The number of prime number in [1,x] is:
The length of representation of each prime in X is polylogarithmical
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Randomized algorithm:
• pick a prime p’ uniformly at random from the set X
• compute Perm(A) mod p’
• if (Perm(A) mod p’ == 0) return Perm(A)==0
• else return Perm(A)!=0
If Per(A)==0 the answer is correct with probability 1
If Per(A)!=0 the answer is correct with probability > ½
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Pick a prime p’ uniformly at random from the set X
• pick p’ uniformly at random from [1, ]
• while(! primality test(p’) ) p’ = pick [1, ]
Primality test: AKS[04]Expected number of attempts: O(logn)
Zero Succinct Permanent (mod p) is NEXP hard (via randomized reduction)
Outline
17Ben Gurion University of the Negev
Searching for Hard Problem
Succinct Permanent (mod p) is NEXP Time Hard
Succinct Permanent (mod p) Has Many Hard Instances
Holographic Based Optical Architecture
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Computing Permanent over
XRPeraRPeraRPeraAPer nn
iiii )()()()( loglog
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1
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4321 tttt
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222 aaaa
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233 aaaa
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244 aaaa
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41
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211 aaaa
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45
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255 aaaa
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Given an answers for (log n +1) matrices chosen at random from the set, compute an answer for matrix A in polynomial time :
solve a system of equations to find
Unique solution exists Computations mod p
naaa log1
2111
naaa log2
2221
nnnn aaa log1log
21log1log1
)( 1APerm
)( 2APerm
)( 1log nAPerm
Outline
20Ben Gurion University of the Negev
Searching for Hard Problem
Succinct Permanent (mod p) is NEXP Time Hard
Succinct Permanent (mod p) Has “Many ” Hard Instances
Holographic Based Optical Architecture
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Optical Device for restricted Succinct Permanent( mod p )
• Solves the instances of the balanced form
• Preprocessing unit: generates and records all matrices that can be represented by balanced small boolean circuits
(holographic based implementation)
• Optical Solver: given an instance outputs an encoded matrix. Forward matrix as an instance to the existing Permanent Solver.
• Applies mod p operation to the result
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Preprocessing Procedure
A A A A
O O O O
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 0 1 1 0 0 1 0
0 1 1 1 0 0 0 1
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Preprocessing Procedure
1 1 1 0 1 0 0 0
1 1 0 1 0 1 0 0
1 0 1 1 0 0 1 0
0 1 1 1 0 0 0 1
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Conclusions
Establishing a computational complexity of Succinct Permanent Problem mod p – NEXP time hard via randomized
reduction – Average case complexity detect a hard instance and compose many hard instances
Optical Solver device – restricted inputs
– existence of the Permanent Solver