Dominique Unruh 3 September 2012

Quantum Cryptography

Dominique Unruh

Dominique Unruh

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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Dominique Unruh

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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Dominique Unruh

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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Dominique Unruh

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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Dominique Unruh

Organizatorial

• Questions?

Dominique Unruh

Quantum Mechanics

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Dominique Unruh Quantum Cryptography

Double Slit Experiment

• Light falls through two slits (S2)

• Light-dark pattern occurs

• Reason: Light is a wave → Interference

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Dominique Unruh Quantum Cryptography

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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Dominique Unruh Quantum Cryptography

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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Dominique Unruh Quantum Cryptography

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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Dominique Unruh

Quantum Computing

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Dominique Unruh

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

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Dominique Unruh

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

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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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Dominique Unruh

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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Dominique Unruh

Today

• No quantum computers (except for toy models)

• Cannot execute quantum algorithms

• Future will tell

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Dominique Unruh

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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Dominique Unruh

Quantum Key Exchange Alice Bob

Polarisation:

Measures

Sends basis

Shared key bits

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Dominique Unruh

Quantum Key Exchange – Attack Alice Bob

Polarisation:

Adversary 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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Dominique Unruh

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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Dominique Unruh

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

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