10 - February, 2010 Jordan Myronuk
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1 10 - February, 2010 Jordan Myronuk
2 Classical Cryptography EPR Paradox] The need for QKD Quantum Bits and Entanglement No Cloning Theorem Polarization of Photons BB84 Protocol Probability of Qubit States Example of QKD Intercept-Resend Attack Image From Digital World Tokyo LINK
3 One-time pad (Vernam, 1917) o Uses each symmetric random key once, and is discarded o There is no encryption method that requires less keys (Shannon, 1947) o Alice and Bob must have a way of securely sharing their keys that are as long as the text to be encrypted. Impractical method to share keys given the large amounts of information to be shared today. Key space increased to the point where it becomes computationally infeasible for an adversary for cryptanalysis. Quantum computation has the ability to factor large numbers efficiently, rendering today's schemes computationally insecure. (Shor, 1994, 1997, Shor s Algorithm)
4 Entangled particles are emitted simultaneously o Entangled Electrons Electron 'a' has spin UP Electron 'b' has spin DOWN Law of Conservation Can be separated by vast distances, light years apart. Observation\Measurement of one of the electrons must be the exact opposite of the other, but Nothing can travel faster than the speed of light due to Einstein's special relativity....a property Einstein described as, "spooky action at a distance"
5 Since an action on an entangled particle has an immediate effect on its complementary particle, the information seems to travel faster than light Observation of an entangled particle influences its state, but prior to the observation, the state can not be known due to superposition. Light behaves as a wave during propagation, but collapses into a photon when conducting measured observation.
6 Peter Shor Shor s Algorithm (1994) Uses a specific number of Qubits to solve Finds the prime factors of an integer N in O((log N) 3 ) Break s RSA Factoring is efficient Requires a Quantum computer which doesn t exist yet
7 A classical bit has 2 states A quantum bit, Qubit, can have 3 states: o Superposition can be in both places at once o The double-slit experiment o Interference Pattern o Wave-Particle duality of light waves-photons o The light photons move through the slits as a wave, but collapse as a photon on the wall. Classical Idea Images from Blacklight Power Inc. LINK
8 An opponent, Oscar, must perform a measurement on a quantum particle such as a photon to extract information. This measurement consumes the photon, and modifies the state of the entangled particles. Since the photon is consumed, the loss of energy can be detected as a loss of signal providing an indicator of eavesdropping. The no cloning theorem states, One cannot duplicate an unknown quantum state while keeping the original in tact. This prevents a man-in-the-middle attack as we will see using probability later. No Cloning Theorem Wootters and Zurek, 1982
9 Alice holds a source that generates individual photons. Bob can detect these photons Alice will encrypt the key using the polarization state of these individual photons. Horizontal-Vertical Basis +45 /-45 Basis Bits are encoded as: Physical polarization States: Our quantum channel to transmit this information could be a laser beam, or through fibre-optic cable.
10 Alice and Bob use two channels. Classical Channel such as Radio, Internet, etc. Quantum Channel Classical channel uses classical bits Oscar could intercept and forward the bits from Alice to Bob. Quantum channel uses Qubits Generated from the H/V process Generated from the +45 /-45 process no-cloning of qubits You cannot determine which generation was used without measurement.
11 Alice sends a photon to Bob encoded in one of the 4 possible states: This process is repeated N times. Bob does not know the basis which had been selected before transmission. If Bob receives a Qubit in the + polarization using his + polarization, then the bits correlate correctly100% of the time However, if Bob receives a + polarization Qubit in the x polarization, then the probability reduces to 50%.
12 When a HV polarization of a photon passes through the HV polarizing mirror, it will pass with 100% probability. With the same mirror, but a ±45 polarization photon is sent, it can be projected into the incorrect basis equal to 50%. It is possible that a photon in another polarization can make it through the polarizing mirror as a projection into the other basis
13 When Bob chooses to measure in the same basis as Alice, he will get the bits correct 100% of the time.
14 When the incorrect basis is chosen, Bob has a 50% chance of guessing the correct Qubit P(Guessing Correct Basis) = 50% P(Interpreting Correct Bit)= 50%
15 For each of Alice s transmitted N Qubits, Bob sends the Qubits randomly to either basis to be decoded. Pr[Correct Basis] = 0.5 Pr[Incorrect Basis] = 1- Pr[Correct Basis] = 0.5 Pr[Matching Alice] = 0.75 Bob communicates back the results of his guessing to Alice over the classical channel. Alice tells Bob for which bits he guessed correctly over the classical channel They discard the incorrect bits and agree on a raw key.
16 Alice wishes to securely share a key with Bob Oscar can listen to the quantum channel Alice and Bob use the classical channel to verify the raw key Oscar must guess like Bob does, which introduces error in the channel. The probability of Bob guessing correctly drops to 62.5% Indicator that an eavesdropper was present
17 Alice Sends Bits Basis + + X X + X X + Bob Receives Bits 0/1 1 0/ Basis X + + X + X X + Bob now uses the classical channel to tell Alice which machine he used to decode each Qubit. Alice responds in the classical channel with the bits that were guessed correctly.
18 Alice Bob Bits Basis + + X X + X X + Bits 0/1 1 0/ /1 1 0 Basis X + + X + + X + Bob discards the bits in red as he guessed incorrectly
19 Alice Bob Bits Basis + + X X + X X + Bits 0/1 1 0/ /1 1 0 Basis X + + X + + X + Alice does the same
20 The Agreed Upon Raw Key: Bits Basis + X + X + In order to verify, Alice can ask Bob a query about a subset of the raw key bits. Alice:What is the 5 th bit? Bob:0 Alice 0=0 is true, the key is secure. Now we will look at an example of when Oscar has eavesdropped on the quantum channel.
21 Suppose Oscar taps the quantum channel and intercepts the Qubits before they reach Bob Oscar has to use the same guesses as Bob to determine the Qubits that were sent. With equal probability, Oscar will process a Qubit in a random bias, and re-transmit the Qubit in the same bias that was used to process the incoming Qubit. 50% of the time, Oscar will be correct
22 Bits 1 1 0/1 1 0/1 Basis + X X X X Since Oscar must also guess, 50% of his guesses will be correct This is reflected in the same raw key as before
23 Bits Basis + X X X X Alice asks Bob for the 5 th bit again Alice sees on her end it s a 1, but Bob s reply is a 0, therefore an eavesdropper was present on the quantum channel and changed the state of a bit by observing it.
24 Oscar is able to block the quantum channel as easily as blocking the laser beam, or cutting the fibre optic cable Force Alice and Bob to use a classical method of cryptography Oscar can generate enough interference on the quantum channel that once the probability that Bob can guess the correct bits drops below 17%, a secure key cannot be established. Csiszár and Körner (1978)
25 [1] Scarani et al. The Security of Practical Quantum Key Distribution. Reviews of Modern Physics, Vol. 81, [2] Mermin, David N. Lecture Notes on Quantum Computation, Cornell University, 2006 [3] A. Einstein, B. Podolsky, N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Institure for Advanced Study, Princeton, NJ [4] Csiszár, I., and J. Körner, IEEE Transactions on Information Theory [5] Wootters, W. K., and W. H. Zurek, Nature. London, 1982
26 1. What does Shor s Algorithm do? 2. What are the states of a Qubit? 3. How does a Man-in-the-middle attack work over a Quantum channel? 4. What does the No-Cloning theorem prevent, and why does it work? 5. If you send Bob the bits below, what is your raw key? Bits Bias X + X + X X X X + + X X Bob s Response: Bits 1 0/ /1 Bias X X + + X X X X + + X + 6. If a specific bit of Bob s raw key is incorrect, what can you determine about the key exchange over the quantum channel?
27 1. Factors large primes efficiently in O(logN) , 1 and BOTH = 3 states 3. Captures Qubits with a guessed bias, and retransmits them on the same guessed bias. 4. Prevents copying of photons because they are destroyed upon measuring them. 5. If you send Bob the bits below, what is your raw key? Bits Bias X + X + X X X X + + X X raw key= An eavesdropper was listening on the quantum channel.
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