Elementary Introduction to
Quantum Cryptography
RJ Irwin
Syracuse University
To be presented…
•
A little Cryptography (Crypto)
•
A little Quantum Mechanics (QM)
•
Application of QM of polarized light to crypto:
–
Bennett/Brassard’s 1984 key distr. protocol (BB84)
•
Mathematics mostly avoided
Basic Crypto Scenario
Alice
Eve
Bob
Eve wants to eavesdrop undetected
Cryptography
•
Goal is to
encode
sender’s message so that
only receiver can
decode
it
•
Encoding/decoding process depends on one
or two
keys
•
Two classical approaches to to cryptography:
–
Symmetric
(or
secret
)
key
(one key)
–
Asymmetric
(or
public
)
key
(two keys)
Symmetric Key Cryptography
•
Alice and Bob use same key
k
for encoding
plaintext
p
into
ciphertext
c
and decoding
c
back into
p
:
E
k
(
p
) =
c
,
D
k
(
c
) =
p
•
If Eve learns
k
, then there is no security
•
How can Alice and Bob share the single key
k
without Eve intercepting it?
Asymmetric Key Cryptography
•
Alice and Bob each have two keys:
–
A
u
,
A
r
(Alice’s public, private keys)
–
B
u
,
B
r
(Bob’s public, private keys)
•
Public keys may be openly communicated
(e.g., they may be posted to a public website)
•
Private keys never shared by their owners
Asymmetric Key Cryptography
•
Sender uses receiver’s public key, receiver
uses own private key:
–
E
Bu
(
p
) =
c
, D
Br
(
c
) =
p
(Alice to Bob)
–
E
Au
(
p
) =
c
, D
Ar
(
c
) =
p
(Bob to Alice)
•
Rivest

Shamir

Adleman (1977) developed the
best

known, most popular public key system
(“RSA”)
Security of Crypto Schemes
•
Symmetric crypto using short keys exposes
ciphertext to
cryptanalytic attack
(e.g., DES)
•
Asymmetric crypto schemes in current use
depend on
unproven
assumptions
–
RSA scheme assumes that prime factorization of
integers is computationally intractable
•
Only
provably secure
scheme is
one

time pad
,
as shown by Claude E. Shannon in the 1940s
One

Time Pad
•
A symmetric key system, also called the
Vernam
cipher
(Gilbert
Vernam
at AT&T, 1918)
•
Key is a stream of random bits as long as the
plaintext to be encoded (key itself called the
“one

time pad” because it
is
used only once
)
•
The big problem, as usual with symmetric
systems, is how to share the key securely
BB84 Quantum Key Distribution
•
Charles Bennett (IBM) and Gilles Brassard
(
U.Montreal
), “Quantum Cryptography: Public
Key Distribution and Coin Tossing”, Int. Conf. on
Computers, Systems & Signal Processing,
Bangalore India, 1984
•
Showed how to use the quantum mechanical
nature of polarized light for distributing a one

time pad with a high degree of security
•
Eavesdropping detectable with high probability
Polarized Light
100% of light
50% of light
50% of light
Diagonal Basis (45º and 135º orthogonal basis members)
Light is composed of photons, 50% light = ½ of photons
Polarized Light
100% of light
50% of light
Diagonal Basis (45º and 135º orthogonal basis members)
no light
Polarized Light
100% of light
50% of light
50% of light
Rectilinear Basis (horiz. and vert. orthogonal basis members)
Polarized Light
Rectilinear Basis (horiz. and vert. orthogonal basis members)
100% of light
50% of light
no light
Polarized Light
Adding another filter can yield counter

intuitive results
100% of light
50% of light
25% of light
12.5% of light
Quantum Mechanics
•
States
,
Operations
and
Measurements
(Observations) as
Vectors in Hilbert Space
,
Unitary Operators
and
Self

adjoint Projection
Operators
, respectively
•
State of an
n

level system is a unit

length
vector in an
n

dimensional Hilbert Space,
H
n
,
called the
state space
Quantum Mechanics
•
Choice of orthonormal basis {
x
1
›,…,
x
n
›}
yields a
physical observable
that can have any
of these
n
values (basis can be chosen freely)
•
In general, a state of
H
n
(when not observed)
is a
superposition
of basis states:
α
1

x
1
›+
. . .
+
α
n

x
n
›
where basis state 
x
i
› is observed with
probability 
α
i

2
and ∑
α
i

2
= 1 (
α
i
complex)
Polarized Light
R basis
D basis
R basis
R = Rectilinear
D = Diagonal
Choice of basis (type of measurement) determines possible outcomes
Thus, observation of state (measurement) can alter state
Heisenberg Uncertainty Principle of Quantum Mechanics
BB84 Quantum Key Distribution
•
Alice and Bob use two channels:
–
A quantum channel
–
A classical channel
•
Quantum channel used first, to transmit a train of
polarized photons; eavesdropping (observing)
disturbs long trains with high probability
•
Classical channel used to complete derivation of
one

time pad from photon train; need not be
secured against
passive
eavesdropping
0
1
1
0
1
1
0
0
1
0
1
1
0
0
1
D
R
D
R
R
R
R
R
D
D
R
D
D
D
R
R
D
D
R
R
D
D
R
D
R
D
D
D
D
R
1
1
1
0
0
0
1
1
1
0
1
Alice’s random bits......
Random
send bases….
Photons Alice
sends….
Random
recv
bases…..
Bits Bob receives……….
BB84 Protocol
R
D
R
D
D
R
R
D
D
D
R
x
x
x
x
x
x
1
1
0
1
0
1
1
0
x
x
Bob
reports bases……..
䅬Ac攺
correct bases…..
Presumed
s
hared
info
Bob
shows some bits..
Alice
confirms them….
QUANTUM
CLASSICAL
1
0
1
1
ONE

TIME
PAD
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