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 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

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
Max Planck postulates that energy, like matter, also comes in discrete quantities
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
Quantum Physics
Quantum Computers