Information-Theoretic Security Framework

• Information-Theoretic Security is a formalism that uses entropy, mutual information, and statistical distances to quantify and guarantee secrecy beyond computational limits.
• It applies to both classical and quantum protocols, including key distribution, wiretap coding, and secure multiparty computations for robust, unconditional security.
• The framework facilitates precise trade-offs between key size, throughput, and leakage, grounded in metrics like Shannon entropy and min-entropy.

An information-theoretic security framework is a rigorous formalism for achieving, analyzing, and parameterizing cryptographic and security properties such that guarantees are grounded in information and entropy measures—rather than computational assumptions or conjectures about adversary power. This approach quantifies secrecy, robustness, authenticity, and privacy using tools such as Shannon entropy, min-entropy, mutual information, statistical distance, and channel capacity. Information-theoretic security (ITS) is characterized by proofs that hold against adversaries with unbounded computational and memory resources, and often admit precise, quantitative trade-offs between key size, throughput, leakage, and protocol complexity. In both classical and quantum cryptographic domains, the information-theoretic paradigm applies from key distribution and wiretap coding to robust multiparty computation, physical-layer security, secure representation learning, key-fusion, and beyond.

1. Core Definitions and Formal Security Criteria

• Entropy Measures Classical R\'enyi entropy of order $\alpha$ for a random variable $X$ over alphabet $\mathcal X$:

$$H_\alpha(X) = \frac{1}{1-\alpha}\log\left(\sum_{x\in\mathcal X}P_X(x)^\alpha\right)$$

Special cases: $H(X)=\lim_{\alpha\to1}H_\alpha(X)$ (Shannon), $H_2(X)=-\log\sum_xP_X(x)^2$ (collision), $H_\infty(X)=-\log\max_xP_X(x)$ (min-entropy).

Quantum extension for density operator $\rho$:

$$S_\alpha(\rho) = \frac{1}{1-\alpha}\log\,\mathrm{Tr}(\rho^\alpha),\qquad S_\infty(\rho) = -\log\lambda_{\max}(\rho)$$

• Security Notions
  • Perfect secrecy: $I(M;C)=0$ (statistical independence of message and ciphertext) (Tyagi et al., 2014).
  • Semantic security: $\max_{P_A} I(A;Z,S)$ or $\max_{a,a'}\|P_{Z,S|A=a} - P_{Z,S|A=a'}\|_1$ is negligible (Wiese et al., 2021).
  • Composable security: Protocol’s distinguishing advantage over an ideal functionality is negligible (Xu et al., 28 Aug 2025).
  • Adversarial indistinguishability in quantum settings: For all (possibly unbounded) adversaries,

  $$|\Pr[C_n(E(pk,x))=1] - \Pr[C_n(E(pk,y))=1]| < 1/p(n)$$

  for all large $n$, where $C_n$ is any quantum circuit, $E$ the encryption algorithm (Pan et al., 2010).

• Secrecy Capacity and Leakage:

  • Wiretap secrecy capacity: For channel $X \to Y_b, Y_e$,

  $$C_s = \sup_{p(x)} \left[ I(X; Y_b) - I(X; Y_e) \right]^+$$

  where $I(\cdot;\cdot)$ is mutual information (Forouzesh et al., 2018, Tyagi et al., 2014). - Key agreement security: For secret key $K$, and adversary’s side information $Z$,

  $\| P_{KZ} - U_K \otimes P_Z \|_1 \leq \delta$

  (statistical distance to idealizes uniformity) (Tyagi et al., 2014).

2. Entropy Amplification and Secrecy Preservation Mechanisms

• Entropy-Preserving Aggregation

  • For independent $s_i \in \{0,1\}^m$, each with $H_\infty(s_i) \geq \gamma$, the XOR $S = \bigoplus_{i=1}^n s_i$ satisfies

  $H_\infty(S) \geq \max(0, n\gamma - (n-1)m)$

  (Xu et al., 28 Aug 2025). - Under quantum side-information $\rho_E$ of max-entropy $S_0(\rho_E)$,

  $$S_\infty\left(S \mid E\right) \geq n\gamma - (n-1)m - S_0(\rho_E)$$

  (Xu et al., 28 Aug 2025).

• Privacy Amplification by Universal Hashing

  • Given $X$ of min-entropy $H_\infty(X)$, universal hash functions $f_S$ produce $K=f_S(X)$ with

  $$\| P_{KS} - U_K \otimes P_S \|_1 \leq \varepsilon + \frac{1}{2}2^{(k-H_\infty^\varepsilon(X))/2}$$

  (Tyagi et al., 2014).

• Confidentiality-Preserving Verification (Polynomial Commitments, Shamir Sharing)

  • Shamir secret sharing hides each $s_i$; commitments $c_i = \mathcal H(s_i \| \hat H_i)$ guarantee any forgery or substitution is detectable except with negligible probability in quantum queries and min-entropy (Xu et al., 28 Aug 2025).

3. Canonical Models and Protocol Architectures

Setting	Channel Model/Structure	Security Metrics/Guarantees
Wiretap Channel	DMC/AWGN/Compound/Degraded MIMO	Secrecy capacity, mutual information leakage
Key Agreement	Noisy correlated randomness, public transcript	Statistical distance, min-entropy
Secret Sharing	Compound wiretap channel, MIMO BC	Secrecy capacity region (layered decoding)
Quantum Protocols	Entanglement-swapping, teleportation	Trace-distance, Holevo accessible info
Physical-Layer Security	Jitter, noise sources (KLJN/PLKG)	Energy balance, statistical indistinguishability
Secure Representation	Encoder $X \to Z$ with MI regularizers	$I(Z;U)$ privacy, $I(X;Z|U)$ utility, robustness (Zhang et al., 2024)

Notable Examples:

• KLJN protocol: Passive eavesdropping is bounded by Second Law of Thermodynamics; active attacks trigger immediate alarms thanks to instantaneous public comparison of current/voltage (Mingesz et al., 2012).
• Quantum public-key encryption: Trace-distinguishability of ciphertexts goes as $O(2^{-(n-l)})$ for $l$-bit per $n$-qubit public key (Pan et al., 2010).
• Distortion-based secrecy: For $m$-point inference $Y$, keeping adversary MSE at $\mathrm{Var}(Y)$ requires only $k \ll \log_2 m$ key bits (Tsai et al., 2017).

4. Security Optimization: Power, Rate, and Leakage Allocation

• Secrecy/Covert Communication via Power Splitting
  • In joint transmission+jammer models, optimize ergodic secrecy rate or detection error subject to SINR, under convexified surrogate objectives using DC programming/SCA (Forouzesh et al., 2018).
  • Unified optimization for secrecy and covertness: allocate Alice's power between data and jamming to maximize

  $$\mathbb{E}\left[ \log_2(1 + \mathrm{SINR}_b) - \log_2(1 + \mathrm{SINR}_e) \right]^+$$

  and/or ensure detection error sum is above threshold (Forouzesh et al., 2018).

• Local Secrecy with Euclidean Geometric Programming

  • Use quadratic approximations of mutual information to transform non-convex secrecy-utility-leakage optimization to a linear program over Lagrange multipliers, grounded in the channel's matrix pencils and generalized eigenvalues (Athanasakos et al., 15 Oct 2025).
  • Defines the secret local contraction coefficient:

  $$\eta_{\mathrm{loc}}^{\mathrm{sec}} = \sup_{\mathbf{L} \perp \sqrt{P_X}} \frac{\mathbf{L}^T V \mathbf{L}}{\mathbf{L}^T \Lambda \mathbf{L}}$$

  for rate-utility vs. leakage tradeoff (Athanasakos et al., 15 Oct 2025).

5. Extensions: Secure Multiparty Computation, Fusion, and Layered Architectures

• MPC without an Honest Majority

  • Secure protocols using only pairwise private channels and broadcast/simultaneous broadcast, statistically simulating the ideal functionality for vote, veto, anonymous message, and others (statistical error $O(n2^{-s})$) (0706.2010).
• Key-Fusing for Secret Outage Resilience
  • Sliding-window key-fusing functions (e.g., iterated XOR): exposure of up to $w-1$ raw keys in a window does not compromise any fused key; secret-outage probability decreases exponentially with window size $w$ (Li et al., 2020).
• Combinatorial Mosaics and Seed Efficiency
  • Use mosaics of balanced incomplete block designs (BIBD) and group-divisible designs (GDD) to construct modular security functions with optimal seed-length vs. color-rate tradeoffs under semantic security (Wiese et al., 2021).

6. Fundamental Limits, Implementation, and Open Challenges

• Security Parameterization
  • $2^{128}$-bit quantum security: attainable with $n=5$ parties, $m=384$-bit shares, min-entropy guarantees, and quantum-resistant commitments; all bounds derived from explicit entropy and commitment formulas (Xu et al., 28 Aug 2025).
  • Privacy amplification via universal hashing and LSCs achieves secrecy rate up to $I(X; Y_b) - I(X; Y_e)$ as blocklength increases (Tyagi et al., 2014, Calmon et al., 2015).
• Physical and Quantum Foundations
  • Security claims for KLJN rest on thermodynamic equilibrium; for QKD and unified quantum frameworks, on nonlocality, teleportation, and the quantum one-time pad (Mingesz et al., 2012, Nadeem, 2015).
• Challenges and Future Work
  • Efficient, scalable, and composable information-theoretic primitives in the presence of imperfections, noise, adversarial channel control, or device side-channels (Mingesz et al., 2012, Sun et al., 2024).
  • Cross-modal threats in foundation models, end-to-end realization of robust bandwidth/noise/signal optimization in adaptive systems (Sun et al., 2024).
  • Tight bounds and practical code constructions for wiretap and multi-user broadcast secrecy in high-dimensional, non-Gaussian, and quantum scenarios.

7. Summary Table: Key Frameworks and Techniques

Paradigm	Building Block	Quantitative Guarantee
Universal Hashing (Tyagi et al., 2014)	2-universal hash, Slepian–Wolf	$\|P_{KS} - U_K \otimes P_S\|_1 \leq \varepsilon$
KLJN (Mingesz et al., 2012)	Johnson noise, Kirchhoff law	$I(\mathrm{key};\mathrm{Eve})=0$ in idealized setup
Quantum Key/Fusion (Xu et al., 28 Aug 2025Li et al., 2020)	Entropy-preserving XOR, Shamir-Sharing	$H_\infty(S|\mathrm{Adv}) \geq$ threshold
Symbol Secrecy (Calmon et al., 2015)	List-source code, MDS code	$\mu_0 = k/n$ fraction with $I(X^J;Y)=0$
Combinatorial Mosaics (Wiese et al., 2021)	BIBD, GDD, affine geometry	Explicit MI/TV bounds given BIBD parameters
Representation Robustness (Zhang et al., 2024)	MI-regularized encoder, adversarial training	$I(Z;U)$ minimal, $I(X;Z|U)$ maximal

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The discipline of information-theoretic security encompasses a spectrum from foundational channel models (e.g., Shannon, Wyner, Csiszár-Körner), through physically motivated protocols and quantum information, to modern constructions for secure computation, key-agreement, secret sharing, coding, and robust machine learning. Security guarantees are always expressed via entropy, mutual information, error exponents, or cryptographically composable statistical distances, yielding provably unconditional, implementation- and adversary-agnostic assurances under explicit, physical or mathematical models.

Principal references: (Tyagi et al., 2014, Xu et al., 28 Aug 2025, Pan et al., 2010, Mingesz et al., 2012, Forouzesh et al., 2018, Zou et al., 2014, Nadeem, 2015, 0706.2010, Athanasakos et al., 15 Oct 2025, Li et al., 2020, Wiese et al., 2021, Zhang et al., 2024, Calmon et al., 2015).