Quantum Key Distribution (BB84)

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BB84 QKD cryptography eavesdropping information-theoretic-security

Core Idea

Quantum key distribution (QKD) enables two parties to establish a shared secret key whose security is guaranteed by the laws of quantum mechanics, not computational hardness assumptions. The BB84 protocol (Bennett-Brassard 1984) works by encoding random bits in one of two conjugate bases (rectilinear or diagonal). An eavesdropper measuring the qubits inevitably disturbs them (by the no-cloning theorem and measurement disturbance), introducing detectable errors. After basis reconciliation and error estimation, the parties distill a secure key. BB84 achieves information-theoretic security — it is provably secure even against adversaries with unlimited computational power, including quantum computers.

Explainer

Classical cryptography faces a fundamental problem: the security of widely used public-key systems (RSA, elliptic curves) rests on assumptions about the computational hardness of certain mathematical problems — assumptions that Shor's algorithm would break. Quantum key distribution offers a qualitatively different kind of security: one based on physics rather than mathematics. The BB84 protocol, proposed by Bennett and Brassard in 1984, uses quantum mechanics to distribute a shared secret key between two parties in a way that any eavesdropping attempt is detectable.

The protocol works as follows. Alice prepares random qubits, each encoding a random bit in a randomly chosen basis: either the Z basis (|0> for 0, |1> for 1) or the X basis (|+> for 0, |-> for 1). She sends these qubits to Bob, who measures each in a randomly chosen basis (Z or X). When their bases match (about 50% of the time), their bit values agree perfectly. When bases mismatch, Bob's result is completely random. Alice and Bob publicly compare their basis choices (not bit values) and keep only the rounds where they used the same basis — this is sifting, producing the raw key.

Eavesdropping detection comes next. Suppose Eve intercepts qubits, measures them, and resends them to Bob (an intercept-resend attack). Eve does not know Alice's basis, so she guesses randomly. When Eve guesses wrong (50% of the time), her measurement disturbs the qubit, and when Bob subsequently measures in Alice's correct basis, he gets a random result instead of Alice's bit. This introduces an error rate of approximately 25% in the sifted key. Alice and Bob sacrifice a random subset of their sifted key bits, compare them publicly, and check the error rate. An error rate significantly above the channel noise threshold indicates eavesdropping, and they abort.

If the error rate is acceptably low, Alice and Bob apply error correction (to fix the remaining errors) and privacy amplification (to eliminate any partial information Eve may have gained). The result is a shorter but provably secure shared secret key. The security proof, formalized by Lo, Chau, Shor, Preskill, and others, shows that any eavesdropping strategy — including sophisticated quantum attacks beyond intercept-resend — is detectable. The proof relies on the no-cloning theorem (Eve cannot copy the qubits and keep them for later analysis) and the information-disturbance tradeoff (gaining information about a quantum state necessarily disturbs it). QKD has been commercially deployed over fiber-optic links and demonstrated via satellite (the Micius experiment), making it the most mature application of quantum information science.

Practice Questions 3 questions

Prerequisite Chain

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIterated Integrals and Fubini's TheoremDouble Integrals in Cartesian CoordinatesDouble Integrals over Rectangular RegionsDouble Integrals in Polar CoordinatesDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsGreen's TheoremSurface Integrals and Flux of Vector FieldsSurface Integrals and Flux of Vector FieldsDivergence Theorem: Flux and OutflowDivergence TheoremElectric FluxGauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and BatteriesKirchhoff's Circuit Laws: Voltage and CurrentDC Circuit Network Analysis MethodsTransient Response in RC CircuitsRC CircuitsLC and RLC CircuitsAC Circuits: FundamentalsImpedance and ReactanceAC Power and ResonanceElectromagnetic WavesThe Electromagnetic SpectrumBlackbody Radiation and Planck's LawPhotoelectric EffectThe Photon: Light as QuantaCompton ScatteringWave-Particle Dualityde Broglie WavelengthHeisenberg Uncertainty PrincipleWavefunction and the Born RuleThe Schrödinger EquationSchrödinger Equation: Time-Dependent FormWavefunctions and Boundary ConditionsBoundary Value Problems in ElectrostaticsParticle in a Box (Infinite Square Well)Quantum NumbersSpin-1/2 SystemsPauli MatricesQuantum GatesNo-Cloning TheoremQuantum Key Distribution (BB84)

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