Genomes Computing. Reminders. Cost of Synthesizing. Encoding the Problem B. Solving HAMPATH with DNA. cs3102: Theory of Computation

Transcription

1 cs3102: Theory of omputation lass 27: NP-omplete esserts (NA, RSA, QP, NSA) Spring 2010 University of Virginia avid Evans Tuesday: PS7 ue Reminders Optional Presentation. If you would like to present or perform your artifact in class on May 4, send me an by 5:00pm on Monday, May 3explaining 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. heck your scores are recorded correctly in ollab ost of Synthesizing 1990: Human Genome Project starts, estimate $3 to sequence one genome ($0.50/base) 2000: Human Genome Project declared success, cost ~$300M Genomes omputing June 2010: omplete Genomics will start offering full-genome sequencing for $5000 ($ /base) Solving HAMPATH with NA A Make up a two random k-nucleotide sequences for each node: (for example, k= 4) A:A 1 = ATT A 2 = gcag : 1 = TGG 2 = actg : 1 = GGT 2 = atgt : 1 = GAT 2 = tcca Upper and lowercase letters are the same, just written this way for clarity. A Encoding the Problem If there is a link between two nodes (X Y), create a nucleotide sequence: X 2 Y 1 A: A 1 = ATT A 2 = gcag : 1 = TGG 2 = actg : 1 = GGT 2 = atgt : 1 = GAT 2 = tcca For each node, create a complement sequence X 1 X 2 (replace A T, G ): ased on Fred Hapgood s notes on Adelman s talk

2 A Encoding the Problem If there is a link between two nodes (X Y), create a nucleotide sequence: X 2 Y 1 A gcagtgg A gcagggt actgggt actggat atgtgat A: A 1 = ATT A 2 = gcag : 1 = TGG 2 = actg : 1 = GGT 2 = atgt : 1 = GAT 2 = tcca For each node, create a complement sequence X 1 X 2 (replace A T, G ): A TGAAcgtc AGtgac GAtaca TAGaggt Solving The Problem Mix up all the link and complement NA strands A gcagtgg A gcagggt actgggt actggat atgtgat A TGAAcgtc AGtgac GAtaca TAGaggt atgtgat GAtaca TAGaggt TGAAcgtc AGtgac actggat gcagggt actgggt TGAAcgtc TGAAcgtc GAtaca actggat Shake it Up! atgtgat GAtaca AGtgac actgggt TGAAcgtc TGAAcgtc TGAAcgtcGAtaca actggat gcagggt actggat TAGaggt A TGAAcgtc gcagggt A GAtaca A ATTgcag Path inding atgttgg AGtgac TGGactg GGTatgt actggat TAGaggt GATtcca Getting the Solution Shake up all the NA to get it to bind Extract strands that start with Aand end with an 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 Weighremaining strands to find ones with the right weight (7 * 8 nucleotides) Select one of these and read its sequence Is hurch-turing Thesis Wrong?!? Time to solve problem with NA computer doesn t scale with input size an shake up any amount of NA in the same amount of time! an NA computers solve undecidable problems? Is TM model robust enough for P to be the same for NA computer? No(at least not like this). an simulate everything (including mixing) with TM. No:NA computer can solve NP-Hard problems in constant time! Volume of NA needed grows exponentially with input size.

3 NA-Enhanced P To solve HAMPATH for 45 vertices, you need ~20M gallons onclusions. 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. ut 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 What Sneakers is really about... what would happen if P = NP. A reakthrough of Gaussian Proportions Worth A+++++ on PS7! ryptosystem Encryption 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. ryptosystem ecryption For efficiency, it should be easy to invert c with kd. For efficiency, it should be easy to invert c with kd. For security, it should be hard to invert c without kd. For security, it should be hard to invert c without kd.

4 Is REAKin NP? Would this reallymean all cryptography is broken? So, what if P = NP? Moore s/kurzweil s/tyson s law (Almost) everything improves exponentially Science s Endless Golden Ageby Neil egrasse 1E+128 1E+120 1E+112 1E+104 1E+96 1E+88 1E+80 1E+72 1E+64 1E+56 1E+48 1E+40 1E+32 1E+24 1E ^n n^100 Point where Excel blows up 1E+195 1E+180 1E+165 1E+150 1E+135 1E+120 1E+105 1E+90 1E+75 1E+60 1E+45 1E+30 1E ^n n^100 1E+300 1E+280 1E+260 1E+240 In practice, if P=NP and computing power continues to improve exponentially, all cryptosystems are eventually broken! 1E+220 1E+200 1E+180 1E+308 Same graph, non-log scale 1E+160 1E+140 8E+307 1E+120 6E+307 1E+100 4E+307 1E+80 1E+60 2E+307 1E E ^n n^

5 There s an unwritten rule in astrophysics: your computer simulation must end before you die. Neil degrasse Tyson What about actual cryptosystems? RSA Public-Key ryptosystem 1978 Adi Shamir Len Adelman RSA [Rivest, Shamir, Adelman 78] Ron Rivest RSA in Perl print pack"*", split/\+/, Until 1997 `echo "16iII*o\U@{$/=$z; Illegal to show [(pop,pop,unpack"h*",<>)]} \EsMsKsN0[lN*1lK[d2%Sa2/d0 this slide to nonnon<x+d*lmla^*ln%0]dsxx++lmln US citizens! /dsm0<j]dsjxp" dc` (by Adam ack) Until Jan 2000: can export RSA, but only with 512 bit keys Now: can export RSA except to embargoed destinations S588 Spring 2005 RSA Me mod E(M) = n Public key: (e, n) () = d mod n Private key: (d, n) p, q are prime n = pq d 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. 28 Security of RSA Given n, how much work is it to find p and q where n = pq? General Number Field Sieve (fastest known factoring algorithm) is in Largest challenge factored so far (Jan 2010): b=768 (232 digits) RSA computing years

6 Factoring Might e Hard omplexity lass QP NP P Factoring? Known to be in NP Notknown to be in P Notknown to be NP- Known to be in QP P = Polynomial time:languages that can be decided by a deterministic TM in Θ(N k )steps. NP = Nondeterministic Polynomial time:languages that can be decided by a nondeterministic TM in Θ(N k )steps. QP = ounded Quantum Polynomial time: languages that can be decided by a quantum TM in Θ(N k )steps with at most 1/3 probability of error Assuming P NP Quantum Physics for ummies Light behaves like botha 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 at Put a live cat in a box with cyanide vial that opens depending on quantum state at is bothdead and alive at the same time until you open the box Quantum omputing Regular bit: eithera 0 or a 1 7 bits can represent any one of 2 7 different states Quantum bit (qubit): in 2 possible states at once 7 qubitsrepresent 2 7 different states (at once!) omputation on qubits: try all possible values at once! Richard Feynman, 1982 Ifyou could do regular TM operations with a Quantum TM, this would make QP= NP. ut you can t! Actual operations are strange. What is Known Today Most Likely Universe What is Unknown Today QP NP P NP QP NP QP NP QP Any of these could be true!

7 Quantum omputers Today Handful of quantum algorithms Shor s algorithm: factoring in P using a quantum computer Grover s algorithm: searching N unsorted entries in O( N) Actual quantum computers 5-qubit computer built by IM (2001) Implemented Shor salgorithm to factor 15 (probably5 * 3) Los Alamos: 7-qubit computer To exceed practical normal computing need > 30 qubits Adding another qubitis more than twice as hard 15 (= 5 * 3) harge Our Pand NPcomplexity classes are robust ut, not to very strange definitions of a step NA and Quantum omputers can modify an unbounded amount of state in one time step The universe is a very strange place indeed If QP=NPit is an even stranger strange place!