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Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices

Carl A. Miller, Yaoyun Shi

arXiv:1402.0489v4quant-ph

TL;DR

Secure, abundant randomness matters, but conventional generators derive outputs deterministically from seeds. This work analyzes randomness expansion using untrusted devices and reports cryptographic-level security, tolerance of constant imprecision, and robust unbounded expansion with two devices.

  • Problem

    Securely generating abundant, high-quality randomness matters, while typical generators deterministically process seeds to produce outputs.

    Source: §1.1 · p. 3

  • Method

    The paper analyzes a one-shot randomness-expansion protocol using untrusted devices and proves security against general quantum adversaries.

    Source: §1.3 · p. 5

  • Results

    A device winning the GHZ game with probability at least 0.985 achieves exponential randomness expansion with probability approaching 1; output distinguishability is negligible, and robust unbounded expansion uses two devices.

    Source: §1.3 · pp. 5–6

  • Takeaways & Limitations

    For distributed key generation, Protocol Rkd adapts Protocol R and has exponentially small soundness and completeness errors for an appropriate parameter range.

    Source: §10.2 · p. 51

    The device count T(n,1/ε) grows super-polynomially for μ<1 or polynomially for μ=1, constraining achievable error quality.

    Source: §1.5 · p. 11

Abstract

from arXiv · show

Randomness is a vital resource for modern day information processing, especially for cryptography. A wide range of applications critically rely on abundant, high quality random numbers generated securely. Here we show how to expand a random seed at an exponential rate without trusting the underlying quantum devices. Our approach is secure against the most general adversaries, and has the following new features: cryptographic level of security, tolerating a constant level of imprecision in the devices, requiring only a unit size quantum memory per device component for the honest implementation, and allowing a large natural class of constructions for the protocol. In conjunct with a recent work by Chung, Shi and Wu, it also leads to robust unbounded expansion using just 2 multi-part devices. When adapted for distributing cryptographic keys, our method achieves, for the first time, exponential expansion combined with cryptographic security and noise tolerance. The proof proceeds by showing that the Renyi divergence of the outputs of the protocol (for a specific bounding operator) decreases linearly as the protocol iterates. At the heart of the proof are a new uncertainty principle on quantum measurements, and a method for simulating trusted measurements with untrusted devices.

1 Background and Summary of Results

This section presents protocols for robust randomness expansion and key distribution using untrusted quantum devices, including cryptographic security, noise tolerance, and small-memory honest implementation. It also covers applications to unbounded expansion, randomness amplification, and quantum key distribution, plus a GHZ-game noise threshold for positive-rate randomness generation.

1.1 The Problem and Its Motivations

  • Classical generators cannot increase randomness, rely on computational assumptions and limited adversaries, and require trust in their implementation, leaving users seeking minimal trust without a satisfactory solution.Insufficient entropy has enabled attacks on DSA secret keys, including keys of over 1% of scanned SSH hosts.Source: §1.1 · pp. 3–4

1.2 Related Problems

  • Untrusted-device randomness expansion belongs to device-independent quantum cryptography and is distinct from randomness amplification, which aims to turn a weak source into near-perfect randomness without additional randomness.The companion work [Chung et al., 2014] studies amplification.Source: §1.2 · p. 5

1.3 Overview of Our Results

  • The protocol tolerates constant noise, including GHZ-game devices winning with probability at least 0.985, needs only constant-size quantum memory per component, and accepts any strongly self-testing binary XOR game.The allowed games include CHSH and GHZ; the memory condition lets components interact between rounds and maintain entanglement for only one game.Source: §1.3 · pp. 5–6

1.4 Related Works

  • Unlike, whose expansion proof is described as non-robust and not cryptographically secure, this work’s QKD adaptation is robust and uses a polylogarithmic seed, improving on the linear seed in.Robust DI-QKD with full security had already been achieved in.Source: §1.4 · p. 6, §1.3 · p. 6

1.5 Technical Statements

  • For any n-player strong self-test, Protocol R yields at least (1−δ)N extractable bits with soundness error K exp(−bqN), while tolerating η′ noise with completeness error exp(−(η−η′)^2qN/3).The theorem requires q and η to be at most positive constants q0 and η0, and 0 < η′ < η.Source: §1.5 · p. 9
  • For any ω ∈ (0, µ), Protocol R with a quantum-proof extractor converts k bits into 2^{k^{µ−ω}} uniform bits with soundness and completeness errors exp(−Ω(k^ω)).With running time T = Θ(2^{k^{µ−ω}}), choosing ω = 1 + λµ for λ > 1 makes the errors exp(−Ω(log^λ T)), negligible in T.Source: §1.5 · p. 9
  • With O(log∗N) iterations and devices, Protocol R expands k uniform bits to N bits with exp(−Ω(k^ω)) error under constant noise; cross-feeding reduces the device count to 2.The two-device result follows by combining the protocol with Chung et al.’s Equivalence Lemma.Source: §1.3 · p. 6, §1.5 · p. 10
  • With Chung et al. [Chung et al., 2014], the protocol converts an (n, k) source into arbitrarily long near-perfect randomness under constant noise, with errors ǫ = exp(−O(k^{ν2})) and 2^{O(log^{1/ν}(n/ǫ))} devices.This guarantee applies for sufficiently large k, n = exp(O(k^{ν2})), and ν ∈ [1/2, µ]; the device count does not depend on output length.Source: §1.5 · p. 10
  • Improving T is an open problem framed as broadening the achievable error quality, not necessarily improving computational efficiency.For output length N, ε = exp(−O(logν N)) still permits a polynomial number of devices; the passage identifies inverse-polynomial or cryptographic error as settings where improving T would broaden applicability.Source: §1.5 · p. 11
  • Directly adapting the expansion protocol for key distribution achieves randomness expansion and key distribution together while reducing the number of untrusted devices from two to one.The alternative is to expand Alice’s randomness first and then use it in the untrusted-device QKD protocol of Vazirani and Vidick.Source: §1.5 · p. 11
  • Protocol Rkd produces a key of rN extractable bits and tolerates any constant noise level η′ < η, with completeness error exp(−Ω((η −η′)2qN)).For any strong self-test, the stated result holds for infinitely many N and q ≤ q0; the initial randomness is O(Nh(q) + log N), and the running time is polynomial in N.Source: §1.5 · p. 11
  • With q = (log2 N)/N, the key-distribution protocol distributes Ω(N) extractable bits from an O(log3 N)-bit seed with error terms achieving cryptographic security.Composing it with a quantum-proof randomness extractor using a polylogarithmic seed also yields untrusted-device QKD from a polylogarithmic seed.Source: §1.5 · p. 11

1.6 Numerical Results

  • For the GHZ game, vG ≥ 0.14 implies a positive linear extraction rate when η < 0.0154.More generally, Protocol R produces π(η/vG)N extractable bits per round when η < vG/2, and this rate is positive when η < 0.11 · vG.Source: §1.6 · p. 12

2 Overview of Proofs

The proof combines a Rényi-divergence induction with an uncertainty principle and a simulation that reduces security against untrusted devices to security for partially trusted ones. A performance-weighted bound supports extractable-randomness guarantees, while the key-distribution argument uses game scores and information reconciliation.

2.1 Quantum Rényi Entropies

  • The proof inductively bounds output Rényi divergence conditioned on the adversary, then uses smooth min-entropy to lower-bound extractable bits.The resulting lower bound for Protocol A′ transfers to Protocol R through the stated reduction.Source: §2.1 · p. 12, §2.3 · p. 14
  • The Rényi uncertainty principle gives dimension-independent constraints for measurements on a qubit entangled with another system.Its proof uses a known matrix inequality for the (2 + 2ε)-Schatten norm.Source: §2.1 · p. 12

2.2 Partially Trusted Measurement Simulation

  • The protocol’s input-output state can be simulated by a partially trusted device, reducing Protocol R’s security proof to Protocol A′.The simulation preserves the joint state with any purifying system up to isomorphism.Source: §2.2 · p. 13

2.3 An Induction Proof With a Weighted Measure of Randomness

  • A weighted bounding operator adds weight to game-loss events, and a uniform negative Rényi-divergence bound enables induction.The weighting lowers the randomness expectation according to device performance.Source: §2.3 · pp. 13–14

2.4 Quantum Key Distribution

  • When the protocol succeeds at sufficiently low noise tolerance, Azuma’s inequality shows the device scores well in both game and generation rounds.The resulting strings differ in at most (1/2 − λ)N places, followed by efficient information reconciliation adapted from and.Source: §2.4 · p. 14

3 Preliminaries

The preliminaries establish notation and operator conventions, define the Rényi-divergence and smooth-min-entropy measures used for randomness, and formalize the quantum devices and simulations used in the paper.

3.1 Notation

  • The notation section fixes sequence and operator conventions, including direct sums, Schatten norms, and base-2 versus natural logarithms.It also identifies h as the Shannon entropy function.Source: §3.1 · pp. 14–15

3.2 Quantum Rényi Divergence

  • The paper uses quantum Rényi divergence to quantify randomness conditioned on quantum side information and relates it to smooth min-entropy, which measures extractable bits.The conditional measure is expressed using −Dα(ρAB∥IA ⊗ σB).Source: §3.2 · pp. 15–16

3.3 Quantum Devices

  • A multi-component device applies transcript-dependent joint unitaries and local binary measurements; any binary device can be simulated in canonical form.The device and protocol simulation definitions require isomorphic joint output-and-purifying-system states.Source: §3.3 · p. 18

4 An Uncertainty Principle

This section studies how measurements on a qubit entangled with another system affect Rényi-entropy quantities. It presents an uncertainty principle for the resulting measurement states and develops its proof using a matrix inequality.

4 An Uncertainty Principle

  • Computational- and Hadamard-basis measurements on an entangled qubit produce subnormalized states of the other system used in the uncertainty analysis.The section studies the map ρ ↦ Tr[ρ^(1+ε)] in this measurement setting.Source: §4 · p. 19
  • The uncertainty principle gives uniform constraints on quantities arising from qubit measurements entangled with a system of arbitrary dimension.The theorem states the existence of a continuous function Π governing the bounds.Source: §4 · p. 19
  • The expression on the left side is the difference in (1 + ǫ)-Renyi entropy between the state ρ+ ⊕ρ− and the state ρ, and the desired limiting condition follows using L’Hospital’s rule.Source: §4 · p. 20

5 The Self-Testing Property of Binary Nonlocal XOR Games

This section relates the geometry of binary XOR game score functions to self-testing strength and develops scoring-operator decompositions for aligned self-tests. It gives decompositions under winnability and strong self-testing assumptions, including a theorem for positively aligned strong self-tests with canonical-form measurements.

5.1 Definitions and Basic Results

  • A game is a self-test exactly when ZG has a maximum with no player angle a multiple of π and all maxima are equivalent up to sign modulo 2π.The definition identifies a self-test as having a unique optimal qubit strategy up to local unitaries.Source: §5.1 · pp. 21–23
  • A self-test is strong exactly when ZG has nonzero Hessian matrices at all its maxima, equivalently giving quadratic control of distance to a maximum.The supplied passage states an equivalent distance bound for points whose ZG value is within ε of its maximum.Source: §5.1 · p. 23
  • For positively aligned strong self-tests, Proposition 5.8 and Corollary 5.9 constrain the image of the semicircle domain and yield a disk-based inequality.The supplied passages omit the displayed boundary curve and inequality, so their precise forms cannot be stated here.Source: §5.1 · p. 24

5.2 Decomposition Theorems

  • Theorem 5.10 gives a constant-gap scoring-operator decomposition for winnable, positively aligned self-tests satisfying its stated additional condition.The supplied statement does not include the decomposition formula or the full additional condition.Source: §5.2 · p. 25
  • Theorem 5.11 establishes a scoring-operator decomposition for positively aligned strong self-tests, with constants δG > 0 and |α| = 1.The supplied theorem statement omits the decomposition formula.Source: §5.2 · p. 26
  • For canonical-form strategies with underlying space (C2 ⊗ W1) ⊗ … ⊗ (C2 ⊗ Wn), Theorem 5.12 gives constants δG > 0 and |α| = 1 for decomposing the scoring operator, with the remaining phase coefficients arbitrary unit-length complex numbers.The theorem assumes G is a positively aligned strong self-test.Source: §5.2 · p. 28

6 Randomness Expansion with Partially Trusted Measurements

This section defines randomness-expansion protocols for trusted and partially trusted measurement devices, then relates the partially trusted protocol to Protocol R. The simulation result transfers security guarantees from Protocol A’ to Protocol R.

6.1 Devices with Trusted Measurements

  • The trusted-measurement protocol uses a biased coin flip g directly as the device input, rather than generating random inputs for game rounds.Its measurements are assumed to perfectly anti-commute.Source: §6.1 · p. 29

6.2 Devices with Partially Trusted Measurements

  • On input 1, a partially trusted device selects T(1), N, or 0 with probabilities v, 1 − v − h, and h, respectively.Here v is the trust coefficient and h the coin flip coefficient; h measures how much the output is determined by a fair coin flip.Source: §6.2 · p. 29

6.3 Entanglement with a Partially Trusted Measurement Device

  • For an entangled system E, the proof bounds the subnormalized state ρP by ρP ≤ (1 − h/2)ρ − vρ1.ρP represents input 1 and output 0; ρ1 is the state if trusted measurement T(1) were applied.Source: §6.3 · p. 32

6.4 Simulation

  • For any positively aligned strong self-test game, Protocol A’ with a partially trusted device simulates Protocol R; the security result also extends beyond positively aligned games.The device parameters are (δG, 2fG).Source: §6.4 · pp. 32–33

7 The Proof of Security for Partially Trusted Devices

This section completes the security proof for Protocol A′ with partially trusted devices, establishing a device-independent linear entropy-rate bound by selecting parameters and minimizing a convex expression.

7.1 Proof Idea

  • The proof discounts randomness in failing game rounds using a 2^-κ coefficient, then applies induction and the Renyi-divergence-to-smooth-min-entropy relationship to bound extractable bits.The unadjusted randomness measure treats passing and failing rounds alike, despite the device’s limited number of allowed failures.Source: §7.1 · pp. 35–36

7.2 One-Shot Results

  • The one-shot results first assume extra information about the device, then give a bound without that information or reliance on its failure parameters.For the first result, the limiting Λ value is π(t0) + h/2 + vt0.Source: §7.2 · pp. 36–38

7.3 Multi-Shot Results

  • Induction extends the one-shot bound across iterations, and a revised bound avoids an unusable divergence term that could be arbitrarily large.The section then uses the multi-shot bound to prepare an operator classical with respect to E and G for the security proof.Source: §7.3 · pp. 38–39

7.4 The Security of Protocol A’

  • Theorem 7.7 gives functions T and F for Protocol A′’s security bound; T is device-independent and tends to π(η/v) as q and κ approach zero.The section identifies T as a lower bound on the protocol’s linear entropy rate, uniform over the particular device used.Source: §7 · p. 34, §7.4 · p. 41
  • For Protocol A’, setting T(v, h, η, q, κ) = R(v, h, η, q, κ, r) and F = 2/r gives (7.44); convexity minimizes (7.51) at s = η/v, yielding π(η/v).The proof also states that equality (7.46) follows immediately.Source: §7.4 · p. 41

8 Randomness Expansion from an Untrusted Device

Protocol R expands randomness using strong self-tests, with security guarantees, a GHZ-game example, and a bound on abort probability under noise. Its output can approach a linear number of extractable bits with exponentially small soundness error.

8.1 The Trust Coefficient of a Strong Self-Test

  • For a strong self-test G, the trust coefficient is the largest δ_G permitting simulation by a partially trusted device with parameters (δ_G, 2f_G).It quantifies the simulation guarantee used in deriving Protocol R's security.Source: §8.1 · p. 42

8.2 The Security of Protocol R

  • N·(1−δ) extractable bits are produced with soundness error K·2^-bqN for suitable parameters and q≤q0.This guarantee holds for any strong self-test G and δ>0.Source: §8.2 · p. 45

8.3 Example: The GHZ game

  • The GHZ game has trust coefficient at least 0.14, and for η<0.0154 Protocol R achieves a positive linear entropy rate when q is sufficiently small.The threshold is 0.11 times the stated trust-coefficient bound.Source: §8.3 · pp. 45–46

8.4 Completeness

  • If the device noise level η′ is below the protocol threshold η, the abort probability is at most exp(−(η−η′)^2qN/3).A device with noise level β has expected score at least w_G−β.Source: §8.4 · p. 47

9 Unbounded Expansion

The section establishes sequential composition for strong untrusted-device extractors, including a two-device cross-feeding construction. The composition lemma bounds the combined errors by sums of the component errors.

9 Unbounded Expansion

  • A cross-feeding protocol alternates between two devices, using each extractor's output as the next extractor's input.The device used at step i is D_(i mod 2).Source: §9 · p. 49
  • For two-device composition, communication from an active device to the inactive device is forbidden until the latter becomes active.Device-adversary operations can replenish consumed entanglement in an honest implementation.Source: §9 · p. 49
  • The composed protocol's soundness error and adjustment completeness error are bounded by the respective sums of component errors under a common noise tolerance.The lemma assumes device-uniform inputs; it also applies to global-uniform inputs when the extractors have no device-adversary interactions.Source: §9 · pp. 49–50

10 Untrusted-device Quantum Key Distribution

This section develops untrusted-device QKD through Protocol Rkd and its soundness analysis, including error control beyond strong self-tests and information reconciliation. It establishes soundness for every rate below a positive threshold, preserves a linear extractable-key yield, and bounds random-bit use.

10.2 The Protocol Rkd

  • Protocol Rkd adapts Protocol R to untrusted-device key distribution, with correctness established by bounding raw-key errors and then reconciling the parties’ keys.The reconciliation step lets Bob correct differences from Alice’s key with small failure probability.Source: §10 · p. 50, §10.2 · p. 51

10.1 Definitions

  • An ME-UD-KD protocol uses device-independent shared randomness and authenticated public communication, with soundness requiring extractable bits and matching keys and completeness bounding honest-device aborts.The protocol outputs a public success-or-abort bit and private strings for Alice and Bob.Source: §10.1 · pp. 50–51

10.2 The Protocol Rkd

  • Protocol R has soundness and completeness error exp(−Ω(qN)); in the ideal state, Alice and Bob’s keys differ in at most a (1/2−λ) fraction, which reconciliation corrects with small failure probability.The proof separates the raw-key error bound from the information-reconciliation step.Source: §10.2 · p. 51

10.3 Error Rate

  • The error analysis extends beyond strong self-tests to f-self-testing-in-probability games, where near-optimal overall winning probability guarantees high winning probability on a designated input.For a strategy winning with probability (1−θ)wG, the designated input is won with probability at least (1−f(θ))wG.Source: §10.3 · p. 51

10.4 Efficient Information Reconciliation

  • The reconciliation design targets efficient recovery when errors stay below one-half, while minimizing communication and shared randomness because communication reduces the min-entropy guarantee.The authors present this subsection as a synthesis of known results rather than an original contribution.Source: §10.4 · p. 55
  • Efficient information reconciliation handles key differences up to (1/2−λ)N with R=O(log(N/ε)) shared bits, M≤(1−c)N+O(log(1/ε)) communication, and T=poly(N, log(1/ε)) computation.The protocol uses binary linear error-correcting codes and approximate universal hashing; Proposition 10.11 establishes its correctness.Source: §10.4 · pp. 55–56
  • For error rates below 1/4, uniquely decodable codes avoid shared randomness, while list-decodable codes can offer higher rate at the cost of randomness.For λ>1/4 with a uniquely decodable code, EIR succeeds with certainty and needs no randomness.Source: §10.4 · p. 56
  • Protocol EIR reconciles binary strings X,Y of length N when |X ⊕ Y| ≤ (1/2 − λ)N.Source: §10.4 · p. 57

10.5 The Security of Protocol Rkd

  • rG = Ω((wG − 1/2)^3) > 0, and every rate r < rG is achievable using an explicit list-decodable-code-based reconciliation protocol.The code family has polynomial list size.Source: §10.5 · p. 58
  • After information reconciliation, Alice has at least rN extractable bits, and the final state is ε′-close to a mixture of an aborting state and a state with that yield.The proof chooses q small enough that O(q) ≤ δ/3, with r = rG − δ.Source: §10.5 · p. 58
  • The soundness proof also bounds the probability that the protocol does not abort while Alice and Bob’s keys disagree.Source: §10.5 · pp. 58–59
  • O(Nh(q) + log N) random bits suffice when qN = Ω(1).The expansion protocol uses O(Nh(q)) bits, while reconciliation uses O(log N/ε) bits for ε = exp(−qN).Source: §10.5 · p. 59

11 Further Directions

The discussion identifies rate and scope extensions, broader security models, and seed/output tradeoffs as open directions. It also flags risks from more complex protocol composition.

11 Further Directions

  • Improving Protocol R’s certified rate and Protocol Rkd’s key rate depends on computing a game’s trust coefficient and improving the limiting function π(x).The authors also suggest that the hybrid technique of Vazirani and Vidick may extend to general binary XOR games.Source: §11 · p. 59
  • Extending Theorem 1.1 beyond strong self-tests would require identifying proof-relevant features or robust self-testing conditions beyond binary XOR games.The authors also identify automatic lifting from classical to quantum security as an open problem.Source: §11 · p. 59
  • The authors propose exploring security based on physical principles beyond quantum mechanics, including non-signaling and information causality.Source: §11 · p. 59
  • The protocols require initial perfect randomness; the Chung-Shi-Wu protocol [Chung et al., 2014] accepts arbitrary min-entropy sources and constant noise, but uses many more non-communicating devices.Handling non-uniform inputs remains an open problem.Source: §11 · p. 59
  • The unbounded protocol can use a fixed seed length that determines its error, unlike classical extractors, which require Ω(log N/ε) seed bits for N output bits and error ε.The authors pose the maximum extractable randomness per device entanglement and the expansion-versus-communication tradeoff as open questions.Source: §11 · p. 60
  • Security under compositions more complex than cross-feeding remains open because reused devices may retain prior-run information, potentially creating memory attacks in sequential QKD.The authors note that this appears not to be a problem for sequential compositions of their randomness-expansion protocol.Source: §11 · p. 60

A.1 The Canonical Form for Two Binary Measurements

The appendix establishes a canonical-form result for two binary measurements by embedding a pair of Hermitian contractions into a larger space. Its proof proceeds through intermediate operator forms and a two-dimensional case.

A.1 The Canonical Form for Two Binary Measurements

  • Any pair of Hermitian contractions admits a unitary embedding into a larger space where the operators take the theorem’s canonical form, while compression recovers the original pair.Source: §A.1 · p. 60
  • The proof reduces the original operator pair through conditions (A) and (B), with (B) imposing a uniform diagonal block form.The lemmas establish each embedding step, including the reduction from (A) to (B) and from (B) to the theorem’s form.Source: §A.1 · pp. 60–61
  • For the two-dimensional case, a suitable basis makes one measurement antidiagonal, completing the canonical-form proof.Source: §A.1 · p. 62

A.2 Smooth Min-entropy and Renyi Divergence

This appendix supplies supporting proofs for Section 3.2, including an operator approximation result and its preservation of classical-quantum structure. The argument also uses data processing for Rényi divergence.

A.2 Smooth Min-entropy and Renyi Divergence

  • For a density operator ρ supported within the support of positive semidefinite σ, Proposition A.5 constructs a positive semidefinite subnormalized ρ′ bounded above by σ and close to ρ in trace norm.Source: §A.2 · p. 62
  • The proof applies data processing for Dα to a two-outcome projection operation and then uses monotonicity to obtain the stated bound.Source: §A.2 · pp. 62–63
  • When ρ and σ are classical-quantum operators on a bipartite system, the approximation ρ′ can also be chosen classical-quantum.Source: §A.2 · p. 63

A.3 Variables and Functions Used in Section 7

This appendix subsection collects functions and limit assertions used in the security proof for Protocol A′. It also records shorthand notation and several parameter descriptions.

A.3 Variables and Functions Used in Section 7

  • The subsection gathers the variables, functions, and limit assertions used in Protocol A′’s security proof.The authors describe the list as a convenience, noting that its statements also appear in the paper’s body.Source: §A.3 · p. 63
  • The list describes a multiplier for the Rényi coefficient and an error parameter for smooth min-entropy.It also specifies that A is a classical register in the cited footnote’s register form.Source: §A.3 · p. 63
  • The notation uses γ and r as shorthand for functions whose full arguments are γ(q, κ, r) and r(v, η, q, κ).The passage says both functions are written simply as γ and r below.Source: §A.3 · p. 63
  • Λ is defined as −log(λ), while Δ is the minimum of Λ over s in [0,1], and T is set equal to R.These definitions are given for the listed parameter arguments; the passage also states a limit for Λ as (q, κ, t) approaches (0, 0, t0).Source: §A.3 · pp. 64–65

A.4 Mathematical Results

This appendix section presents mathematical propositions and their proofs. One proof uses logarithms, while another uses compactness to control minima near a boundary point.

A.4 Mathematical Results

  • Proposition A.7 concerns continuous functions near a point in the closure of U, and its proof takes logarithms of both sides of (A.20).The supplied proposition passage gives the setup, while the proof passage identifies this logarithmic step.Source: §A.4 · pp. 65–66
  • Proposition A.8 uses compactness of V and continuity to show that minima near z are controlled by the minimum of the extended function at z.The proof extends f continuously to the boundary point, uses a finite cover of V, and concludes the stated equality (A.21).Source: §A.4 · p. 66

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