23-07-2012, 02:23 PM
Quantum cryptography
cryptography.ppt (Size: 379 KB / Downloads: 63)
Introduction
Spawned during the last century
Describes properties and interaction between matter at small distance scales
Quantum state determined by(among others)
Positions
Velocities
Polarizations
Spins
qubits
Notation
Bra/Ket notation (pronounced “bracket”)
From Dirac 1958
Each state represented by a vector denoted by a arrow pointing in the direction of the polarization
Simplified Bra/Ket-notation in this presentation
Representation of polarized photons:
horizontally:
vertically:
diagonally: and
Key distribution - BB84
Alice sends a sequence of photons to Bob.Each photon in a state with polarization corresponding to 1 or 0, but with randomly chosen basis.
Bob measures the state of the photons he receives, with each state measured with respect to randomly chosen basis.
Alice and Bob communicates via an open channel. For each photon, they reveal which basis was used for encoding and decoding respectively. All photons which has been encoded and decoded with the same basis are kept, while all those where the basis don't agree are discarded.
Eavesdropping
Eve has to randomly select basis for her measurement
Her basis will be wrong in 50% of the time.
Whatever basis Eve chose she will measure 1 or 0
When Eve picks the wrong basis, there is 50% chance that she'll measure the right value of the bit
E.g. Alice sends a photon with state corresponding to 1 in the {,} basis. Eve picks the {, } basis for her measurement which this time happens to give a 1 as result, which is correct.
Eves problem
Eve has to re-send all the photons to Bob
Will introduce an error, since Eve don't know the correct basis used by Alice
Bob will detect an increased error rate
Still possible for Eve to eavesdrop just a few photons, and hope that this will not increase the error to an alarming rate. If so, Eve would have at least partial knowledge of the key.
Detecting eavesdropping
When Alice and Bob need to test for eavesdropping
By randomly selecting a number of bits from the key and compute its error rate
Error rate < Emax assume no eavesdropping
Error rate > Emax assume eavesdropping(or the channel is unexpectedly noisy)Alice and Bob should then discard the whole key and start over
Noise
Noise might introduce errors
A detector might detect a photon even though there are no photons
Solution:
send the photons according to a time schedule.
then Bob knows when to expect a photon, and can discard those that doesn't fit into the scheme's time window.
There also has to be some kind of error correction in the over all process.
Error correction
Suggested by Hoi-Kwong Lo. (Shortened version)
Alice and Bob agree on a random permutation of the bits in the key
They split the key into blocks of length k
Compare the parity of each block. If they compute the same parity, the block is considered correct. If their parity is different, they look for the erroneous bit, using a binary search in the block. Alice and Bob discard the last bit of each block whose parity has been announced
This is repeated with different permutations and block size, until Alice and Bob fail to find any disagreement in many subsequent comparisons
cryptography.ppt (Size: 379 KB / Downloads: 63)
Introduction
Spawned during the last century
Describes properties and interaction between matter at small distance scales
Quantum state determined by(among others)
Positions
Velocities
Polarizations
Spins
qubits
Notation
Bra/Ket notation (pronounced “bracket”)
From Dirac 1958
Each state represented by a vector denoted by a arrow pointing in the direction of the polarization
Simplified Bra/Ket-notation in this presentation
Representation of polarized photons:
horizontally:
vertically:
diagonally: and
Key distribution - BB84
Alice sends a sequence of photons to Bob.Each photon in a state with polarization corresponding to 1 or 0, but with randomly chosen basis.
Bob measures the state of the photons he receives, with each state measured with respect to randomly chosen basis.
Alice and Bob communicates via an open channel. For each photon, they reveal which basis was used for encoding and decoding respectively. All photons which has been encoded and decoded with the same basis are kept, while all those where the basis don't agree are discarded.
Eavesdropping
Eve has to randomly select basis for her measurement
Her basis will be wrong in 50% of the time.
Whatever basis Eve chose she will measure 1 or 0
When Eve picks the wrong basis, there is 50% chance that she'll measure the right value of the bit
E.g. Alice sends a photon with state corresponding to 1 in the {,} basis. Eve picks the {, } basis for her measurement which this time happens to give a 1 as result, which is correct.
Eves problem
Eve has to re-send all the photons to Bob
Will introduce an error, since Eve don't know the correct basis used by Alice
Bob will detect an increased error rate
Still possible for Eve to eavesdrop just a few photons, and hope that this will not increase the error to an alarming rate. If so, Eve would have at least partial knowledge of the key.
Detecting eavesdropping
When Alice and Bob need to test for eavesdropping
By randomly selecting a number of bits from the key and compute its error rate
Error rate < Emax assume no eavesdropping
Error rate > Emax assume eavesdropping(or the channel is unexpectedly noisy)Alice and Bob should then discard the whole key and start over
Noise
Noise might introduce errors
A detector might detect a photon even though there are no photons
Solution:
send the photons according to a time schedule.
then Bob knows when to expect a photon, and can discard those that doesn't fit into the scheme's time window.
There also has to be some kind of error correction in the over all process.
Error correction
Suggested by Hoi-Kwong Lo. (Shortened version)
Alice and Bob agree on a random permutation of the bits in the key
They split the key into blocks of length k
Compare the parity of each block. If they compute the same parity, the block is considered correct. If their parity is different, they look for the erroneous bit, using a binary search in the block. Alice and Bob discard the last bit of each block whose parity has been announced
This is repeated with different permutations and block size, until Alice and Bob fail to find any disagreement in many subsequent comparisons