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 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

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.

Breaking RSA Everyone has N and Kp but not P or Q Factoring N is hard! P and Q are key to determining relationship between Ks and Kp Factoring N is hard! Current record is a 576-bit prime. Bits Pentiums (1 year) Memory 430 1 ~ 760 2.1x105 4GB 1020 3.4x108 170GB 1620 1.6x1015 120TB

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

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.

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

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

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

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

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

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

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