Quantum Quantum Cryptography, Or How Alice Outwits

Quantum Crypto ?? Quantum Cryptography or How Alice Outwits Eve Cani Samuel J. Lomonaco, Jr. Dept. of Comp. Sci. & Electrical Engineering University of Maryland Baltimore County Baltimore, MD 21250 Email: [email protected] WebPage: http://www.csee.umbc.edu/~lomonaco L-O-O-P This work is supported by: Introducing Alice & Bob ! • The Defense Advance Research Projects ob Alice hr Bob Agency (DARPA) & Air Force Research T Laboratory (AFRL), Air Force Materiel Command, USAF Agreement Number F30602-01-2-0522. • The National Institute for Standards and Technology (NIST) Sender Receiver • The Mathematical Sciences Research Eve Bah ! Institute (MSRI). Humbug ! • L-O-O-P The L-O-O-P Fund. • The Institute of Scientific Interchange Eavesdropper Introducing Alice & Bob Key Idea b ! Alice ro RobertoBob Th Quantum cryptography provides a new mechanism enabling the parties communicating with one another to: Sender Receiver Automatically detect eavesdropping. EvePia Bah ! Consequently, it provides a means of Humbug ! Spia determining when an encrypted communication has been compromised. Eavesdropper 1 The Dilemma Alice Takes a Cryptography Course How can I outwit Eve How do I prevent ??? Eve from eavesdropping ??? ? ? ? ? ? ? Alice Alice Classical Shannon The Bit Classical Decisive World Individual 0 or 1 Classical Bits Can Be Copied In Out Cryptographic Copying Machine Systems 2 A Classical Cryptographic Communication System Catch 22 Eavesdropper Il cane che si morde la coda Eve Transmitter Receiver Alice Bob Info Insecure Info Encrypter Decrypter Source Channel Sink There are perfectly good ways to Key communicate in secret provided P = Plaintext C = Ciphertext P = Plaintext we can already communicate in secret … Secure Channel Classical Crypto Systems Types of Communication Security • Practical Secrecy (Circa 106 BC) Ciphertext breakable after x years CHECK LIST Examples: Data Encryption Standard (DES), Catch 22 Solved ? NO Examples: Data Encryption Standard (DES), • Advanced Data Encryption Standard (AES) • Authentication ? NO • Eavesdropping Detection ? NO • Perfect Security (Shannon, 1949) Ciphertext C without key gives no information about plaintext P Prob() P|C= Prob ()P An Example of Perfect Security Difficulties The Vernam Cipher, a.k.a., the One-Time-Pad Consider a random sequence of bits • PROBLEM: Long random bit sequences == must be sent over a secure channel Key K K12 K K n Encrypting algorithm • CATCH 22: There are perfectly good ways =+ CPKii imod 2 to communicate in secret provided we can communicate in secret … P = 0110 0101 1101 K = 1010 1110 0100 KEY PROBLEM in CRYPTOGRAPHY: ⊕= • C 1100 1011 1001 We need some way to securely communicate key • Perfectly secure if key K is unknown • Easy to decode with Key = K 3 Objective of All Crypto Systems: Safety Objective of All Crypto Systems: Safety Old Idea: New Idea: Unconditional Security Computational Security The crypto system can resist any cryptanalitic attack no matter The crypto system is unbreakable how much computation is involved. because of the computational cost of cryptanalysis, but would succumb to an attack with unlimited computation. Computational Security Computational Security (Diffie-Hellman, circa 1970) For example, the crypto system: For example, the crypto system: Public Key Crypto Systems … Example: RSA 30 • Requires 10 years to be broken on the Public Phone • E fastest known computer EB Directory C 100 C • Or, requires 10 bits of memory to break Insecure Channel • Or, requires 1030 euros to break • Encrypter Decrypter D C DB P P System computationally safe implies safe for Eavesdropper all practical purposes Info Eve Info Source Sink Idea comes from a field in computer science called Computational Complexity. Transmitter Receiver Alice Bob Public Key Crypto Systems Alice Takes a Quantum Mechanics Course CHECK LIST • Catch 22 Solved ? Yes & No •Authentication ? Yes •Eavesdropping Detection ? No Alice 4 Introducing the Quantum Bit … The Qubit The Look Here Indecisive Quantum Individual World Can be both 0 & 1 at the same time ! Quantum Representations of Qubits Quantum Representations of Qubits 1 Example 1. A spin- particle Example 2. The polarization state of a photon 2 Vertical Horizontal Polarization Polarization 1 = 0 =↔ Spin Up Spin Down 1 0 1 0 Where does a Qubit live ? H = Def. A Hilbert Space is a vector space H over together with an inner Home product −− ,: HH × → such that A Qubit is a quantum += + += + system whose state is 1) uuv12,,, uvuv 1 2 & vu ,,, 12 u vu 1 vu 2 2) uv,,λλ= uv represented by a Ket ∗ 3) uv,,= vu lying in a 2-D Hilbert ∈ 4) ∀ Cauchy sequu,,… in H , lim un H 12 n→∞ Space H The elements of H will be called kets, and will be denoted by label 5 Superposition of States “Collapse” of the Wave Function Qubit A typical Qubit is ??? αα01+= e 01 iv is c =+αα !! e 0101 ! Observer d h n s I o o 22 h αα+= W where 011 2 b | o i The above Qubit is in a Superposition of states r a P | 0 and 1 = i It is simultaneously both 0 and 1 !!! Another Activity in Quantum Village: Measurement MeasurementMeasurement Measurement Connecting Quantum Village to the Classical World Group of Friendly Physicists Another Activity in Quantum Village: Observables ??? Measurement Measurement What does our observer Measurement actually observe ? Observables = Hermitian Operators O A HH→ where T = Group of Angry Physicists OOA A 6 ??? Observables (Cont.) ??? Observables (Cont.) ??? What does our observer observe ? What does our observer actually observe ? The state of an n-Qubit register can be written in the eigenket basis as Ψ= α ϕ ϕ ∑ i ii Let i be the eigenkets of O A , and let a i denote the corresponding eigenvalues , i.e., = α 2 So with probability p ii , the observer ϕϕ= a O Aia ii observes the eigenvalue i , and !! h os Caveat: We only consider observables whose ho eigenkets form an orthonormal basis of H ϕ W i Measurement Revisited Important Feature of Observable MacroWorld Eigenvalue Observable MacroWorld λ Quantum Mechanics j O It is important to mention that: Physical =ψψ Out Prob Pj In Reality Philosopher We cannot completely P ψ Turf ψ = j ψ j ψψP control the outcome of BlackBox j quantum measurement Q. Sys. Quantum Q. Sys. State World State O = λ P where ∑ j jjSpectral Decomposition More Dirac Notation Hilbert Space of morphisms from H to * More Let HH= Hom(), Dirac * We call the elements of H Bra’s, and Notation denote them as label 7 More Dirac Notation Dirac Notation (Cont.) * There is a dual correspondence between and H H • Consider a Quantum System in the state † B t r ψ Ket Ke ψψ↔ a * • Suppose we measure many of these Hermitian There exists a bilinear map HH×→ • states with the observable A Operator defined by ()ψψ()∈ 12 • Then the average value of all these which we more simpy denote by measurements w.r.t. A is: Avg. ψψ| of A 12 ψψ()AAA== ψψ|| BraBra-c-KetKet Heisenberg’s Uncertainty Principle = 1 The No-Cloning Theorem Definition. Observables A and B are compatible if []AB,0=−= AB BA Dieks, Wootters, Zurek Otherwise, A and B are incompatible. ∆= − Let AA A In Out Copying Heisenberg’s Uncertainty Principle Machine 221 2 ()∆∆≥AB() [] AB, 4 ()∆ 2 A is the Standard Deviation. It is a measure of the uncertainty of the observable A . Particle vs Wave Picture of Matter Young’s 2-slit Experiment An Example of E E Heisenberg’s E E B L Uncertainty O E C Principle K 8 Particle vs Wave Picture of Matter Particle vs Wave Picture of Matter Young’s 2-slit Experiment Young’s 2-slit Experiment B E L O C E K E E E Particle not observed An interferenceBut a wave patternobserved appears Particle vs Wave Picture of Matter Application of Heisenberg’s Uncetainty Principle Young’s 2-slit Experiment Observables X Position Operator = 1 O P Momentum Operator b se er ve Note: X and P are incompatible observables; for: []XP,0=− i ≠ Therefore, by Heisenberg’s Uncertainty Principle: 2211 ()∆∆≥XP() [] XP, = 44 What happens if we observe which of the two slits each electron passes ? Uncertainty Uncertainty the two slits each electron passes ? in Position in Momentum The interference pattern disappears !! Ergo, to know precisely which of the two slits the Wave not observed; electron passed through, forces the momentum to be But a particle is observed ! uncertain Alice Daydreams Alice Has an Idea How can I But How ??? outwit Eve How do I prevent ??? Idea: Couldn’t I somehow Eve from use Heisenberg’s Uncertainty Principle to eavesdropping ??? detect Eve’s eavesdropping ? ? ??? ? ? ? ? Alice Alice 9 Alice Bob Bob Alice Bob Eve What if the evil Eve tries to listen in ??? What if I use the the electron gun to send Bob a message, i.e., an interference pattern ??? Aha! Bob knows the evil Eve is listening in !!! A Quantum Crypto System for the Alice Invents the BB84 BB84 Protocol Quantum Crypto Protocol Two-Way Communication Public Channel BB84 = Bennett-Brasard 1984 Second Stage Second Stage Alice Eve Bob First Stage First Stage Quantum Channel One-Way Communication Two Bases of 2-D Hilbert Space H The Quantum Channel • The vertical and horizontal polarization states • Alice will communicate over the quantum and ↔ channel by sending 0’s and 1’s, each encoded form a basis of H which we will call the as a quantum polarization state of an individual vertical/horizontal (V/H) basis photon. • Reminder: We note that the polarization state of an individual photon is an element • The slanted polarization states ψ of a 2-D Hilbert space H . and also form a basis of H which we will call the oblique basis 10 Quantum Channel Encoding Conventions Using Heisenberg’s Uncertainty Principle • For the V/H basis , Alice & Bob agree to communicate via the following quantum alphabet • Because of Heisenberg’s uncertainty principle, "1" = Alice & Bob know that observations with respect to the basis are incompatible with "0"=↔ observations with respect to the basis.

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