Circular Evaluation Protocols Explained

• Circular Evaluation Protocols are computational frameworks that map linear or angular data to a circular domain using techniques such as the Angular Probability Integral Transform (APIT).
• They enhance secure quantum communication by implementing cyclic transmission routes in semi-quantum secret sharing protocols, reducing resource demands and simplifying error detection.
• These protocols improve statistical independence tests and temporal knowledge base completion by integrating joint evaluation strategies over closed, cyclical domains.

Circular evaluation protocols refer to computational and statistical methodologies in which the key objects of evaluation—data, quantum resources, or predictions—are indexed on a circular domain or traverse a closed, cyclical route among participants or through the space of possible instants. The concept encompasses both circularly-structured statistical procedures, such as those found in circular statistics, and circular transmission architectures in quantum cryptography. Several notable applications arise from circular probability integral transforms for independence testing, as well as circular transmission strategies in semi-quantum cryptographic secret sharing.

1. Angular Probability Integral Transform in Circular Evaluation

The angular probability integral transform (APIT) is a procedure for mapping an absolutely continuous random variable $X$ to an angle uniformly distributed on the unit circle. Given a probability distribution function $F_X$, the APIT is computed as $\theta_U = 2\pi F_X(X)$. This transformation enables all linear or angular data to be embedded onto the circular domain $(0, 2\pi]$.

Circular evaluation protocols leveraging the APIT depend on key properties of the circular uniform distribution, including stability under summation. If $U_1, U_2, \ldots, U_d$ are independently and identically distributed as uniform variables on the circle, their sum modulo $2\pi$ is also circularly uniform. The characteristic function $\psi(t)$ of the circular uniform variable vanishes for all integer $t \neq 0$, underscoring its analytic tractability for probabilistic modeling.

2. Circular Transmission Methodology in Quantum Protocols

Circular transmission, as proposed in circular semi-quantum secret sharing (SQSS) protocols (Ye et al., 2022), describes a transmission sequence where quantum particles travel from a quantum source (Alice) to classical parties (Bob, Charlie), who perform prescribed operations such as measurement or reflection, and then return the particles to the source. Unlike tree-type arrangements in previous SQSS schemes, this circular route minimizes distribution complexity, consolidates security verification to the final reception, and exploits the distinct transmission stages for tailored security proofs.

The protocol can be instantiated with single particles prepared randomly in one of the four states $\{|0\rangle, |1\rangle, |+\rangle, |-\rangle\}$, and provides options for classical parties to either perform measurements in the computational basis or simply reflect or reprepare states in a structured, circular fashion. This eliminates the reliance on entangled or product states, and in certain protocol variants releases the classical participants from measurement duties entirely.

3. Circular Statistical Evaluation for Independence Testing

Utilizing circular evaluation, independence testing in circular–linear or circular–circular domains is conducted by transforming data variables via APIT, then assessing the circular uniformity of their sum or difference modulo $2\pi$. Under the null hypothesis of independence, these transformed aggregates should follow the circular uniform distribution. Empirical procedures typically involve:

• Computing empirical cumulative distribution functions for $X_1$ and $X_2$.
• Applying APIT to obtain $\theta_{U_1}$, $\theta_{U_2}$.
• Aggregating the angles, $[\theta_{U_1} \pm \theta_{U_2}]~\mathrm{mod}~2\pi$.
• Testing circular uniformity via established tests (Rayleigh test for unimodal deviations; Pycke test for multimodal).

Performance evaluation leverages flexible circular families such as nonnegative trigonometric sums (NNTS) distributions,

$$f_\theta (\theta; M, c) = \frac{1}{2\pi} \left\| \sum_{k=0}^M c_k e^{ik\theta} \right\|^2,$$

with normalization $\sum_{k=0}^M |c_k|^2 = 1$ and $M = 0$ corresponding to circular uniformity.

4. Security and Robustness of Circular Protocols

Circular evaluation in cryptographic protocols supports enhanced robustness against classical attacks. In SQSS, particles traversing the cyclic route are subject to systematic error tests by the originating quantum party. Security analysis considers cases such as measure-resend, intercept-resend, and entangle-measure attacks.

Security checks utilize the fact that measurement basis mismatches induce disturbance in particle states prepared in non-orthogonal bases, leading to a detection probability (e.g., 25% for specific error classes). Theoretical analysis confirms that for entangle–measure attacks, probe unitaries used by adversaries must evolve without encoding dependence on participants’ secret bits in the absence of detectable error, as framed by expressions like:

$$U_E(|s\rangle_A |E\rangle) = \alpha|0\rangle_A |E_0\rangle + \beta|1\rangle_A |E_1\rangle.$$

If the final probe state is independent of the participants’ choices, Eve gains no information.

The protocols require only single-particle resources and are compatible with current technologies such as single-photon sources, linear optics, and standard photon detectors.

5. Joint Evaluation of Entities and Time in Temporal Knowledge Bases

Circular evaluation in temporal knowledge base completion (TKBC) protocols (Jain et al., 2020) addresses the structured nature of time intervals in joint prediction of entities and durations. Unlike prior pointwise evaluations, whole intervals $T$ are scored:

$$\phi(s, r, o, T) = \sum_{t \in T} \phi(s, r, o, t),$$

and training employs a margin-based loss proportional to the complement of averaged intersection-over-union (aeIoU):

$$\Delta(T^{\mathrm{tr}}, T) = 1 - \mathrm{aeIoU}(T^{\mathrm{tr}}, T).$$

Greedy coalescing and exhaustive interval search are used to recover intervals maximizing the scoring function, with filtering to exclude previously observed temporal intervals in queries. Link prediction scores aggregate across intervals and are normalized via softmax:

$$\Pr(o \mid s, r, T) = \frac{\phi(s, r, o, T)}{\sum_{o'} \phi(s, r, o', T)}.$$

This structured protocol refines model assessment, ensuring generalization is not overestimated due to temporal overlap between training and test cases.

6. Practical Applications and Implications

Circular evaluation protocols are employed across domains requiring circular symmetry or closed-loop transmission pathways:

Domain	Circular Protocol Role	Application Example
Circular statistics	APIT-based independence tests	Wind direction–pollutant level analysis, protein angles
Quantum cryptography	SQSS circular transmission for robustness	Secure key sharing with semi-quantum parties
TKBC	Interval reasoning integrated with entity prediction	Event forecasting, historical data analysis

The approach facilitates model comparison, detailed error analysis, and supports generalized methodologies for groupwise or multivariate independence, as well as extensions to scenarios with variable uncertainty in start and duration times. Protocols designed with circular evaluation improve reliability and accuracy in both statistical inference and cryptographic security.

7. Comparison with Classical and Tree-Type Protocols

In contrast to tree-type secret sharing and classical linear statistical procedures, circular evaluation protocols confer several advantages:

• Resource Efficiency: Use of single particles over entangled or product states simplifies hardware requirements in quantum protocols.
• Error Localization: Cyclic transmission allows the source to directly assess round-trip integrity and detect anomalies with high probability.
• Distinct Security Proofs: Transmission order enables differentiated analysis for dishonest participants at different transmission points.
• Integrated Temporal Evaluation: In TKBC settings, joint entity-time filtering ensures fairness and avoids the inflation of predictive metrics.

Such protocols therefore represent a substantial development in both theoretical and experimental domains of circular statistics and circular quantum communication, with documented enhancements in both security and statistical power.