Dominique Unruh 3 September 2012

Quantum Cryptography

Dominique Unruh

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Organization

• Lecture: Tuesday 10.15am• Practice: Wednesday 10.15am– Problem solving as a group

• (sometimes switched)

• Homework: Due after approx. one week• 50% needed for exam

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Organizatorial

• Black board lecture (except today)• Material:– Board photos– Lecture notes (short)– Book: Nielsen, Chuang, “Quantum Computation

and Quantum Information” (not required)

• Deregistering: Not after deadline

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Scope of the lecture

• No physics (almost)– Do you need electrodynamics to understand

Turing-machines?– Mathematical abstraction of quantum

computation/communication• Intro to Quantum

computation/communication• Selected topics in quantum crypto

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Requirements

• No physics needed• Some crypto background recommended– (To have a context / the big picture)

• Some linear algebra will be used– You should not be afraid of math– Can do recap during tutorial ask!!!

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Organizatorial

• Questions?

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Quantum Mechanics

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Double Slit Experiment

• Light falls through two slits (S2)

• Light-dark pattern occurs

• Reason: Light is a wave → Interference

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Double Slit Experiment

• Send a single photon at a time• Photon either goes through left or right

path• After a while, interference pattern occurs• Each photon “interferes with itself”

→ Physicists puzzled• Solution: Quantum mechanics:– Photon takes both ways in superposition

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Superposition

• If two situations are possible, nature “does not always decide”– Both situations happen “in superposition”– (Doesn’t need to make sense now)

• Only when we look, “nature decides”

• Schrödinger’s cat

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Quantum Mechanics

• Superposition: Several things happen “at once”

• Our intuition is classical, we cannot understand this

• Mathematical notions allow to handle QM, even if we do not understand it

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Quantum Computing

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Church-Turing Thesis

• Turing: Definition of Turing-machines• Church-Turing thesis:

→ Turing-Machine characterises physical computability

Usually: Efficient = polynomial-time

Any physically computable function can be computed by a Turing machine

efficiently

efficient

Strong

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Randomized algorithms

• 1970s: Solovay-Strassen primality test• No deterministic test known (at that time)• Polynomial identity:

No deterministic test today

Any efficiently physically computable function can be computed by an efficient

Turing machine

probabilistic

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Enters: The Quantum Computer

• Strong Church-Turing extended once– Perhaps has to be extended again

• Feynman 1982:– Simulating quantum systems difficult for TMs– Quantum system can simulate quantum system

• Probabilistic Church-Turing thesis wrong?– Unknown so far… But seems so…

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Quantum Algorithms

• Deutsch-Jozsa 1992:– Testing whether function is balanced or constant– No practical relevance– Shows: Quantum Computers more powerful than

classical• Shor 1994:– Factorization of integers

• Grover 1996:– Quadratic speed-up of brute-force search

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Today

• No quantum computers(except for toy models)

• Cannot execute quantum algorithms• Future will tell

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Quantum Cryptography

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Quantum Key Exchange

• Bennet, Brassard 1984:– Key exchange using quantum communication

• Idea:– Measurement destroys state→ Adversary cannot eavesdrop unnoticed

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Quantum Key ExchangeAlice Bob

Polarisation:

Measures

Sends basis

Shared key bits

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Quantum Key Exchange – AttackAlice Bob

Polarisation:

measuresAdversary measures→ Bit destroyed→ Alice+Bob: different keys→ Attack detected

Changed by measurement

Caution: This is only the intuition.Security analysis much more involved.

(Took 12 additional years…)

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Quantum Key Exchange

• Idea proposed 1984• First security proof: Mayers 1996• Possible with today’s technology– Single photon sources– Polarisation filters

• No complexity assumptions– Impossible classically

• Details later in lecture

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Quantum Cryptography

• Any cryptography using quantum– Key exchange– Bit commitment– Oblivious transfer– Zero knowledge– Signatures

• Often: Quantum Crypto = Key Exchange– Other applications often ignored

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End of Intro