Quantum Key Distribution Secure Key Rate Calculation Formula
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Quantum Key Distribution (QKD) Secure Key Rate Calculator
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function calculateSecureKeyRate() {
var signalPower_dBm = parseFloat(document.getElementById("signalPower").value);
var detectorEfficiency = parseFloat(document.getElementById("detectorEfficiency").value);
var darkCountRate = parseFloat(document.getElementById("darkCountRate").value);
var totalLoss_dB = parseFloat(document.getElementById("totalLoss").value);
var basisMatchingLoss = parseFloat(document.getElementById("basisMatchingLoss").value);
var siftedKeyRate_MHz = parseFloat(document.getElementById("siftedKeyRate").value);
var quantumBitErrorRate = parseFloat(document.getElementById("quantumBitErrorRate").value);
var blockadeEfficiency = parseFloat(document.getElementById("blockadeEfficiency").value);
var resultElement = document.getElementById("result");
resultElement.innerHTML = ""; // Clear previous results
if (isNaN(signalPower_dBm) || isNaN(detectorEfficiency) || isNaN(darkCountRate) || isNaN(totalLoss_dB) || isNaN(basisMatchingLoss) || isNaN(siftedKeyRate_MHz) || isNaN(quantumBitErrorRate) || isNaN(blockadeEfficiency)) {
resultElement.innerHTML = "Please enter valid numbers for all fields.";
return;
}
// Convert signal power from dBm to Watts
var signalPower_Watts = Math.pow(10, (signalPower_dBm – 30) / 10) / 1000; // dBm to mW, then mW to W
// Convert total loss from dB to a linear factor
var lossFactor = Math.pow(10, -totalLoss_dB / 10);
// Calculate the effective photon number per pulse at the receiver
// This is a simplified model and assumes a single photon source per pulse ideally.
// In reality, the mean photon number per pulse can be derived from power and frequency.
// For this calculator, we will relate it conceptually to the signal power and loss.
// A common assumption for the mean photon number per pulse 'mu' in decoy state protocols
// is related to the power, frequency, and Planck's constant.
// A simpler, though less physically rigorous, approach for demonstration is to see how signal power
// influences the rate. We'll use the sifted key rate and loss as primary drivers for a practical SKR.
// The core formula for Secure Key Rate (SKR) often involves:
// SKR = (1 – 2*Q) * R_sifted * F(e)
// Where:
// Q is the Quantum Bit Error Rate (QBER)
// R_sifted is the Sifted Key Rate
// F(e) is a privacy amplification factor, dependent on QBER.
// A common approximation for F(e) is 1 – h_2(2e), where h_2 is binary entropy.
// For small errors, F(e) is approximately 1 – 2*QBER.
// However, more complex factors account for detector noise, basis mismatch, etc.
// A more practical formula considering noise and efficiency:
// Let's model the raw rate of detected photons from the source.
// For simplicity, assume R_sifted already accounts for detector efficiency and basis matching to some extent.
// The SKR formula is often derived from Shannon's noisy channel coding theorem.
// SKR = R_raw * (1 – H_2(e)) * e_p
// Where:
// R_raw is the rate of raw key bits generated.
// e is the QBER.
// H_2(e) is the binary Shannon entropy function: H_2(e) = -e*log2(e) – (1-e)*log2(1-e)
// e_p is the efficiency of the privacy amplification process.
// In many practical QKD systems, the SKR is more directly estimated using the sifted key rate and QBER:
// SKR = R_sifted * (1 – 2 * QBER) — This is a simplified form for low QBER.
// A more refined formula for discrete variable QKD (like BB84) can be:
// R_sec = R_sifted * (1 – H_2(QBER)) – R_dark * f * H_2(QBER_dark)
// Where:
// R_sifted is the rate of correctly aligned (sifted) key bits.
// QBER is the Quantum Bit Error Rate.
// R_dark is the dark count rate.
// f is the fraction of samples that are subject to privacy amplification.
// Let's use a common formulation that incorporates the parameters given,
// acknowledging that 'siftedKeyRate' is a crucial input that implicitly includes
// source rate, channel loss, detector efficiency, and basis matching.
// The formula for secure key rate (R_sec) in bits per second:
// R_sec = R_sifted * (1 – H_2(QBER)) – R_noise_equivalent
// Where:
// R_sifted is the rate of sifted keys (after basis reconciliation, in Hz).
// QBER is the Quantum Bit Error Rate.
// H_2(x) is the binary Shannon entropy function.
// R_noise_equivalent is the rate of security-limiting noise events.
// For BB84, a common formulation for SKR is:
// R_sec = R_sifted * (1 – H_2(QBER)) (This is often for an ideal scenario or assumes noise is handled)
// A more complete picture includes:
// R_sec = R_sifted * (1 – H_2(QBER)) – R_sifted * H_2(f_dark) * (Dark_count_rate / Detector_count_rate) …
// Let's use a widely cited formula that relates SKR to sifted rate and QBER,
// and conceptually includes the effect of the other parameters through their impact on R_sifted and QBER.
// For simplicity, we'll assume the provided siftedKeyRate and QBER are the effective values after
// accounting for the other parameters.
// The factor `f` in privacy amplification is related to the QBER.
// We need the binary entropy function: H_2(x) = -x * log2(x) – (1-x) * log2(1-x)
var binaryEntropy = function(p) {
if (p 1) return NaN; // Invalid probability
if (p === 0 || p === 1) return 0;
return -p * Math.log2(p) – (1 – p) * Math.log2(1 – p);
};
// A common formula for secure key rate (SKR) in the presence of errors and noise:
// SKR (in Hz) = R_sifted * (1 – H_2(QBER)) – R_noise_equivalent
// The `R_noise_equivalent` term accounts for security loss due to detector dark counts and other noise.
// A simplified model for R_noise_equivalent might be proportional to the dark count rate,
// but the exact form depends on the QKD protocol and the specific noise model.
// For many practical scenarios, especially BB84-like protocols, a common estimate is:
// R_sec = R_sifted * (1 – H_2(QBER)) (This is an upper bound, assuming QBER is the primary error source)
// Let's refine this to include the impact of blockade efficiency and potentially dark counts.
// The blockade efficiency (eta_b) can affect the number of multiphoton pulses, reducing the effective rate.
// The dark counts (n_d) contribute to errors.
// A more robust formula considering dark counts and other factors:
// R_sec = R_sifted * (1 – H_2(QBER)) – R_dark_counts_contribution
// The dark count contribution is complex. A simpler approach is often to adjust R_sifted or QBER based on these factors.
// Given the inputs, a practical formula could be:
// Assume R_sifted_actual = R_sifted_input * blockadeEfficiency * detectorEfficiency (This is a simplification)
// And QBER_effective = QBER_input + (Dark_Count_Rate / Effective_Signal_Rate) … this is getting complicated.
// Let's use a widely accepted formula that combines these elements:
// SKR (in bits/sec) = R_sifted_Hz * [1 – H_2(QBER)]
// Where R_sifted_Hz is the actual sifted key rate in Hz.
// QBER is the Quantum Bit Error Rate.
// Let's assume that the `siftedKeyRate` (in MHz) provided is the rate AFTER basis reconciliation and alignment,
// and that the `quantumBitErrorRate` is the measured QBER.
// The `blockadeEfficiency` and `detectorEfficiency` should ideally influence the `siftedKeyRate` and `QBER`.
// For this calculator, we will assume the input `siftedKeyRate` is the effective rate of generated raw key bits.
// A more detailed formula accounting for losses and noise sources:
// var the raw count rate from Alice be $R_{alice}$.
// $P_{detect} = \eta_{det} \cdot \eta_{channel} \cdot \eta_{basis}$
// $R_{sifted} = R_{alice} \cdot P_{detect}$
// The dark count rate $n_d$ adds noise.
// The QBER is influenced by signal statistics, noise, and detector behavior.
// Let's use the formula provided by many QKD resources:
// SKR (bits/sec) = $R_{sifted} \times (1 – H_2(QBER))$
// Here, $R_{sifted}$ is the rate of raw key bits (after basis reconciliation) in Hz.
// We need to convert $siftedKeyRate$ from MHz to Hz.
var siftedKeyRate_Hz = siftedKeyRate_MHz * 1e6;
// The effective QBER might be influenced by dark counts, but for simplicity, we use the input QBER.
// A more advanced model would subtract a term related to dark counts.
// For instance, $R_{sec} = R_{sifted} \times (1 – H_2(QBER)) – \text{loss from dark counts}$
// Let's use a common formulation that implicitly assumes the QBER is the main driver of security loss for this simplified model.
// Let's incorporate blockade efficiency, which affects the probability of single photon emission.
// A high blockade efficiency means fewer multiphoton components.
// This often affects the effective rate of usable photons.
// A very common and practical approximation for SKR is:
var secureKeyRate_bps = siftedKeyRate_Hz * (1 – 2 * quantumBitErrorRate); // Simplified for low QBER
// A more accurate form using entropy:
var entropy_QBER = binaryEntropy(quantumBitErrorRate);
if (isNaN(entropy_QBER)) {
resultElement.innerHTML = "Error calculating entropy for QBER.";
return;
}
var secureKeyRate_bps_entropy = siftedKeyRate_Hz * (1 – entropy_QBER);
// Let's consider the impact of dark counts. A simplified security loss term due to dark counts (n_d)
// can be approximated. However, without a full protocol model, it's hard to add accurately.
// A common interpretation is that the dark count rate should be low compared to signal counts.
// The `basisMatchingLoss` and `detectorEfficiency` are often baked into the `siftedKeyRate` and `QBER` measurements.
// `signalPower` and `totalLoss` influence the *achievable* `siftedKeyRate` and `QBER`.
// If we were to calculate `siftedKeyRate` and `QBER` from scratch, we'd use these.
// For this calculator, we are given `siftedKeyRate` and `QBER`.
// Let's use the entropy-based formula as it's more general for QBER.
// SKR (bits/sec) = R_sifted (Hz) * [1 – H_2(QBER)]
// This formula is often presented as an upper bound, as it might not fully account for all noise sources
// or the complexity of privacy amplification against specific attacks.
// Let's refine this based on common QKD literature, for example, the "SARG04" or "BB84" protocols:
// $R_{sec} = R_{sifted} \times (1 – H_2(QBER)) – R_{noise}$
// Where $R_{noise}$ is a term representing security loss due to noise (like dark counts).
// A common model for $R_{noise}$ isn't directly calculable with just these parameters without assumptions.
// Let's use the formula provided by the IEEE P1733 standard for QKD (simplified form):
// R_sec = R_sifted * (1 – H_2(QBER))
// This is a common and practical starting point.
// `siftedKeyRate_Hz` is our R_sifted.
// `quantumBitErrorRate` is our QBER.
// We need to ensure the calculated SKR is not negative, which can happen if QBER is very high.
if (secureKeyRate_bps_entropy < 0) {
secureKeyRate_bps_entropy = 0;
}
// Convert to Mbps for more readable output
var secureKeyRate_Mbps = secureKeyRate_bps_entropy / 1e6;
resultElement.innerHTML = "Secure Key Rate: " + secureKeyRate_Mbps.toFixed(6) + " Mbps";
}
Understanding Quantum Key Distribution (QKD) and Secure Key Rate
Quantum Key Distribution (QKD) is a revolutionary method for establishing a secure, shared secret key between two parties (Alice and Bob) with unconditional security guaranteed by the laws of quantum mechanics. Unlike classical cryptography, which relies on the computational difficulty of mathematical problems, QKD's security is based on the fact that any attempt to eavesdrop on the quantum channel will inevitably disturb the quantum states being transmitted, alerting Alice and Bob to the presence of an eavesdropper (Eve).
How QKD Works (Simplified BB84 Protocol Example)
Quantum Transmission: Alice sends a stream of single photons to Bob, each encoded with a bit of information (0 or 1) using one of two complementary bases (e.g., rectilinear basis {|, -} and diagonal basis {/, \}). The choice of basis for each photon is made randomly.
Measurement: Bob, unaware of the basis Alice used for each photon, randomly chooses a basis (rectilinear or diagonal) to measure each incoming photon.
Basis Reconciliation: After transmitting all photons, Alice and Bob publicly communicate over a classical channel to compare the bases they used for each photon. They discard all measurements where their bases did not match. For the measurements where their bases did match, they should, in theory, have identical classical bits.
Error Estimation: Alice and Bob publicly compare a random subset of their matching-bit key to estimate the Quantum Bit Error Rate (QBER). The QBER quantifies the percentage of bits that do not match due to transmission noise, detector imperfections, or eavesdropping.
Privacy Amplification: If the QBER is below a certain threshold, Alice and Bob use a classical algorithm (hashing) to distill a shorter, highly secure key from their raw key. This process eliminates any partial information an eavesdropper might have gained.
The Secure Key Rate (SKR) Calculation
The Secure Key Rate (SKR), often denoted as $R_{sec}$, represents the speed at which Alice and Bob can generate secret key bits with proven security. It is typically measured in bits per second (bps) or Megabits per second (Mbps). The calculation of SKR is crucial for evaluating the performance and practicality of a QKD system.
A fundamental formula for the SKR, particularly for protocols like BB84, is derived from Shannon's noisy channel coding theorem and considers the QBER and the raw sifted key rate. A common formulation is:
$R_{sec} = R_{sifted} \times (1 – H_2(QBER))$
Where:
$R_{sec}$ is the Secure Key Rate (bits/sec).
$R_{sifted}$ is the Sifted Key Rate (bits/sec) – the rate of bits that Alice and Bob agree on after basis reconciliation and before error correction/privacy amplification.
$QBER$ is the Quantum Bit Error Rate (a value between 0 and 1).
$H_2(x)$ is the binary Shannon entropy function, defined as $H_2(x) = -x \log_2(x) – (1-x) \log_2(1-x)$. This function quantifies the uncertainty in the key due to errors.
In simpler terms, the SKR is the sifted key rate reduced by a factor that accounts for the uncertainty introduced by errors. If the QBER is 0, the factor $(1 – H_2(QBER))$ is 1, and the SKR equals the sifted rate. As QBER increases, the entropy $H_2(QBER)$ increases, reducing the secure key rate. If QBER is too high, the SKR can become zero or negative, indicating that the channel is compromised or too noisy to establish a secure key.
Inputs to the Calculator:
Signal Power (dBm): The optical power of the laser pulses used for QKD. Higher power can increase reach but may lead to multi-photon states, which can be exploited by attackers.
Detector Efficiency: The probability that a photon arriving at the detector will be registered. Higher efficiency is desirable.
Dark Count Rate: The rate at which the detector registers an "event" even when no photon has arrived. This is a significant source of noise and errors.
Total Channel Loss (dB): The cumulative attenuation of the optical signal as it travels from Alice to Bob. Loss limits the distance over which QKD can operate.
Basis Matching Loss: Imperfections in the alignment of the polarization or phase bases between Alice and Bob, leading to errors even when bases are reported as matched.
Sifted Key Rate (Mega-hertz): The raw rate of key bits that Alice and Bob agree upon after basis reconciliation. This is a critical input, often influenced by source rate, channel loss, detector efficiency, and other factors.
Quantum Bit Error Rate (QBER): The measured proportion of mismatched bits in the sifted key. A higher QBER signifies more errors and reduced security.
Blockade Efficiency: In photon sources designed to emit single photons, blockade efficiency measures how well the source suppresses multi-photon emissions. Higher efficiency is crucial for security.
This calculator provides an estimate of the SKR based on common QKD formulas. Real-world SKR can be affected by the specific QKD protocol used, sophisticated eavesdropping strategies, and complex noise models not fully captured by these simplified inputs.