Quantum computING & CRYPTOLOGY

1. S. Aras Kubilay CS 532 Network Security Quantum computING & CRYPTOLOGY

2. Roadmap • Introduction • Quantum Computers • Quantum Computers & Cryptology • Closing Comments

3. Introduction • What is quantum computing? • Collective name for storing, representing and manipulating data in a “quantum computer”.. • .. which is essentially still a hypothetical device on par with teleportation and laser beam weapons.

4. Introduction • With one crucial difference: • Various scientific, governmental and military institutions worldwide are actually funding billions of dollars for making quantum computers a reality. • But why?

5. Quantum Computers • Research suggests that quantum computers are likely to be much faster than any other computational model put forth so far. • That includes, besides traditional transistor-based Von Neumann architecture, experimental designs such as optical and biological computers. • Exponentially faster in some cases.

6. Quantum Computers • Quantum computers work on an atomic level • That is roughly 200 times smaller than Intel’s brand new 45nm architecture. • Furthermore, quantum computers are based on “quantum binary digits” (qubits) just as traditional computers are based on bits. • Qubits have some fundementally unique properties.

7. Quantum Computers • A qubit is essentially an atom showing quantum-mechanical behaviour. • Just as a regular bit, qubits are also used to represent 1 / 0 values, usually denominated by the up-spin or down-spin of the atom. • Spin: An integral quality of all elemental particles and related to orbital angular momentum. • Let’s suffice to say that it exists and is either up or down :)

8. Quantum Computers • However unlike bits, qubits benefit from quantum superposition. • A bit in classical mechanics has some exact probability (usually 0.5) to have either 0 or 1 value. • A qubit in quantum mechanics has a probability distribution function of having any of those values at any given time. • Thus a pair of qubits can have 4 superpositional states while three qubits can have 8 states and so on.

9. Quantum Computers • The fundamental difference about all this is: • At any given time: • n bits can be in one of the 2^n states. • n qubits can be in up to 2^n states simultaneously. • Suggests an incredible potential in parallel computing power.

10. Quantum Computers • As if all that weren’t enough, there is yet another advantage of qubits over bits. • Some qubit pairs may be in quantum entanglement, which is a phenomenon that links the quantum states of two spatially seperated particles. • Which is to say that we can modify or read two qubits in a single action without ever touching the second one.

11. Quantum Computers • So in the end we have computers that are smaller, inherently parallel and distributed. • Due to size of atoms, quantum superpositional states and quantum entanglement, respectively. • However there are still limits to the capabilities of quantum computers, thankfully for us computer security people :) • Let’s see them.

12. Quantum Computers & Cryptology • One famous idea about quantum computers & cryptology: “If realized, a quantum computer can simply try all possible key combinations in parallel and crack any key of infinite size in one single stroke through brute-force.” • True or false?

13. Quantum Computers & Cryptology • Let’s see what quantum computers can do. • Shor’s Algorithm: • While the exact specifications are way out of scope, we will see a simplified overview. • Problem definition: For a non-prime positive integer N, find an integer p that divides N and 1 < p < N. • Sounds familiar?

14. Quantum Computers & Cryptology • Shor’s Algoritm (cont’d) • Pick a random number i < N. • Compute gcd(i, N) through traditional methods. • Euclidean Algorithm etc. • If gcd(i, N) = 1 stop, otherwise: • Use quantum computing to find period r such that f(x) = a^x mod N, and f(x + r) = f(x) • Quantum superposition for efficient calculation. • gcd(i^(r/2) +/- 1, N) is a factor of N. • If r is odd or i^(r/2) ≡ -1 (mod N), restart with different i.

15. Quantum Computers & Cryptology • Shor’s Algorithm, although still mostly academical, was later improved by other researchers. • Up to 8 times faster (David M., Queensland) • The profound meaning of this algorithm is that quantum computers are exponentially better at the factorization problem, rendering brute-force solutions feasible. • O((log N)^3) vs classical O(2^((log N)^1/3)). • So what if factorization can be done in polynomial time?

16. Quantum Computers & Cryptology • The security of public-key cryptograhpy methods (most notably RSA) depend on the infeasibilty of the factorization problem. • In RSA, it is impractically difficult to try and find the prime factors p and q for N. • However, a powerful enough quantum computer can factorize and thereby crack any RSA implementation. • Would increasing key size solve this problem?

17. Quantum Computers & Cryptology • What about “good old” symmetric key cryptography? • We have established that quantum computers cannot instantly try infinite key possibilities, but can they exhaust practical key-size spaces in a reasonable time? • A classical brute-force attack against a symmetric crypto key is O(2^N) for N-bit keys. • So a 256-bit key is reasonable secure while 1024-bit is pretty solid.

18. Quantum Computers & Cryptology • However, if quantum computers can somehow do it in polynomial time, it again becomes a futile race of key size vs. computer power like in PKC. • This was a major concern for the industry, so researchers from IBM and Microsoft together with Berkeley and Montreal Universities have conducted an in-depth research based on quantum Turing machines*. • You may remember from some other courses that Turing machines are equivalents of any computer in terms of computational capabilities. *: Bennett, Bernstein, Brassard, Vazirani. Strength and Weaknesses of Quantum Computing. (1996)

19. Quantum Computers & Cryptology • This joint research revealed that.. • .. a brute force quantum attack against symmetric cryptosystems is bound by O(2^(N/2)). • Later work on Grover’s search algorithm, which is proven to be optimal, has confirmed this finding, with some very specific cases showing quadratic performance gain. • Since there is no exponential gain, keys can be easily guarded against quantum brute-force attacks by simply doubling the key size.

20. Closing Comments • Quantum computing is not the panacea it’s sometimes made to look like. • However, it possesses unique properties and therefore challanges some of the established security measures, PKC chief among them. • Symmetric key systems are likely to hold their own agaisnt quantum cryptanalytic attacks.

21. Closing Comments • Although practicle quantum computers are probably decades away, especially short-term precautions must be taken while long-term methodologies develop. • Studies show promise with one-time algortihms with doubled key sizes. • Such as Lamport digital signatures.

22. Closing Comments • Any questions? • (No theoretical physics, please :) ) • References • Bennett, Bernstein, Brassard, Vazirani. Strength and Weaknesses of Quantum Computing. (1996) • David McMahon. Quantum Computing Explained. (2007) • Nakahara, Ohmi. Quantum Computing: From Linear Algebra to Physical Realizations. (2008) • And of course, Wikipedia.