Quantum cryptography CS415 Biometrics and Cryptography UTC/CSE
Introduction Light waves are propagated as discrete particles known as photons. Light waves are propagated as discrete particles known as photons. Polarization of the light is carried by the direction of the angular momentum, or spin of the photons. Polarization of the light is carried by the direction of the angular momentum, or spin of the photons.
Polarized photons Polarization can be modeled as a linear combination of basis vectors vertical ( ) and horizontal ( ) Polarization can be modeled as a linear combination of basis vectors vertical ( ) and horizontal ( ) A quantum state of a photon is described as a vector quantum cryptography often uses photons in 1 of 4 polarizations (in degrees): 0, 45, 90, 135 ψ b a
Properties of Quantum Information Heisenberg Uncertainty Principle (HUP) If there is a particle, such as an electron, moving through space, it is impossibly to measure both its position and momentum precisely.
A polarization filter A polarization filter is a material that allows only light of a specified polarizatio direction to pass. A polarization filter is a material that allows only light of a specified polarizatio direction to pass. A photon will either pass or not pass through a polorization filter, but if it emerges it will be aligned with the filter regardless of its initial state. There are no partial photons. A photon will either pass or not pass through a polorization filter, but if it emerges it will be aligned with the filter regardless of its initial state. There are no partial photons.
Polarization by a Filter Unpolarized light Vertical aligned filter Vertically polarized light Filter tilted at angle q Unpolarized light enters a vertically aligned filter, some light is absorbed and the remainder is polarized in the vertical direction. A second filter tilted at some angle q absorbs some of the polarized light and transmits the rest, giving it a new polarization.
Polarization by a Filter Unpolarized light Vertical aligned filter Vertically polarized light Filter tilted at angle q If the first one is the generator from Alice, a vertical polarized light is generated. There is a certain probability that the photon will pass through the second filter. The probability depends on the angle q. The angle increases from 0 to 90 degree, and the probability decreases from 1 to 0. When q is 45 degree, the probability is precisely 50%.
Polarization by a Filter Transmitting light polarization and measurements determine the polarization of the outgoing light. TransmittingMeasurementOutgoing Alice transmits 1 (+45 degree) Bob Measures with -45 degree filter Photos are always blocked Bob Measures with 90 degree filter 45% photons blocked 45% photons pass Bob transmit 0 (0 degree) Bob Measures with -45 degree filter 45% photons blocked 45% photons pass Bob Measures with 90 degree filter Photos are always blocked Perpendicular blocked; Otherwise some pass
More examples
Quantum Cryptography
Better Name – Quantum Key Distribution (QKD) – It’s NOT a new crypto algorithm! Better Name – Quantum Key Distribution (QKD) – It’s NOT a new crypto algorithm! Two physically separated parties can create and share random secret keys. Two physically separated parties can create and share random secret keys. Allows them to verify that the key has not been intercepted. Allows them to verify that the key has not been intercepted.
Quantum Key Distribution Requires two channels one quantum channel (subject to adversary and/or noises) one public channel (authentic, unjammable, subject to eavesdropping)
BB84 QKD protocol uses polarization of photons to encode the bits of information – relies on “uncertainty” to keep Eve from learning the secret key. uses polarization of photons to encode the bits of information – relies on “uncertainty” to keep Eve from learning the secret key. Bennett: “Quantum cryptography using any two nonorthogonal states”, Physical Review Letters, Vol. 68, No. 21, 25 May 1992, pp Bennett: “Quantum cryptography using any two nonorthogonal states”, Physical Review Letters, Vol. 68, No. 21, 25 May 1992, pp Charles H. Bennett an IBM Fellow at IBM Research Gilles Brassard Canada Research Chair in Quantum Information processing
Properties of Quantum Information Quantum “no-cloning” theorem: an unknown quantum state cannot be cloned. Measurement generally disturbs a quantum state one can set up a rectilinear measurement or a circular (diagonal) measurement a circular (diagonal) measurement disturbs the states of those diagonal photons having 0/90
Properties of Quantum Information
BB84 Alice transmits short bursts. The polarization in each burst is randomly modulated to one of four states (horizontal, vertical, left-circular, or right-circular). Alice transmits short bursts. The polarization in each burst is randomly modulated to one of four states (horizontal, vertical, left-circular, or right-circular). Bob measures photon polarizations in a random sequence of bases (rectilinear or diagonal). Bob measures photon polarizations in a random sequence of bases (rectilinear or diagonal). Bob tells the sender publicly what sequence of bases were used. Bob tells the sender publicly what sequence of bases were used. Alice tells the receiver publicly which bases were correctly chosen. Alice tells the receiver publicly which bases were correctly chosen. Alice and Bob discard all observations not from these correctly-chosen bases. Alice and Bob discard all observations not from these correctly-chosen bases. The observations are interpreted using a binary scheme: left-circular or horizontal is 0, and right- circular or vertical is 1. The observations are interpreted using a binary scheme: left-circular or horizontal is 0, and right- circular or vertical is 1.
BB84 representing the types of photon measurements: representing the types of photon measurements: + rectilinear O circular representing the polarizations themselves: representing the polarizations themselves: < left-circular > right-circular | vertical − horizontal Probability that Bob's detector fails to detect the photon at all = 0.5. Probability that Bob's detector fails to detect the photon at all = 0.5. Reference:
BB84 – No Eavesdropping A B: | >−<>||−− >−<>||−−< Bob randomly decides detector: Bob randomly decides detector: ++++O+O+OO+O+++++O+O ++++O+O+OO+O+++++O+O For each measurement, P(failure to detect photon) = 0.5 For each measurement, P(failure to detect photon) = 0.5 The results of Bob's measurements are: The results of Bob's measurements are: − >− − − −<< ||| B A: types of detectors used and successfully made (but not the measurements themselves): B A: types of detectors used and successfully made (but not the measurements themselves): + O+ +OO O+ +OO +++ Alice tells Bob which measurements were of the correct type: Alice tells Bob which measurements were of the correct type: − − < | ( key = ) − − < | ( key = ) Bob only makes the same kind of measurement as Alice about half the time. Given that the P(B detector fails) = 0.5, you would expect about 5 out of 20 usable shared digits to remain. In fact, this time there were 4 usable digits generated. Bob only makes the same kind of measurement as Alice about half the time. Given that the P(B detector fails) = 0.5, you would expect about 5 out of 20 usable shared digits to remain. In fact, this time there were 4 usable digits generated.
BB84 – With Eavesdropping A B: − − <−<|<−|−< Eavesdropping occurs. Eavesdropping occurs. To detect eavesdropping: Bob only makes the same kind of measurement as Alice about half the time. Given that the P(B detector fails) = 0.5, you would expect about 5 out of 20 usable shared digits to remain. Bob only makes the same kind of measurement as Alice about half the time. Given that the P(B detector fails) = 0.5, you would expect about 5 out of 20 usable shared digits to remain. A B: reveals 50% (randomly) of the shared digits. A B: reveals 50% (randomly) of the shared digits. B A: reveals his corresponding check digits. B A: reveals his corresponding check digits. If > 25% of the check digits are wrong, Alice and Bob know that somebody (Eve) was listening to their exchange. If > 25% of the check digits are wrong, Alice and Bob know that somebody (Eve) was listening to their exchange. NOTE – 20 photons doesn’t provide good guarantees of detection. NOTE – 20 photons doesn’t provide good guarantees of detection.
DARPA Quantum Network
Pros & Cons Nearly Impossible to steal Nearly Impossible to steal Detect if someone is listening Detect if someone is listening “Secure” “Secure” Distance Limitations Distance Limitations Availability Availability vulnerable to DOS vulnerable to DOS keys can’t keep up with plaintext keys can’t keep up with plaintext
Quantum cryptology
Key distribution Alice and Bob first agree on two representations for ones and zeroes Alice and Bob first agree on two representations for ones and zeroes One for each basis used, { , } and { , }. One for each basis used, { , } and { , }. This agreement can be done in public This agreement can be done in public Define 1 = 0 = 1 = 0 = Define 1 = 0 = 1 = 0 =
Key distribution - BB84 1. 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. 2. Bob measures the state of the photons he receives, with each state measured with respect to randomly chosen basis. 3. 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 Eve has to randomly select basis for her measurement Her basis will be wrong in 50% of the time. Her basis will be wrong in 50% of the time. Whatever basis Eve chose she will measure 1 or 0 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 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. 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.
Eavesdropping Alice’s basis Alice’s bit Alice’s photon Eve’s basis Correct Eve’s photon Eve’s bit Correct {,}{,}{,}{,}1 {,}{,}{,}{,}Yes1Yes { , } No1Yes 0No 0 {,}{,}{,}{,}Yes0Yes No1No 0Yes 1 {,}{,}{,}{,}No1Yes 0No Yes1Yes 0 {,}{,}{,}{,}No1No 0Yes yes0Yes
Eves problem Eve has to re-send all the photons to Bob 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 Will introduce an error, since Eve don't know the correct basis used by Alice Bob will detect an increased error rate 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. 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 When Alice and Bob need to test for eavesdropping By randomly selecting a number of bits from the key and compute its error rate By randomly selecting a number of bits from the key and compute its error rate Error rate < E max assume no eavesdropping Error rate < E max assume no eavesdropping Error rate > E max assume eavesdropping (or the channel is unexpectedly noisy) Alice and Bob should then discard the whole key and start over Error rate > E max 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 Noise might introduce errors A detector might detect a photon even though there are no photons A detector might detect a photon even though there are no photons Solution: Solution: send the photons according to a time schedule. 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. 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. There also has to be some kind of error correction in the over all process.
Error correction Suggested by Hoi-Kwong Lo. (Shortened version) Suggested by Hoi-Kwong Lo. (Shortened version) 1. Alice and Bob agree on a random permutation of the bits in the key 2. They split the key into blocks of length k 3. 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 4. This is repeated with different permutations and block size, until Alice and Bob fail to find any disagreement in many subsequent comparisons
Privacy amplification Eve might have partial knowledge of the key. Eve might have partial knowledge of the key. Transform the key into a shorter but secure key Transform the key into a shorter but secure key Suppose there are n bits in the key and Eve has knowledge of m bits. Suppose there are n bits in the key and Eve has knowledge of m bits. Randomly chose a hash function where h(x): {0,1\} n {0,1\} n-m-s Randomly chose a hash function where h(x): {0,1\} n {0,1\} n-m-s Reduces Eve's knowledge of the key to 2 –s / ln2 bits Reduces Eve's knowledge of the key to 2 –s / ln2 bits
Encryption Key of same size as the plaintext Key of same size as the plaintext Used as a one-time-pad Used as a one-time-pad Ensures the crypto text to be absolutely unbreakable Ensures the crypto text to be absolutely unbreakable
What to come Theory for quantum cryptography already well developed Theory for quantum cryptography already well developed Problems: Problems: quantum cryptography machine vulnerable to noise quantum cryptography machine vulnerable to noise photons cannot travel long distances without being absorbed photons cannot travel long distances without being absorbed
Summary The ability to detect eavesdropping ensures secure exchange of the key The ability to detect eavesdropping ensures secure exchange of the key The use of one-time-pads ensures security The use of one-time-pads ensures security Equipment can only be used over short distances Equipment can only be used over short distances Equipment is complex and expensive Equipment is complex and expensive