Recent from talks
Nothing was collected or created yet.
The BB84 protocol, named after its inventors Charles Bennett and Gilles Brassard in 1984, is a prepare-and-measure Quantum key distribution (QKD) protocol, in which, one party (e.g. Alice) performs the encoding by preparing the quantum states, and the other party (e.g., Bob) measures them.[1] The BB84 QKD scheme is the first quantum cryptography protocol, and has become one of the most well-studied QKD protocols.[2] The protocol is provably secure assuming a perfect implementation, relying on two conditions: (1) the quantum property that information gain is only possible at the expense of disturbing the signal if the two states one is trying to distinguish are not orthogonal (see no-cloning theorem); and (2) the existence of an authenticated public classical channel.[3] As such, the security of the BB84 protocol is fundamentally based on the principle that two non-orthogonal quantum states cannot be perfectly distinguished. This inherent limitation means that the states cannot be reliably copied, thereby ensuring a robust framework for secure quantum communication. The BB84 QKD protocol is usually explained as a method of securely communicating a private key from one party to another for use in one-time pad encryption.[4] The proof of BB84 QKD scheme depends on a perfect implementation. Side channel attacks exist, taking advantage of non-quantum sources of information. Since this information is non-quantum, it can be intercepted without measuring or cloning quantum particles.[5] The BB84 protocol provides a significant advancement in the field of quantum cryptography and represents a pioneering step toward achieving secure communication in the quantum era.[6]
Overview
[edit]BB84 QKD system transmits individual photons through a fiber optic cable, with each photon representing a bit of data (zero or one). Polarizing filters on the sender's side set each photon's orientation, while the receiver uses beam splitters to read it. The sender and receiver then compare their photon orientations, with the matching set becoming the cryptographic key.[7] However, encoding withother degrees of freedom, e.g., phase, is also possible, and the procedures are similar.[8]
Description
[edit]
In the BB84 scheme, Alice wishes to send a private key to Bob. She begins with two strings of bits, and , each bits long. She then prepares an -qubit state written as:
where and are the -th bits of and respectively. Together, give us an index into the following four qubit states:
Note that the bit is what decides which basis is encoded in (either in the computational basis or the Hadamard basis). The qubits are now in states that are not mutually orthogonal, and thus it is impossible to distinguish all of them with certainty without knowing .
Alice sends over a public and authenticated quantum channel to Bob. Bob receives a state , where represents both the effects of noise in the channel and eavesdropping by a third party we'll call Eve. After Bob receives the string of qubits, both Bob and Eve have their own states. However, since only Alice knows , it makes it virtually impossible for either Bob or Eve to distinguish the states of the qubits. Also, after Bob has received the qubits, we know that Eve cannot be in possession of a copy of the qubits sent to Bob, by the no-cloning theorem, unless she has made measurements. Her measurements, however, risk disturbing a particular qubit with probability 1/2 if she guesses the wrong basis.
Bob proceeds to generate a string of random bits of the same length as and then measures the qubits he has received from Alice, obtaining a bit string . At this point, Bob announces publicly that he has received Alice's transmission. Alice then knows she can now safely announce , i.e., the bases in which the qubits were prepared. Bob communicates over a public channel with Alice to determine which and are not equal. Both Alice and Bob now discard the bits in and where and do not match.
From the remaining bits where both Alice and Bob measured in the same basis, Alice randomly chooses bits and discloses her choices over the public channel. Both Alice and Bob announce these bits publicly and run a check to see whether more than a certain number of them agree. If this check passes, Alice and Bob proceed to use information reconciliation and privacy amplification techniques to create some number of shared secret keys. Otherwise, they cancel and start over.
See also
[edit]- SARG04
- E91 – quantum cryptographic communication protocol
References
[edit]- ^ Bennett, Charles H.; Brassard, Gilles (2014-12-04). "Quantum cryptography: Public key distribution and coin tossing". Theoretical Computer Science. Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84. 560: 7–11. arXiv:2003.06557. doi:10.1016/j.tcs.2014.05.025. ISSN 0304-3975.
- ^ Branciard, Cyril; Gisin, Nicolas; Kraus, Barbara; Scarani, Valerio (2005). "Security of two quantum cryptography protocols using the same four qubit states". Physical Review A. 72 (3) 032301. arXiv:quant-ph/0505035. Bibcode:2005PhRvA..72c2301B. doi:10.1103/PhysRevA.72.032301. S2CID 53653084.
- ^ Scarani, Valerio; Bechmann-Pasquinucci, Helle; Cerf, Nicolas J.; Dušek, Miloslav; Lütkenhaus, Norbert; Peev, Momtchil (2009). "The security of practical quantum key distribution". Rev. Mod. Phys. 81 (3): 1301–1350. arXiv:0802.4155. Bibcode:2009RvMP...81.1301S. doi:10.1103/RevModPhys.81.1301. S2CID 15873250.
- ^ Quantum Computing and Quantum Information, Michael Nielsen and Isaac Chuang, Cambridge University Press 2000
- ^ Dixon, A. R., Dynes, J. F., Lucamarini, M., Fröhlich, B., Sharpe, A. W., Plews, A., Tam, W., Yuan, Z. L., Tanizawa, Y., Sato, H., Kawamura, S., Fujiwara, M., Sasaki, M., & Shields, A. J. (2017). Quantum key distribution with hacking countermeasures and long term field trial. Scientific Reports, 7, 1978.
- ^ Brassard, Gilles (2016). Freivalds, Rūsiņš Mārtiņš; Engels, Gregor; Catania, Barbara (eds.). "Cryptography in a Quantum World". SOFSEM 2016: Theory and Practice of Computer Science. Berlin, Heidelberg: Springer: 3–16. arXiv:1510.04256. doi:10.1007/978-3-662-49192-8_1. ISBN 978-3-662-49192-8.
- ^ "What Is Quantum Cryptography? | IBM". www.ibm.com. 2023-11-29. Retrieved 2024-09-25.
- ^ Gisin, Nicolas; Ribordy, Grégoire; Tittel, Wolfgang; Zbinden, Hugo (2002-03-08). "Quantum cryptography". Reviews of Modern Physics. 74 (1): 145–195. arXiv:quant-ph/0101098. doi:10.1103/RevModPhys.74.145.
- ^ Migdał, Piotr; Jankiewicz, Klementyna; Grabarz, Paweł; Decaroli, Chiara; Cochin, Philippe (2022). "Visualizing quantum mechanics in an interactive simulation - Virtual Lab by Quantum Flytrap". Optical Engineering. 61 (8) 081808. arXiv:2203.13300. doi:10.1117/1.OE.61.8.081808.
Background
Quantum Information Fundamentals
A qubit, or quantum bit, serves as the fundamental unit of quantum information, analogous to the classical bit but with distinct properties that enable quantum computation and communication. Unlike a classical bit, which exists definitively in one of two states—0 or 1—a qubit can occupy a superposition of both states simultaneously, allowing it to encode more complex information. This capability arises from the principles of quantum mechanics, where the qubit's state is represented in a two-dimensional Hilbert space. Quantum superposition permits a qubit to exist as a linear combination of basis states, embodying multiple potential outcomes until measured. In this framework, information encoding leverages the interference of these superimposed states, providing an exponential advantage in processing power compared to classical systems for certain tasks. Mathematically, a general qubit state can be expressed aswhere and are complex amplitudes satisfying , ensuring probabilistic normalization. The squared magnitudes and represent the probabilities of measuring the qubit in state or , respectively. Quantum measurement fundamentally alters the qubit's state through wavefunction collapse, projecting it onto one of the basis states depending on the chosen measurement basis. This process is inherently probabilistic and irreversible, with the outcome determined by the Born rule, where the superposition resolves into a definite classical result. The basis in which measurement occurs dictates the possible outcomes, highlighting the context-dependent nature of quantum information extraction. The no-cloning theorem asserts that it is impossible to create an identical copy of an arbitrary unknown quantum state, a direct consequence of the linearity of quantum evolution and the superposition principle.[8] Formally, if a unitary operation could clone any input state to produce , it would fail for superpositions like , as the output would incorrectly include cross terms such as , violating linearity.[8] This prohibition extends to all non-orthogonal states, preventing perfect replication without prior knowledge of the state.[8] In the context of quantum security, the theorem implies that quantum information cannot be intercepted and duplicated without detection, forming a cornerstone for protocols in quantum key distribution.[8]
Historical Context
The foundations of quantum cryptography, including the BB84 protocol, trace back to innovative ideas in the late 1960s and early 1970s, when physicist Stephen Wiesner developed the concept of conjugate coding while a graduate student at Columbia University.[9] Wiesner's work, which explored the use of quantum states for secure information transmission through complementary orthogonal modes like polarization, remained unpublished until 1983 but circulated informally among researchers and laid crucial groundwork for quantum-based secure communication. This idea built on emerging quantum information concepts from the 1970s, shifting focus from classical cryptographic limitations toward leveraging quantum mechanics' inherent uncertainties for privacy.[9] The BB84 protocol was formally proposed in 1984 by Charles H. Bennett and Gilles Brassard during the IEEE International Conference on Computers, Systems and Signal Processing in Bangalore, India, marking the first practical quantum key distribution scheme. Their collaboration, initiated in 1979, culminated in the protocol's description in the conference proceedings, where it was presented as a method for two parties to generate a shared secret key resistant to eavesdropping.[9] This development occurred amid Cold War-era imperatives for unbreakable secure communications, as governments and researchers sought alternatives to vulnerable classical encryption amid escalating technological espionage concerns.[10] Subsequent refinements strengthened BB84's theoretical foundations, including the 1988 introduction of privacy amplification by Bennett, Brassard, and Jean-Marc Robert, which addressed partial information leakage through public discussion to distill a highly secure key.[11] A key milestone came in 1989 with the first experimental demonstration by Bennett, Brassard, and colleagues, who successfully transmitted a 403-bit key over 32.5 cm using polarized photons in free space, validating the protocol's feasibility despite early technological constraints.[12] The no-cloning theorem, established in 1982, further motivated BB84 by proving the impossibility of perfectly copying unknown quantum states, underpinning its security against interception.[9]Protocol Operation
Photon Encoding and Transmission
In the BB84 protocol, information is encoded onto individual photons using their polarization states as the quantum carriers. Alice prepares single photons with polarizations corresponding to binary bits in one of two non-orthogonal bases: the rectilinear basis, where horizontal polarization (0°) represents bit 0 and vertical polarization (90°) represents bit 1; and the diagonal basis, where +45° polarization represents bit 0 and -45° (or 135°) polarization represents bit 1.[1] This choice of bases ensures that measurements in the incorrect basis yield random outcomes due to the quantum superposition of polarization states.[1] To generate the key, Alice first creates a random binary string of bits, typically using a pseudorandom number generator or quantum random source. For each bit, she independently selects the encoding basis (rectilinear or diagonal) at random with equal probability, then polarizes the corresponding photon accordingly.[1] The photons are transmitted sequentially over a quantum channel to Bob, such as an optical fiber for guided transmission or free-space optics for line-of-sight links, preserving the polarization states to the extent allowed by the channel's fidelity.[1] A typical experimental setup for photon preparation at Alice's side involves a laser source, such as a pulsed diode laser operating at wavelengths like 1550 nm for low-loss fiber transmission, attenuated to produce weak pulses approximating single-photon emission.[13] The laser output passes through a polarizer or electro-optic modulator (e.g., a Pockels cell) to set the desired polarization based on the bit and basis choice, followed by a beam splitter or attenuator to ensure the mean photon number per pulse is much less than 1 (e.g., 0.1–0.5 photons on average).[14] This configuration allows for high-speed encoding, with clock rates up to 1.25 Gbit/s in fiber-based systems.[14] In practice, ideal single-photon sources are challenging to implement, so weak coherent pulses from attenuated lasers are commonly used as an approximation, though this introduces multi-photon emission risks that can compromise security.[15] To mitigate such vulnerabilities, decoy-state methods employ additional pulse intensities (e.g., vacuum, weak decoy, and signal states) alongside the standard BB84 pulses, enabling estimation of multi-photon contributions without altering the core encoding process.[15]Basis Selection and Measurement
In the BB84 protocol, upon receiving each photon from Alice, Bob independently selects a measurement basis at random, choosing between the rectilinear basis (corresponding to horizontal and vertical polarizations) or the diagonal basis (corresponding to 45° and 135° polarizations) with equal probability. This selection is performed without any prior knowledge of Alice's encoding basis, ensuring that the probability of their bases matching for any individual photon is exactly 50%. The random choice is generated locally by Bob using a secure random number generator to maintain the protocol's security properties.[1] Bob then measures the polarization of the photon in his chosen basis, which causes the photon's quantum state to collapse onto one of the two basis states via projective measurement. This yields a deterministic binary outcome: conventionally, horizontal or 0° polarization corresponds to bit 0, while vertical or 90° polarization corresponds to bit 1 in the rectilinear basis, with analogous assignments in the diagonal basis. If Bob's basis matches Alice's, the measurement faithfully recovers Alice's intended bit value with near-perfect fidelity in the absence of noise, as the photon arrives in an eigenstate of Bob's measurement observable. Conversely, if the bases mismatch, the measurement outcome is completely random relative to Alice's bit—yielding 0 or 1 with 50% probability each—due to the incompatibility of the bases, which destroys the original superposition and any encoded information, in accordance with the principles of quantum mechanics.[1] Practical implementations are susceptible to environmental noise in the quantum channel, such as depolarization or scattering, which can induce bit flips in the measurement outcomes independently of the selected basis. These errors would contribute to the quantum bit error rate (QBER) if bases match.[14]Key Extraction
Sifting Procedure
After the quantum transmission phase, Alice and Bob engage in a classical post-processing step known as sifting to filter their raw data and generate a shared sifted key. This procedure relies on an authenticated public channel to compare their basis choices without revealing the actual bit values. Specifically, Bob announces the basis he used for each measurement, and Alice reveals the basis she selected for each corresponding qubit transmission. They then identify the positions where their bases match—either both rectilinear (e.g., horizontal/vertical polarizations) or both diagonal (e.g., 45°/135° polarizations)—and discard all bits from positions where the bases differ. The retained bits, for which the bases aligned, form the initial sifted key, which Alice and Bob now hold in common.[1] The sifting process follows a straightforward algorithm to ensure efficiency and security:- Alice generates and publicly sends her sequence of basis choices (a string of 'R' for rectilinear or 'D' for diagonal) corresponding to the transmitted qubits.
- Bob compares this with his own basis sequence and identifies the matching positions, retaining only the measurement outcomes (bits) from those positions in his record.
- Alice performs the same comparison on her end, keeping her original bits only from the matching positions. At this stage, no bit values are exchanged; only the positions are reconciled.
