Cs3102: Theory of Computation Class 27: NP-Complete Desserts (DNA, RSA, BQP, NSA) Spring 2010 University of Virginia David Evans.

cs3102: Theory of Computation

Class 27: NP-Complete Desserts(DNA, RSA, BQP, NSA)

Spring 2010University of VirginiaDavid Evans

Reminders

Tuesday: PS7 DueOptional Presentation. If you would like to present or perform your artifact in class on May 4, send me an email by 5:00pm on Monday, May 3 explaining what you would like to do and how much time you think you need for this.

You’ll get a handout that will help you prepare for the final Tuesday.

Check your scores are recorded correctly in Collab

Cost of Synthesizing1990: Human Genome Project starts, estimate $3B to sequence one genome ($0.50/base)

2000: Human Genome Project declared success, cost ~$300M

June 2010: Complete Genomics will start offering full-genome sequencing for $5000 ($0.00000083/base)

Last class

Today

Genomes Computing

Solving HAMPATH with DNA

Make up a two random k-nucleotide sequences for each node: (for example, k = 4)

Based on Fred Hapgood’s notes on Adelman’s talk

A: A1 = ACTT A2 =

gcagB: B1 = TCGG B2 =

actgC: C1 = GGCT C2 =

atgtD: D1 = GATC D2 =

tcca

A

B

C

D

Upper and lowercase letters are the same, just written this way for clarity.

Encoding the ProblemA: A1 = ACTT A2 = gcagB: B1 = TCGG B2 = actgC: C1 = GGCT C2 = atgtD: D1 = GATC D2 = tcca

If there is a link between two nodes (XY), create a nucleotide sequence: X2Y1

A

B

C

D

For each node, create a complement sequence X1X2

(replace A T, G C):

Encoding the ProblemA: A1 = ACTT A2 = gcagB: B1 = TCGG B2 = actgC: C1 = GGCT C2 = atgtD: D1 = GATC D2 = tcca

If there is a link between two nodes (XY), create a nucleotide sequence: X2Y1

A

B

C

D

AB gcagTCGGAC gcagGGCTBC actgGGCTBD actgGATCCD atgtGATC

For each node, create a complement sequence X1X2

(replace A T, G C):

A’ TGAAcgtcB’ AGCCtgacC’ CCGAtacaD’ CTAGaggt

Solving The ProblemMix up all the link and complement DNA strands

AB gcagTCGGAC gcagGGCTBC actgGGCTBD actgGATCCD atgtGATC

A’ TGAAcgtcB’ AGCCtgacC’ CCGAtacaD’ CTAGaggt

TGAAcgtc AGCCtgac

gcagGGCT

TGAAcgtc TGAAcgtcactgGGCT

CTAGaggtCCGAtaca

CCGAtaca

actgGATC

actgGATC

atgtGATC

Shake it Up!

TGAAcgtc

AGCCtgac

gcagGGCT

TGAAcgtc TGAAcgtc

actgGGCT

CTAGaggt

CCGAtaca

CCGAtaca

actgGATC

actgGATC

atgtGATC

Path Binding

A

B

C

D

ACTTgcag

TCGGactg

GATCtcca

GGCTatgt

A’TGAAcgtc

gcagGGCTAB

B’CCGAtaca

atgtTCGG BC

C’AGCCtgac

D’CTAGaggt

actgGATC CD

Getting the SolutionShake up all the DNA to get it to bindExtract strands that start with A and end with D

Can do this with chemical binding on start/end tags: remove all strands that do not start with A, and then remove all strands that do not end with D

Weigh remaining strands to find ones with the right weight (7 * 8 nucleotides)

Select one of these and read its sequence

Is Church-Turing Thesis Wrong?!?

• Time to solve problem with DNA computer doesn’t scale with input size– Can shake up any amount of DNA in the same

amount of time!• Can DNA computers solve undecidable

problems?• Is TM model robust enough for P to be the

same for DNA computer?

No (at least not like this). Can simulate everything (including mixing) with TM.

No: DNA computer can solve NP-Hard problems in constant time! Volume of DNA needed grows exponentially with input size.

DNA-Enhanced PC

To solve HAMPATH for 45 vertices, you need ~20M gallons

Conclusions. For thousands of years, humans have tried to enhance their inherent computational abilities using manufactured devices. Mechanical devices such as the abacus, the adding machine, and the tabulating machine were important advances. But it was only with the advent of electronic devices and, in particular, the electronic computer some 60 years ago that a qualitative threshold seems to have been passed and problems of considerable difficulty could be solved. It appears that a molecular device has now been used to pass this qualitative threshold for a second time.

Len Adleman

“A Breakthrough of Gaussian

Proportions”

Worth A+++++ on PS7!

What Sneakers is really about...

If P = NP all* cryptography is (in theory) broken!

*Not quite all. All cryptography where there are less key bits than total message bits. Information-theoretic crypto (one-time pad) is still perfectly secure.

what would happen if P = NP.

CryptosystemEncryption Decryption

For security, it should be hard to invert c without kd.

For efficiency, it should be easy to invert c with kd.

Cryptosystem

For security, it should be hard to invert c without kd.

For efficiency, it should be easy to invert c with kd.

Is BREAK in NP?

So, what if P = NP?

Would this really mean all cryptography is broken?

Moore’s/Kurzweil’s/Tyson’s law

Science’s Endless Golden Age by Neil DeGrasse

(Almost) everything improves

exponentially

2 3 4 5 6 7 8 9 10111213141516171819201

100000000000000

1E+028

1E+042

1E+056

1E+070

1E+084

1E+098

1E+112

1E+126

1E+140

2^nn^100

2 9 16 23 30 37 44 51 58 65 72 79 86 93100

1

1E+020

1E+040

1E+060

1E+080

1E+100

1E+120

1E+140

1E+160

1E+180

1E+200

2^nn^100

2 77 1522273023774525276026777528279029771

1E+031

1E+062

1E+093

1E+124

1E+155

1E+186

1E+217

1E+248

1E+279

2^nn^100

01E+3072E+3073E+3074E+3075E+3076E+3077E+3078E+3079E+3071E+308 Same graph, non-log scale

In practice, if P=NP and computing power continues to improveexponentially, all cryptosystems are eventually broken!

Point where Excel blows up

What about actual cryptosystems?

“There’s an unwritten rule in astrophysics: your computer simulation must end before you die.”

Neil deGrasse Tyson

RSA Public-Key Cryptosystem 1978

Ron RivestLen Adelman

Adi Shamir

RSA [Rivest, Shamir, Adelman 78]

CS588 Spring 2005 28

RSA in Perl

print pack"C*", split/\D+/, `echo "16iII*o\U@{$/=$z; [(pop,pop,unpack"H*",<>)]} \EsMsKsN0[lN*1lK[d2%Sa2/d0 <X+d*lMLa^*lN%0]dsXx++lMlN /dsM0<J]dsJxp"|dc`(by Adam Back)

Until 1997 – Illegal to show

this slide to non-US citizens!

Until Jan 2000: can export RSA, but only with 512 bit keysNow: can export RSA except to embargoed destinations

RSA

E(M) = Me mod nD(C) = Cd mod n

n = pq p, q are primed is relatively prime to (p – 1)(q –

1)ed 1 (mod (p – 1)(q – 1))

Key property: if you know p and q, it is easy to compute d.

Public key: (e, n)Private key: (d, n)

Security of RSAGiven n, how much work is it to find p and q where n = pq?

Largest challenge factored so far (Jan 2010): b=768 (232 digits) RSA-7682000-computing years

General Number Field Sieve (fastest known factoring algorithm) is in

Factoring Might Be Hard

PNP

NP-Complete

Assuming P NP

Factoring?

Known to be in NPNot known to be in P Not known to be NP-CKnown to be in BQP

Complexity Class BQP

P = Polynomial time: languages that can be decided by a deterministic TM in (Nk) steps.

NP = Nondeterministic Polynomial time: languages that can be decided by a nondeterministic TM in (Nk) steps.

BQP = Bounded Quantum Polynomial time: languages that can be decided by a quantum TM in (Nk) steps with at most 1/3 probability of error

Quantum Physics for Dummies

Light behaves like both a wave and a particle at the same time

A single photon is in many states at once but can’t “observe” its state without forcing it into one state

Schrödinger’s CatPut a live cat in a box with cyanide vial that opens depending on quantum state

Cat is both dead and alive at the same time until you open the box

Quantum ComputingRegular bit: either a 0 or a 1

7 bits can represent any one of 27 different states

Quantum bit (qubit): in 2 possible states at once7 qubits represent 27 different states (at once!)

Computation on qubits: try all possible values at once!

Richard Feynman, 1982

If you could do regular TM operations with a Quantum TM, this would make QP = NP. But you can’t! Actual operations are strange.

What is Known Today

What is Unknown Today

NP BQP NPBQP NPBQPAny of thesecould be true!

Most “Likely” Universe

NPPBQP

NP-Complete

Quantum Computers Today

Handful of quantum algorithmsShor’s algorithm: factoring in P using a quantum computerGrover’s algorithm: searching N unsorted entries in O( N)

15 (= 5 * 3)Actual quantum computers5-qubit computer built by IBM (2001)

Implemented Shor’s algorithm to factor 15 (probably 5 * 3)

Los Alamos: 7-qubit computerTo exceed practical normal computing need > 30 qubits

Adding another qubit is more than twice as hard

Charge

• Our P and NP complexity classes are robust– But, not to very strange definitions of a “step”– DNA and Quantum Computers can modify an

unbounded amount of state in one time step• The universe is a very strange place indeed

If BQP=NP it is an even stranger strange place!