QUANTUM COMPUTERS AND CRYPTOGRAPHY

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Nov 21, 2013 (3 years and 6 months ago)

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QUANTUM COMPUTERS AND CRYPTOGRAPHY

Matthew Guidry

The Fundamentals of
Cryptography


One of the fundamentals of cryptography is
that keys selected for various protocols that
are computationally infeasible for an attacker
to compute given the same public
information.



Consider for example the RSA Assumption

The RSA Assumption


the RSA assumption states that given :


a large number
n

=
p*q


p and q are primes


e

such that
GCD(e,
Φ
(n)) = 1


ciphertext
C


It is computationally infeasible to compute


the original message
M

such that


C = M
e
mod N

Computational Infeasibility


Many cryptographic protocols rest on the
assumption that secret keys are
computationally infeasible to compute.



However, the Quantum Computer may be
able to increase the power of current
computing methods exponentially. This
exponential increase would actually make
these problems feasible.

The Effects of Moore’s Law


Computers have become more and more
powerful following Moore’s Law, which states




Every 18 months the number of transistors which can
be fit within one square inch doubles.



If this trend continues unabated, by 2015
transistors will roughly be the size of single
atoms and molecules. At this size the laws of
physics which governed classic computers give
way to the laws of quantum mechanics.

The Basics of a Quantum Computer


A current computer has bits which represent
0 and 1 based on electrical signals.



In a Quantum Computer these could be
replicated by atoms in the excited or
grounded state. However, given the multiple
properties of quantum mechanics it would
allow that other states to be inferred at the
same time.

The Qubit


The basic building block of a Quantum
Computer is the qubit



quantum
” + “
bit
” =
qubit



Classical bits and quantum bits share the
same property, once measured they will only
reveal one of two possible outcomes.

The Qubit


The difference between qubits and normal
bits is not in the possible
answers

inferred
from the states, it is in the possible number
questions

that can be asked of them



Qubits exhibit two very special properties of
superposition

and
quantum entanglement

Superposition


Measuring a qubit which is in a superposition
forces a collapse of the wave function thus
putting the qubit back into a single state as a
result of the measurement.



Before measuring that qubit it can be seen as
being in many different states. The
explanation is difficult to explain, but
consider the qubit to contain many
answers

it
just depends on which
question

is asked.

Superposition


A simple but fitting explanation of the qubit:



Consider the effect of polarization in
sunglasses:


Superposition


Now imagine a combination


of the two.











Or perhaps 3
-
D?




Superposition


These different orientations can be observed
based on the spin
-
up or spin
-
down,
horizontal or vertical representation, and
other properties that the ions would exhibit.


The important fact to the Computer Scientist
:

The representation.


The state of a qubit alone can be thought of as a unit vector in a two
-
dimensional vector space with ba.sis
{ |0>, |1> }
.
Here
|0>
and
|1>

are
orthogonal vectors
.

Superposition


More on the representation:


The qubit may be in a superposition
x|0> + y|1>

of
the two states. The complex amplitudes
x

and
y

determine which state we will see if we make a
measurement. When an observer measures a qubit
in this superposition, the probability that the
observer will see state

|0>

is
|x|
2

and the probability
of seeing
|1>

is

|y|
2
. Note that because
x|0> + y|1>

is
a unit vector, the sum
|x|
2

+ |y|
2

must be equal to
1

Quantum Entanglement


The property of quantum entanglement is
unique to qubits



Two qubits that are passed along in a system
will have an effect on each other’s respective
states



The state of this system is no longer a
Cartesian product of the individual spaces,
but now a Tensor Product of the spaces.

Quantum Entanglement


This implies that the number of dimensions in
the combined space is the product rather
than the sum of the numbers of dimensions
in each of the component space.



The more qubits which are used within a
system, the more states that system could
have and the number of states possible would
grow
exponentially
.


Superposition and Quantum Entanglement


It is mostly through the attractiveness of
these two properties that quantum
computers hold such promising prospect.



Further because a qubit or a system of qubits
can be in a superposition of states, an
operator applied to such a system can
operate on all the states simultaneously

Quantum Computing
vs

Cryptography


Most cryptographic methods such as the
Discrete Logarithm problem rely on the
computation infeasibility of the problem


Consider
Shor’s

1994 Algorithms:


Peter
Shor

created an algorithm to factoring n
-
digit
numbers in bounded
-
probability polynomial time on
a quantum computer and another to compute
discreet logarithms quickly


This algorithm sparked most of the current
interest in Quantum Computers in

Possible Cryptographic Defenses in the
Age of the Quantum Computer


It is important to note that the full potential
of quantum computers is not actually known!
(at least not publically)



The possible methods that could be
implemented with these fantastic machines
will remain just that, “possible”



Possible Cryptographic Defenses in the
Age of the Quantum Computer


Since the qubits are in a superposition of
answers, a measure of the result will not
always give the desired answer.



It is possible that the probability for getting
the correct answer is much lower than
imagined and Quantum Computers are little
better than today’s computers.

Possible Cryptographic Defenses in the
Age of the Quantum Computer


It is probable that most of the cryptographic
functions used for security would have to be
strengthened, at the very least .



Others may have to be completely abandoned.



However, with these new computing powers at
their disposal it is also possible that
cryptographers will develop new methods


as the older ones are being broken.


The Current State of the Quantum
Computer??


As for the current state of this future computer?’



The manipulation of the atoms would be done using an
ion trap, and scientists have thus far been able to trap a
single atom; however, the biggest challenge lies in being
able to orchestrate the millions of atoms needed to run a
quantum computer



Currently Scientists have been able to create a machine
with a couple qubits inside of it, however, these are just a
small scale of what is foreseen to come.

Questions??


These new computers offer many exciting
possibilities and it will be interesting to see if
they pan out to be as fruitful as has been
promised…

Sources


[1] Quantum Information: Joining the Foundations of Physics and
Computer Science



[2] Internet Article: http://arstechnica.com/science/guides/2010/01/a
-
tale
-
of
-
two
-
qubits
-
how
-
quantum
-
computers
-
work.ars . by Joseph B.
Altepeter
, 2010


[3] Marco A.
Barreno
. “The Future of Cryptography Under Quantum
Computers”. Dartmouth College Computer Science Technical Report.
2002


[4] Ion trap in a Semiconductor Chip, D. Stick, W. K.
Hensinger
, S.
Olmschenk
, M. J. Madsen, K. Schwab and C. Monroe,

Nature Physics

advance online publication, 2005


[5] Peter W.
Shor
. “Algorithms for quantum computation: Discrete
logarithms and factoring”. In Proceedings of the 35th Annual IEEE
Symposium on Foundations of Computer Science, pages 124
-
134. IEEE
Computer Society Press, 1994.


[6] Eleanor
Rie_el

and Wolfgang
Polak
. “An Introduction to Quantum
Computing for Non
-
Physicists”.
arXiv:quant
-
ph/9809016, 1998.