 # Matthew Guidry. The Fundamentals of Cryptography  One of the fundamentals of cryptography is that keys selected for various protocols that are computationally.

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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   Quantum Information: Joining the Foundations of Physics and Computer Science   Internet Article: http://arstechnica.com/science/guides/2010/01/a- tale-of-two-qubits-how-quantum-computers-work.ars. by Joseph B. Altepeter, 2010   Marco A. Barreno. “The Future of Cryptography Under Quantum Computers”. Dartmouth College Computer Science Technical Report. 2002   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   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.   Eleanor Rie_el and Wolfgang Polak. “An Introduction to Quantum Computing for Non-Physicists”. arXiv:quant-ph/9809016, 1998.

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