15. Mathematics of Trustworthy and Emerging Computing

Introduction: From Correct Computation to Responsible Computation

Classical computer science often asks whether an algorithm is correct, efficient, and terminates. Contemporary computing must ask additional questions: Can the computation be trusted? Does it preserve privacy? Does it distribute benefits and risks fairly? Can its decisions be explained? Is it secure against adversaries, including adversaries equipped with quantum computers? These questions have transformed mathematical foundations from a study of computation alone into a study of computation embedded in social, organizational, and technological systems.

Trustworthy computing does not mean that a system is infallible. It means that the system’s behavior, limitations, security properties, and social consequences are sufficiently understood, controlled, and governed for its intended use. Mathematics provides the language for this work. Probability quantifies uncertainty; optimization formalizes competing objectives; statistics measures error and disparity; information theory describes leakage; cryptography constructs security guarantees; formal methods establish properties of programs; and quantum theory changes the model of computation itself.

This chapter integrates these areas. It focuses on artificial intelligence, privacy, fairness, explainability, secure computation, post-quantum cryptography, and quantum computing. The emphasis is not merely on definitions, but on relationships and trade-offs. For example, improving privacy may reduce model accuracy; enforcing fairness may alter utility across groups; increasing explainability may require simplifying a model; and deploying a powerful cryptographic protocol may introduce substantial computational cost. At postgraduate level, such trade-offs must be analyzed rather than treated as isolated advantages and disadvantages.

Learning Objectives

After studying this chapter, the learner should be able to:

1. Explain the mathematical foundations of trustworthy artificial intelligence.

2. Distinguish privacy, security, fairness, transparency, interpretability, and explainability.

3. Use probability, statistics, information theory, and optimization to analyze model risk.

4. Describe differential privacy, secure multiparty computation, homomorphic encryption, and zero-knowledge proofs.

5. Compare major fairness definitions and identify when they are mathematically incompatible.

6. Evaluate explanation methods and recognize their limitations.

7. Explain the threat posed by quantum algorithms to classical public-key cryptography.

8. Describe the principles of post-quantum cryptography and quantum computation.

9. Apply a governance framework to an AI-enabled computing system.

10. Analyze ethical and organizational trade-offs in the design and deployment of emerging technologies.

1. A Mathematical View of Trust

A computational system can be represented as a mapping from inputs to outputs. For a deterministic program, this may be written as

\[

f: X \rightarrow Y.

\]

For a machine-learning system, a more realistic representation includes a data distribution, model parameters, and uncertainty:

\[

\hat{Y}=f_{\theta}(X), \qquad (X,Y)\sim P_{X,Y}.

\]

Trust requires more than low average prediction error. A system may achieve high accuracy while failing systematically for a minority population, exposing sensitive information, or producing confident predictions in unfamiliar situations. A useful risk decomposition is

\[

R(f)=\mathbb{E}_{(X,Y)\sim P}[L(f(X),Y)],

\]

where \(L\) is a loss function. In practice, risk is multidimensional. One may define

\[

R_{total}=\alpha R_{accuracy}+\beta R_{privacy}+\gamma R_{security}+\delta R_{fairness}+\epsilon R_{operational},

\]

where the coefficients express the priorities and constraints of a particular application. This expression is not a universal law; it is a governance model that makes assumptions explicit.

Trust is therefore conditional. A medical diagnostic model and a recommendation system may use similar learning algorithms, but their acceptable error rates, explanation requirements, validation procedures, and liability structures differ. Mathematical analysis helps identify these differences rather than hiding them behind a single performance score.

Uncertainty, calibration, and model risk

Accuracy measures the proportion of correct predictions, but it does not indicate whether predicted probabilities are reliable. A classifier is calibrated if, among cases assigned probability \(p\), approximately a proportion \(p\) are positive. Formally,

\[

P(Y=1\mid \hat{P}=p)\approx p.

\]

A model that assigns 0.9 probability to an event should experience that event in roughly 90 percent of comparable cases. Calibration is important in decision-making because users often interpret probabilities as levels of confidence.

Model risk arises when a model is incorrectly specified, trained on unsuitable data, used outside its domain, or interpreted beyond the evidence it supports. Sources include sampling bias, measurement error, concept drift, data leakage, adversarial manipulation, and unjustified causal assumptions. A mathematically sophisticated model can still be organizationally unsafe if its operating conditions are not monitored.

2. Privacy as a Quantifiable Property

Privacy concerns the control and protection of information about individuals or groups. Security protects systems and data against unauthorized actions; privacy also concerns legitimate collection, inference, use, retention, and disclosure. A database can be securely stored yet violate privacy if it collects excessive information or enables sensitive inferences.

Differential privacy

Differential privacy provides a formal guarantee about the effect of one individual’s data on an algorithm’s output. Let \(D\) and \(D'\) be neighboring datasets differing in one person. A randomized mechanism \(M\) is \(\varepsilon\)-differentially private if, for every output set \(S\),

\[

P[M(D)\in S]\leq e^{\varepsilon}P[M(D')\in S].

\]

The parameter \(\varepsilon\), called the privacy budget, controls privacy loss. Smaller values generally provide stronger privacy but may reduce statistical utility. Differential privacy protects against an adversary who may possess arbitrary auxiliary information; it does not require assumptions about what the adversary already knows.

A common mechanism adds calibrated noise to a query. If a numerical query has global sensitivity

\[

\Delta f=\max_{D,D'}\|f(D)-f(D')\|_1,

\]

the Laplace mechanism releases

\[

M(D)=f(D)+Lap\left(\frac{\Delta f}{\varepsilon}\right).

\]

For a counting query, sensitivity is often one. Repeated releases consume a privacy budget. Under sequential composition, mechanisms satisfying \(\varepsilon_1,\ldots,\varepsilon_k\) privacy provide approximately \(\sum_i\varepsilon_i\) privacy under basic composition. This makes privacy an organizational resource that must be allocated deliberately.

Differential privacy is not identical to anonymity, encryption, or access control. Encryption protects data during storage or transmission, while differential privacy limits what can be learned from released results. A system may need all these mechanisms.

Privacy in machine learning

Machine-learning models can memorize training examples and reveal information through model outputs. Privacy-preserving learning approaches include data minimization, federated learning, secure aggregation, differential privacy, and encrypted inference. Federated learning keeps raw data on local devices, but model updates can still leak information. Consequently, federated systems often combine secure aggregation with differential privacy.

The central design problem is utility-privacy optimization. If \(U\) denotes utility and \(P\) denotes privacy protection, deployment may seek to maximize

\[

U-\lambda P_{loss},

\]

subject to legal, ethical, and operational constraints. The parameter \(\lambda\) reflects the cost assigned to privacy loss and should not be chosen solely by engineers.

3. Fairness, Bias, and Statistical Trade-offs

Fairness in computational systems concerns whether outcomes, treatment, and opportunities are acceptable across relevant groups. Let \(A\) be a protected attribute, \(Y\) the true outcome, and \(\hat{Y}\) a prediction.

Demographic parity requires

\[

P(\hat{Y}=1\mid A=a)=P(\hat{Y}=1\mid A=b).

\]

This condition focuses on selection rates. Equal opportunity instead requires equality of true-positive rates:

\[

P(\hat{Y}=1\mid Y=1,A=a)=P(\hat{Y}=1\mid Y=1,A=b).

\]

Equalized odds requires equality of both true-positive and false-positive rates conditional on the actual outcome. Predictive parity requires comparable positive predictive values across groups.

These definitions are not interchangeable. If base rates differ between groups, it is generally impossible to achieve calibration, equalized error rates, and demographic parity simultaneously, except in special cases. This is a mathematical incompatibility, not merely a failure of implementation. Therefore, fairness requires a normative decision about which errors and opportunities matter most in the application.

Bias can enter through the problem definition, sampling, measurement, labels, feature selection, model training, threshold choice, or deployment context. Removing a protected attribute does not necessarily remove discrimination because correlated variables may act as proxies. Conversely, retaining the attribute may enable explicit auditing or group-specific calibration. The correct choice depends on the purpose, law, and governance process.

Fairness metrics should be examined with confidence intervals and subgroup sample sizes. Apparent disparities may be unstable when groups are small, while aggregate equality may conceal intersectional harms. A mature audit reports uncertainty rather than presenting every measured difference as definitive.

4. Explainability and the Limits of Interpretation

Interpretability refers to how readily a human can understand the internal operation of a model. Explainability often refers more broadly to methods that communicate why a particular output was produced. A linear model may offer transparent coefficients, whereas a deep neural network may require post hoc explanations.

Local explanations describe one prediction; global explanations characterize model behavior more generally. Feature attribution methods assign contributions to input variables. In an additive form,

\[

f(x)\approx \phi_0+\sum_{i=1}^{d}\phi_i,

\]

where \(\phi_i\) represents the contribution attributed to feature \(i\). Such attributions can be useful, but they depend on a background distribution, feature dependence assumptions, and the chosen explanation method.

Counterfactual explanations ask what minimal change would alter an outcome:

\[

\min_{x'} d(x,x') \quad \text{subject to } f(x')=y_{desired}.

\]

The distance function should reflect feasible and ethically acceptable changes. Telling a person to change an immutable characteristic is not a meaningful explanation. Counterfactuals can also reveal recourse barriers: two individuals may receive the same adverse decision but require very different changes to obtain approval.

Explanations can be faithful, understandable, actionable, or legally sufficient; these properties may conflict. A simple explanation may be persuasive but inaccurate. A technically faithful explanation may be too complex for a decision-maker. Explanations should therefore be validated for fidelity, stability, completeness, and user comprehension. They must not be treated as proof that a model is fair or causal.

5. Secure Computation and Cryptographic Trust

Cryptography transforms mathematical hardness and carefully defined protocols into security properties. Modern trustworthy computing increasingly requires computation on data that should remain hidden from the computing party.

Secure multiparty computation

Secure multiparty computation allows parties to compute a function over their private inputs without revealing those inputs beyond the agreed output. For parties with inputs \(x_1,\ldots,x_n\), the goal is to compute

\[

f(x_1,\ldots,x_n)

\]

while limiting additional information. Security may be defined against semi-honest participants, who follow the protocol but inspect messages, or malicious participants, who may deviate arbitrarily. Protocols commonly use secret sharing, oblivious transfer, garbled circuits, and zero-knowledge techniques.

Homomorphic encryption

Homomorphic encryption permits computation on ciphertexts. If \(E\) is encryption and \(D\) decryption, an additive homomorphic scheme may satisfy

\[

D(E(m_1)\odot E(m_2))=m_1+m_2.

\]

Fully homomorphic encryption supports arbitrary computations through combinations of additions and multiplications, although present implementations can be computationally expensive. It is attractive for cloud inference because a client can encrypt data, the server can evaluate a model, and the client can decrypt the result.

Zero-knowledge proofs

A zero-knowledge proof enables a prover to demonstrate that a statement is true without revealing the underlying witness. A sound and complete protocol should satisfy:

• completeness: an honest prover convinces an honest verifier of a true statement;
• soundness: a false statement cannot be convincingly proved except with negligible probability;
• zero knowledge: the verifier learns no useful information beyond the statement’s validity.

These properties support identity claims, verifiable computation, and privacy-preserving credentials. The phrase “zero knowledge” does not mean that no information is transmitted; it means that the proof reveals no additional witness information under the security definition.

6. Post-Quantum Cryptography

Quantum computing threatens widely used public-key systems because Shor’s algorithm can solve integer factorization and discrete logarithm problems in polynomial time on a sufficiently capable quantum computer. RSA, Diffie–Hellman, and elliptic-curve cryptography therefore require migration planning. Grover’s algorithm provides a quadratic speedup for unstructured search, affecting symmetric-key security; increasing key sizes can partly compensate.

Post-quantum cryptography develops classical algorithms believed to resist quantum attacks. Major families include lattice-based, code-based, hash-based, and multivariate constructions. Security is based on problems such as learning with errors, decoding random linear codes, or finding preimages of cryptographic hash functions. No security claim is absolute: confidence depends on cryptanalysis, parameter choices, implementation quality, and the absence of side channels.

The “harvest now, decrypt later” threat is especially important. An adversary may record encrypted traffic today and decrypt it when quantum capability becomes available. Organizations holding long-lived sensitive data should therefore assess cryptographic inventory, data lifetimes, protocol dependencies, and migration costs. Cryptographic agility—the ability to replace algorithms without redesigning an entire system—is a governance and engineering requirement.

7. Quantum Computing as a New Mathematical Model

A classical bit is represented by either 0 or 1. A qubit is described by a unit vector in a two-dimensional complex Hilbert space:

\[

|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,

\qquad |\alpha|^2+|\beta|^2=1.

\]

The squared magnitudes determine measurement probabilities. Multiple qubits are represented using tensor products. Two qubits may occupy entangled states that cannot be represented as independent states of each qubit. Quantum gates are unitary transformations, preserving norm, and measurement changes the state probabilistically.

Quantum algorithms exploit superposition, interference, and entanglement. Their advantage does not arise from simply evaluating every possibility and reading all answers simultaneously. Measurement limits the accessible information, so algorithms must arrange interference to amplify desirable outcomes. Shor’s algorithm addresses factoring and discrete logarithms; Grover’s algorithm accelerates search; quantum simulation is expected to be important for physical and chemical systems.

Quantum error correction addresses the fragility of quantum states. A logical qubit is encoded across multiple physical qubits so that certain errors can be detected and corrected without directly measuring the encoded information. Fault-tolerant computation requires error rates below suitable thresholds and substantial overhead. Thus, theoretical speedups do not automatically imply near-term practical superiority.

8. A Governance Framework for Trustworthy Systems

Trustworthy computing requires lifecycle governance rather than a final compliance check. A practical framework can be organized into six linked activities:

1. Purpose and necessity: Define the decision, affected stakeholders, legitimate purpose, and whether automation is necessary.

2. Data and assumptions: Document provenance, consent or legal basis, representativeness, measurement quality, and known limitations.

3. Model and security analysis: Evaluate accuracy, calibration, robustness, privacy loss, fairness metrics, attack surfaces, and cryptographic assumptions.

4. Human and organizational controls: Specify review authority, escalation, contestability, access control, audit logging, and accountability.

5. Deployment monitoring: Track drift, subgroup performance, privacy incidents, security events, explanation quality, and unexpected uses.

6. Retirement and learning: Define conditions for suspension, model replacement, data deletion, and post-incident review.

Documentation should include a model card or equivalent technical report, a data statement, risk assessment, evaluation protocol, intended-use boundaries, and a record of decisions. Independent review is valuable when systems affect rights, employment, education, health, credit, or public safety.

Governance is not opposed to innovation. It makes innovation sustainable by reducing uncontrolled risk and clarifying who is responsible when assumptions fail. Technical teams, legal specialists, domain experts, affected communities, senior management, and auditors may all possess information that the others lack.

An Illustrative Case: Privacy-Preserving Credit Assessment

Background

A consortium of regional banks wishes to improve credit-risk estimation for small businesses. Each bank possesses valuable transaction histories, but commercial confidentiality and privacy obligations prevent direct pooling of raw data.

Situation and problem

The existing models perform well for established firms but poorly for newer firms and businesses in underrepresented regions. The consortium must improve prediction while avoiding discriminatory effects and preventing disclosure of customer information. It also wants to use a cloud provider for computation.

Analysis

The project has several interacting risks. Pooling data may create privacy and re-identification risks. Historical repayment labels may reflect unequal access to credit rather than pure creditworthiness. A model optimized for aggregate loss may reject too many firms in a particular region. Cloud computation creates a confidentiality risk, while explanations are required for adverse decisions.

The consortium defines utility as calibrated default-risk prediction and defines fairness through subgroup analysis of false-positive and false-negative rates, with domain justification for the selected criterion. It recognizes that no single metric proves fairness. It also establishes a threat model distinguishing honest-but-curious cloud operators from malicious participants.

Approach and strategy

The banks first minimize and standardize the features used. They apply secure multiparty computation or secure aggregation to calculate shared statistics without exchanging raw records. Model training uses federated learning supplemented by differential privacy, with a documented privacy budget. Sensitive attributes are retained in a protected audit process so that subgroup performance can be measured, even though they are not necessarily used as predictive features.

The cloud provider receives encrypted or protected intermediate values, and access is divided among organizational roles. A post hoc explanation system provides feature contributions and feasible counterfactuals, while a human credit officer reviews borderline and high-impact cases. The model is calibrated separately only where legally and operationally justified, and all threshold decisions are documented.

Outcome

The resulting system improves evaluation coverage for newer firms without exposing the raw consortium database. Its aggregate accuracy is not the only success criterion: privacy loss, subgroup error rates, calibration, explanation fidelity, and operational review time are reported together. Some disparity remains, leading the consortium to revise data collection and offer an appeal process rather than claiming that the model is unbiased.

Learning points

The case illustrates that trustworthy computing is a systems problem. Privacy technologies cannot correct biased labels; fairness metrics cannot secure a cloud interface; explanations cannot substitute for human accountability; and a technically accurate model may still be unsuitable if affected firms lack recourse. The correct approach combines mathematical guarantees, empirical testing, organizational controls, and stakeholder judgment.

Challenges, Limitations, and Common Errors

A frequent mistake is to optimize a single metric. High accuracy can coexist with poor calibration, privacy leakage, or discriminatory errors. Another mistake is to treat a mathematical guarantee as universal: differential privacy depends on the mechanism and privacy budget; cryptographic security depends on assumptions and implementation; fairness depends on the selected definition and population; and explanations depend on the model and method.

Data drift creates a further challenge. A model validated on one distribution may fail when economic conditions, user behavior, sensors, or institutional policies change. Monitoring should distinguish random variation from structural drift. Security attacks may also be adaptive: an adversary can probe a model, poison training data, exploit weak authentication, or target implementation side channels.

There are difficult resource trade-offs. Privacy-preserving computation may increase latency and energy use. Post-quantum algorithms may require larger keys or signatures. Human review improves accountability but can become a rubber stamp if workload and authority are poorly designed. Explainability may expose attack information or create false confidence. Ethical review can delay deployment, but the cost of unexamined failure may be greater.

Organizations should avoid vague claims such as “the system is unbiased,” “the data are anonymous,” or “the algorithm is secure.” Claims should specify the population, threat model, metric, confidence level, assumptions, and operating conditions.

Current Directions and Research Questions

Research is moving toward unified evaluation of accuracy, robustness, privacy, fairness, and energy consumption. Important topics include private and federated foundation models, verifiable machine learning, robust statistics under distribution shift, causal fairness, confidential computing, quantum-resistant protocol migration, and formal verification of learning-enabled systems.

A major open problem is compositional trust. Individual components may be secure or fair in isolation while their combination creates new risks. Another is governance under uncertainty: organizations must make decisions before standards, evidence, or technology are fully mature. Research is also needed on human-centered explanations, participatory system design, algorithmic recourse, and methods for measuring harms that are not captured by conventional loss functions.

The future of trustworthy computing will therefore depend on both mathematical innovation and institutional capacity. No algorithm can determine the social meaning of fairness, the acceptable level of privacy loss, or the appropriate allocation of responsibility without normative and organizational input.

Best Practices for Professional Practice

• Define intended use, prohibited use, stakeholders, and failure consequences before selecting a model.
• State threat models, privacy parameters, fairness criteria, and cryptographic assumptions explicitly.
• Measure subgroup performance, calibration, robustness, and uncertainty rather than aggregate accuracy alone.
• Use privacy-enhancing technologies in combination with data minimization and access governance.
• Treat explanations as evidence to be validated, not as automatic proof of fairness or causality.
• Maintain cryptographic inventories and design for algorithmic agility.
• Introduce independent review, audit logs, appeals, incident response, and model retirement criteria.
• Monitor systems after deployment for drift, abuse, unequal impact, and changing legal or organizational requirements.

Chapter Summary

Trustworthy and emerging computing extends conventional correctness with requirements for privacy, security, fairness, explainability, robustness, accountability, and social legitimacy. Probability and statistical learning quantify uncertainty and model risk. Differential privacy limits the influence of any individual record, while secure multiparty computation, homomorphic encryption, and zero-knowledge proofs enable protected computation and verification. Fairness is expressed through alternative metrics whose incompatibility often requires normative choices. Explainability supports understanding and recourse but does not guarantee causality or justice. Quantum algorithms threaten important classical public-key systems, motivating post-quantum cryptography and cryptographic agility. Quantum computing itself is grounded in Hilbert spaces, unitary transformations, measurement, entanglement, and error correction. Effective governance combines technical evaluation with documentation, human oversight, monitoring, stakeholder participation, and clear accountability.

Key Terms

Differential privacy: A formal privacy guarantee limiting how much an individual’s participation changes an algorithm’s output.

Privacy budget: A parameter, commonly \(\varepsilon\), that quantifies cumulative privacy loss.

Model risk: The possibility of harm caused by incorrect assumptions, data, implementation, or use of a model.

Calibration: Agreement between predicted probabilities and observed frequencies.

Demographic parity: Equality of prediction or selection rates across protected groups.

Equal opportunity: Equality of true-positive rates across groups.

Equalized odds: Equality of both true-positive and false-positive rates across groups.

Explainability: The provision of information intended to clarify how or why a system produced an output.

Interpretability: The degree to which a model’s operation can be understood by humans.

Counterfactual explanation: An explanation describing the smallest feasible change that would alter an outcome.

Secure multiparty computation: Protocols enabling joint computation while protecting private inputs.

Homomorphic encryption: Encryption that permits specified computations directly on ciphertexts.

Zero-knowledge proof: A protocol proving a statement without revealing its underlying witness.

Post-quantum cryptography: Cryptographic methods designed to resist attacks by quantum computers.

Cryptographic agility: The capacity to replace cryptographic algorithms with limited system disruption.

Qubit: The basic unit of quantum information, represented as a normalized complex state vector.

Superposition: A linear combination of basis states describing a quantum state before measurement.

Entanglement: A non-separable correlation between quantum systems.

Quantum error correction: Methods for protecting logical quantum information from physical errors.

Algorithmic recourse: The ability of an affected person to pursue feasible changes or challenge a decision.

Review Questions

Short Answer Questions

1. Why is calibration important in a risk-sensitive prediction system?

2. State the formal idea behind differential privacy.

3. Distinguish demographic parity from equal opportunity.

4. What is the purpose of a zero-knowledge proof?

5. Why does quantum computing create a need for cryptographic agility?

Long Answer Questions

1. Explain how privacy, fairness, and utility can conflict in machine-learning system design.

2. Describe differential privacy, including sensitivity, privacy budget, and composition.

3. Compare secure multiparty computation, homomorphic encryption, and zero-knowledge proofs.

4. Explain why fairness criteria may be mathematically incompatible when group base rates differ.

5. Discuss the mathematical foundations and practical limitations of quantum computing.

Analytical / Application Questions

1. A hiring model has equal selection rates across two groups but substantially different false-positive rates. Which fairness criterion is satisfied, which may be violated, and what additional evidence should be collected?

2. Design a privacy and governance strategy for a hospital consortium that wants to train a shared diagnostic model without pooling raw patient records.

3. An organization stores encrypted legal records that must remain confidential for fifty years. Analyze the risks from quantum computing and propose a migration plan.

22. Multiple Choice Questions with Answers

Question 1

A model assigns a probability of 0.8 to a group of cases. Which observation best supports calibration?

A. Exactly 80 percent of those cases have the positive outcome

B. The model has 80 percent accuracy overall

C. The model has equal selection rates across groups

D. The model uses eight input features

Correct Answer: A. Exactly 80 percent of those cases have the positive outcome

Question 2

What does epsilon in epsilon-differential privacy primarily control?

A. The number of features in a model

B. The privacy loss associated with an individual’s data

C. The encryption key length

D. The model’s classification accuracy

Correct Answer: B. The privacy loss associated with an individual’s data

Question 3

A system has equal true-positive rates across protected groups but unequal selection rates. Which conclusion is most justified?

A. It satisfies equal opportunity, but demographic parity may not hold

B. It satisfies all fairness definitions

C. It satisfies demographic parity but not equal opportunity

D. It cannot be evaluated without removing the protected attribute

Correct Answer: A. It satisfies equal opportunity, but demographic parity may not hold

Question 4

Which technique is specifically designed to compute a joint function while keeping each party’s input private?

A. Model calibration

B. Secure multiparty computation

C. Demographic parity

D. Quantum measurement

Correct Answer: B. Secure multiparty computation

Question 5

What is the main limitation of treating a feature-attribution explanation as proof of fairness?

A. Attributions cannot be computed for any model

B. An explanation may describe model behavior without establishing equitable outcomes

C. Fairness never involves statistical analysis

D. Feature attributions always reveal raw training data

Correct Answer: B. An explanation may describe model behavior without establishing equitable outcomes

Question 6

Why are RSA and elliptic-curve cryptography threatened by a sufficiently capable quantum computer?

A. Quantum measurement automatically reveals private keys

B. Shor’s algorithm efficiently addresses the underlying factoring or discrete-logarithm problems

C. Grover’s algorithm breaks every symmetric and asymmetric system completely

D. Quantum computers eliminate the need for authentication

Correct Answer: B. Shor’s algorithm efficiently addresses the underlying factoring or discrete-logarithm problems

Question 7

Which statement best describes homomorphic encryption?

A. It permits selected computations on ciphertexts whose result can later be decrypted

B. It guarantees equal error rates across demographic groups

C. It proves a statement without transmitting any protocol messages

D. It converts a classical bit into a qubit

Correct Answer: A. It permits selected computations on ciphertexts whose result can later be decrypted

Question 8

In the illustrative credit-assessment case, why were protected demographic attributes retained for auditing?

A. To guarantee that the model would be fair

B. To measure subgroup outcomes and detect disparities

C. To increase the encryption key size

D. To eliminate the need for human review

Correct Answer: B. To measure subgroup outcomes and detect disparities

Question 9

Which statement about fairness criteria is most accurate when group base rates differ?

A. All major criteria can always be satisfied simultaneously

B. Some criteria can be mathematically incompatible, requiring a normative choice

C. Fairness can be established by omitting protected attributes

D. Only aggregate accuracy is relevant

Correct Answer: B. Some criteria can be mathematically incompatible, requiring a normative choice

Question 10

What is the primary purpose of quantum error correction?

A. To increase the number of classical training examples

B. To protect logical quantum information from physical errors

C. To enforce demographic parity

D. To reduce differential privacy loss

Correct Answer: B. To protect logical quantum information from physical errors