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

Roadmap Introduction Quantum Computers Quantum Computers & Cryptology Closing Comments

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.

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?

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.

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.

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 :)

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.

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.

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.

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.

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?

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?

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.

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?

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?

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.

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)

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.

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.

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.

Closing Comments Any questions? References (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.