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Cryptography and Quantum Computing

Cryptography and Quantum Computing. Brent Plump November 17, 2004. RSA Encryption. Rivest, Shamir and Adelman Developed in 1977 Asymmetric Encryption Two keys: public and private Each key can encrypt a message that only the other key can decrypt. Keys.

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Cryptography and Quantum Computing

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  1. Cryptography and Quantum Computing Brent Plump November 17, 2004

  2. RSA Encryption • Rivest, Shamir and Adelman • Developed in 1977 • Asymmetric Encryption • Two keys: public and private • Each key can encrypt a message that only the other key can decrypt

  3. Keys • Start with two large primes, P and Q • N = PQ • N is known to everyone • Kp is a relative prime to (P-1)(Q-1) • Ks is chosen where KpKs mod (P-1)(Q-1) = 1

  4. Encoding/Decoding • c = mKs mod N • m = cKp mod N • Decoding c verifies that the owner of Ks actually sent the message. • Usually messages are encrypted twice • First with senders private key • Second with recipients public key • Recipient decrypts with their private key, then with senders public key.

  5. Breaking RSA • Everyone has N and Kp but not P or Q • P and Q are key to determining relationship between Ks and Kp • Factoring N is hard! • Current record is a 576-bit prime.

  6. Hard Problems • The only way to solve it is to guess answers repeatedly and check them • There are n possible answers to check • Every possible answer takes the same amount of time to check • There are no clues about which answers might be better. Generating possibilities randomly is just as good as checking them in some special order

  7. Quantum Physics • 1900 - Max Planck postulates that energy, like matter, also comes in discrete quantities • 1921 - Max Born suggests that the probability of finding an electron in a given region depends on the intensity of its wave function there.

  8. Classical Physics Problems • Classical physics states that you can predict what will happen in the future if you measure enough properties (ex. billiard ball) • Mirror Problem: • Classical physics: A mirror reflects 95% of light energy and absorbs 5% • Quantum physics: 19 of every 20 photons is reflected • What happens to each photon is genuinely unpredictable. There is no way to predict the outcome. • Quantum physics states you only know the probabilities of the outcomes

  9. Superposition • Wave function describes the probabilities for different states in the future • Superposition is a combination of all possible states and their probabilities • Superposition exists (is coherent) until the item is observed • When an item is observed, the wave function collapses and the item takes a single, classical state

  10. Building a Quantum Computer • Takes advantage of superposition and evolving wave functions to perform calculations • Force the wave function to a desired result by decreasing the probability of incorrect results • Decoherence is the enemy; nothing can observe the q-bit until you want the result

  11. q-bits • Collapses to either a logical 1 or 0 • While superposition is coherent, may exist in both 1 and 0 states • Calculate by changing the probability of getting a 1 or a 0 • Early q-bit attempts isolated charged particles with magnetic fields

  12. IBM’s Quantum Computer • Uses NMR and chloroform • q-bit: Spin of a Hydrogen nucleus relative to magnetic field • Billions of tiny computers so complete avoidance of decoherence is not important • “Program” is a series of radio frequency pulses

  13. IBM’s Results • Built a 7 q-bit quantum computer • Able to use Shor’s algorithm to factor a 7-bit integer

  14. References • Encryption • http://www.comp.nus.edu.sg/~cs3235/2004-semesterI/foils7.4.pdf • http://www.rsasecurity.com/rsalabs/node.asp?id=2096 • http://www.youdzone.com/rsa.html • Quantum Physics • http://www.newscientist.com/hottopics/quantum/inthebeginning.jsp • http://arxiv.org/abs/quant-ph/0012069 • Quantum Computers • http://www.almaden.ibm.com/st/quantum_information/ • http://alumni.imsa.edu/~matth/quant/299/paper/index.html • http://en.wikipedia.org/wiki/Quantum_computer • http://domino.research.ibm.com/comm/pr.nsf/pages/news.20011219_quantum.html • http://www.media.mit.edu/physics/publications/papers/98.06.sciam/0698gershenfeld.html

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