COMPSCI 290.2: Computer Security
“Quantum Cryptography”
including
Quantum Communication
Quantum Computing
Page 1
Quantum Communication
NOT used to encrypt data!
Goal, instead, is to detect eavesdroppers
Can be used to exchange a private key
Page 2
Uncertainty Principle
In quantum mechanics, certain pairs of properties of
particles cannot both be known simultaneously, e.g.,
– Position and momentum of an electron
(Heisenberg)
If a measurement determines (with precision) the
value of one of the properties, then the value of
the other cannot be known
Page 3
Photon Spin (Polarization)
Photons can be given either “rectilinear’’ or “diagonal’’
spin as they travel down a fiber.
Rectilinear:
or
Diagonal:
or
Measuring rectilinear spin with a rectilinear filter
yields polarization of photon.
blocked
Page 4
What if the wrong filter is used?
or
(blocked)
(equal probability)
Page 5
Quantum Key Exchange
The goal is to enable Alice and Bob to agree on a
private key, even in the face of an eavesdropper,
Eve.
Like Diffie-Hellman, the protocol is still susceptible
to a “man-in-the-middle” attack.
But unlike Diffie-Hellman, the protocol does not
depend on the difficulty of computing discrete
logarithms or any other computational problem.
Page 6
BB84 Protocol (Bennet and Brassard)
1. Alice sends Bob a stream of photons randomly polarized
in one of 4 polarizations:
bit encoding: 0 1
0 1
2. Bob measures the photons in random orientations
e.g.:
x + + x x x + x (orientations used)
\ | - \ / / - \ (measured polarizations)
1 0 1 1 0 0 1 1 (encoded bit values)
3. Bob tells Alice in the open what orientations he used,
but not what bit values he measured
4. Alice sends Bob in the open a list of positions at which
the orientations are correct
Page 7
Detecting an Eavesdropper
Alice selects some subset of k of the shared bits
and reveals them to Bob in the open.
If Bob notices any differences, then Eve must have
changed a bit by guessing the wrong polarization
when eavesdropping.
Eve has little hope of guessing the same polarization
as Bob all k times. Each measurement has a ¼
chance of changing a bit value. The probability of
not changing any values is (3/4)k – which can be
very small if k is chosen large enough
Page 8
In the “real world”
In April 2014 China began installing a 2000kilometer quantum communications link between
Beijing and Shanghai
Page 9
Quantum Computers
The state of a computer consists of the contents of
its memory and storage, including values of
registers (including the program counter), memory,
disk contents, etc.
In a conventional computer each memory “unit” holds
one value (e.g., 0 or 1) at a time. Computation
consists of a sequence of state transitions.
But in a quantum computer, a memory unit holds a
“superposituion” of possible values.
Page 10
Qubit (somewhat simplified)
A single quantum “bit” which is 1 with probability p
and 0 with probability 1-p.
When measured, outcome is either 0 or 1.
Measuring a qubit changes its value! If outcome is 0,
p is set to 0, if outcome is 1, p is set to 1.
A qubit could be implemented using a photon to carry
a horizontal or vertical polarization.
Page 11
Quantum Entanglement
Suppose two bits have value 00 with probability ½
and 11 with probability ½. If the bits are
separated and measured at different locations,
the measurements must yield the same values.
E.g., if first measurement is 0, second must also
be 0.
Entanglement also allows multiple states (e.g., 00 vs.
11) to be acted on simultaneously.
Difficulty in building a quantum computer is
maintaining quantum entanglement in the face of
environmental noise (quantum decoherence).
Page 12
More on Qubits
A qubit is a superposition of two basis states, and
, which can be thought of as north and south poles
of a unit sphere.
https://commons.wikimedia.org/wiki/File:Bloch_sphere.svg
I.e., qubit is
, where v0 and v1 are complex
numbers such that |v0|2 + |v1|2 = 1. (|v0|2 and |v1|2
are probabilities of qubit being 0 or 1)
can be written as
Page 13
Quantum Gates
Qubits are manipulated with quantum logic gates.
Gates are just multiplications by unitary matrices.
Hadamard matrix
like H, e.g.,
maps
to
and
to
Page 14
Factoring Large Primes
In 1994 Peter Shor showed that a quantum computer
can factor a number n in O(log3 n) time.
A similar result holds for solving the discrete
logarithm problem.
If a large-enough quantum computer can be built,
then RSA and Diffie-Hellman key-exchange will no
longer be secure.
(But largest number factored with this algorithm as
of 2015 was 21!)
Page 15
Details of Shor’s Algorithm
1. Pick a random number 1 < a < n
2. If a is a factor of n, what a lucky guess!
3. Use a quantum circuit to find smallest r > 1 such
that ar = 1 mod n
4. If r is odd or ar/2 = -1 mod n, go back to step 1
5. GCD(ar/2 + 1, n) and GCD(ar/2 - 1, n) are factors
of n
Example: n = 15, a = 7, r = 4
a1 = 7 mod n, a2 = 4 mod n, a3 = 13 mod n, a4 = 1 mod n
a4/2+1 = 50, a4/2-1 = 48
GCD(50,15) = 5, GCD(48,15) = 3
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Controversial Quantum Computer
D-Wave Systems, Inc., purports to build a quantum
computer based on a 128-qubit chipset.
No convincing demonstration of speed-up over
conventional computer yet.
Unresolved debate about whether there is actually
quantum entanglement among the qubits.
(Evidence seems to be leaning towards yes?)
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