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Beyond Explicit Generators: Distribution-Free Linear-Decomposition Attacks on Public-Key Encryption

Ziyan Chen, Ding-Xuan Zhou

arXiv:2608.20798v1cs.CR

TL;DR

The paper asks whether public sampling and evaluation can expose a secret linear transport on future ciphertexts without requiring full-span recovery. It develops distribution-free sampled-span guarantees and applies them to a 2024 twisted–skew group-ring PKE, where public protocol samples yield plaintext recovery and an IND–CPA break. The analysis is scoped to exact linear transport and sampled-span coverage rather than every possible cryptanalytic algorithm.

  • Problem

    Existing linear-decomposition and learning results do not characterize future-ciphertext coverage from arbitrary public sampling paired with an unknown secret transport or connect it directly to PKE security.

  • Method

    The paper models publicly generated pairs under a fixed exact linear map, analyzes rank-increasing samples as a stable compression set, and converts sampled-span coverage into key-transport recovery.

  • Results

    Public protocol samples attack the twisted–skew PKE's CTSP, yielding plaintext recovery and an IND–CPA break, while experiments confirm the transport identity and end-to-end attack.

  • Takeaways & Limitations

    Publicly samplable exact linear key transport with polynomial effective dimension cannot provide IND–CPA security when the transported value determines the decryption-critical payload.

  • Takeaways & Limitations

    The generic analysis assumes exact fixed linear transport and does not directly cover nonlinear, essentially hidden-randomness-dependent, or approximately linear relations.

Abstract

from arXiv · show

Linear-decomposition attacks can break public-key schemes without recovering the secret algebraic action: when a target public state lies in a known linear span, its decomposition coefficients transfer through the unknown action to reveal the shared value. We study a setting in which the adversary uses only the public sampling-and-evaluation oracle available to honest participants, the induced distribution is arbitrary, and the goal is to attack future ciphertexts rather than recover the full algebraic span. We model public paired samples under a fixed secret linear transport and define the sampled-orbit dimension as the effective dimension of the encryption distribution. We prove distribution-free one-shot recovery, a high-probability certificate for the future-ciphertext coverage of a sampled span, and the optimal sampled-span complexity $m^\star_{\mathrm{span}}(r,\varepsilon,δ) =Θ((r+\log(1/δ))/\varepsilon)$. These results yield a generic impossibility theorem: publicly samplable linear key transport with polynomial sampled-orbit dimension is incompatible with IND--CPA security when the transported value determines the decryption payload. We apply the framework to the 2024 probabilistic PKE from twisted--skew group rings. Its underlying Computational Twisted--Skew Problem admits a sampler-only linear attack using independently generated public protocol samples, yielding plaintext recovery and constant IND--CPA advantage. Experiments verify the linear transport and end-to-end recovery, and show that high future-ciphertext coverage may precede recovery of the full algebraic span.

1 Introduction

The paper asks whether public sampling and evaluation alone can recover the effect of a secret linear map on future ciphertexts under arbitrary, non-uniform distributions. It develops distribution-free guarantees and applies them to show sampler-only attacks against a twisted–skew group-ring PKE.

  • Motivation: Linear-decomposition attacks transfer public linear-relation coefficients through an unknown secret map, recovering shared values without recovering the map itself.The target must lie in the known span of observed public states.
  • Problem: The paper replaces full-span recovery with future-ciphertext coverage under the arbitrary distribution induced by honest public sampling.A sampled span can cover likely future targets even when rare support directions remain unseen.
  • Framework: The framework models paired samples under a fixed secret linear transport without giving the adversary the map or an explicit basis for the reachable subspace.The public paired sampler supplies observations of the form (Xρ, Yρ) with Yρ = AXρ.
  • Results: 2rμ−1 public samples give one-shot transported-value recovery with probability at least 1/2 for any target distribution μ.This guarantee requires no uniformity or anti-concentration assumption on μ.
  • Results: A stable-compression analysis gives high-probability future coverage, while the sampled-span optimality claim is limited to that coverage problem rather than all cryptanalysis.With confidence 1−δ, a learned span covers at least a 1−ε fraction of future encryption randomness.
  • Cryptographic consequence: Polynomial sampled-orbit dimension makes publicly samplable exact linear key transport incompatible with IND–CPA security when the transported value determines decryption-critical payload information.The paper applies this consequence to the 2024 twisted–skew PKE, obtaining a sampler-only CTSP attack and explicit plaintext-recovery consequences.

2 Preliminaries and Attack Setup

The paper abstracts honest public-key transport as paired samples under a fixed secret linear map, then turns that interface into a sampler-only attack. Linear relations among sampled public states transfer through the unknown map, enabling recovery whenever a target lies in the sampled span.

  • Distribution model: The sampled-orbit distribution μ may be arbitrary, and the attacker targets future samples rather than reconstructing the entire algebraic span.Its effective dimension is the sampled orbit dimension, while the learned space is the span of public samples.
  • Sampler-only attack: The adversary samples public pairs, tests whether a target lies in their span, and uses Gaussian elimination to obtain decomposition coefficients.Only rank-increasing samples need be retained for the learned span.
  • Honest public linear key transport: The framework models public samples as paired values (X_i,Y_i) with Y_i=A X_i for a fixed secret linear map.The public procedure need not reveal A, an algebraic generating set, or a basis for the reached subspace.
  • Sampler-only attack: Linear coefficient transfer recovers the target transported value without recovering the secret map A.The same coefficients expressing X★ through sampled X_i express Y★ through the paired Y_i.
  • One-shot recovery: With m=2r_μ−1 public samples, one-shot recovery succeeds with probability at least 1/2; more generally, m=O(r_μ/ε) gives failure probability at most ε.These guarantees require neither uniformity nor anti-concentration of μ.

3 Main Results

The main results provide distribution-free guarantees for how sampled spans cover future ciphertext randomness, establish optimal sample complexity, and derive an IND-CPA impossibility theorem for recoverable payloads. The theorem applies when public paired sampling and linear algebra are efficient and the sampled-orbit dimension is polynomially bounded.

  • 3.1 A distribution-free future-coverage certificate: A stable compression rule retaining rank-increasing samples yields a high-probability certificate for the future-sample failure rate.With probability at least 1−δ over training samples, recovery covers at least a 1−R_μ(W_m) fraction of fresh samples.
  • 3.2 Optimal sample complexity: m*=Θ((r+log(1/δ))/ε) is the optimal distribution-free sampled-span complexity up to integer rounding.The matching lower bound shows that the O(r_μ/ε) dependence cannot be improved for arbitrary sampling distributions.
  • 3.3 IND-CPA impossibility: The payload condition covers masking schemes and KEM–DEM encryption when the transported value enables key derivation and payload decryption.The general ciphertext model permits additional encryption randomness beyond the public transport component.
  • 3.3 IND-CPA impossibility: Polynomial sampled-orbit dimension and efficient paired sampling imply a probabilistic polynomial-time adversary with non-negligible IND-CPA advantage.The conclusion applies when the transported value determines recoverable payload information.
  • 3.3 IND-CPA impossibility: After preprocessing, the same sampled span can recover transported values and payloads for at least a 1−ε fraction of future randomness, except for negligible recovery error.This is a stronger future-decryption interpretation than the constant-success one-shot bound.

4 Application to Twisted–Skew Group-Ring Protocols

The framework breaks the 2024 twisted–skew group-ring PKE using only independently generated public protocol samples and finite-field linear algebra. The attack yields explicit CTSP and plaintext-recovery guarantees, including constant IND-CPA advantage under the proposed parameterization.

  • 4.2 Sampler-only CTSP solution: The public protocol supplies exactly the paired samples needed for the sampler-only attack, using public sampling and evaluation without explicit generators or a precomputed basis.The attack therefore applies directly to the CTSP instance underlying the construction.
  • 4.1 Twisted–skew linear transport: The twisted–skew construction’s fixed-parenthesis decryption-critical map is F_q-linear despite generally nonassociative multiplication.The proof uses F_q-bilinearity with fixed multiplication order and does not invoke associativity.
  • 4.2 Sampler-only CTSP solution: 8n−1 independently generated public samples recover a fresh CTSP target with probability at least 1/2 when |G|=2^n.The bound follows from the ambient F_q-dimension 4n and is a worst-case ambient-dimension guarantee.
  • 4.4 Plaintext recovery and IND–CPA consequence: 8n−1 self-generated encryptions of zero give plaintext-recovery probability at least 1/2 and IND-CPA advantage at least 1/4 under the stated convention.The training encryptions use neither the secret key nor a decryption oracle.
  • 4.5 Relation to prior linear-decomposition attacks: The effective rank may be smaller than the ambient bound, while high future-ciphertext coverage can be certified without uniformity, coordinate independence, or anti-concentration.The paper’s contribution is specifically a sampler-only, distribution-free attack on this exact construction.

5 Experimental Evaluation

The evaluation validates the fixed linear transport and demonstrates end-to-end plaintext recovery from public samples. Native experiments show a sharp threshold near the sampled-orbit rank, while synthetic non-uniform samplers achieve high future coverage without full-rank recovery.

  • 5.1 Does the implementation realize the attack model?: Exact finite-field tests found no violation in linearity, paired-sample, or encryption/decryption identities.The validation uses full ring-element equality rather than approximate numerical tolerances.
  • 5.2 Does the sampler-only attack recover plaintexts?: At m = r_pk, native-sampler plaintext recovery reached 95.47%, 94.81%, and 94.48% for n = 20, 23, 32, respectively.One further sample raised all three rates above 99.6%, and two further samples gave 100% in recorded trials.
  • 5.2 Does the sampler-only attack recover plaintexts?: Every recorded learned-span hit produced the exact challenge plaintext, so span membership triggered concrete decryption using only public encryptions.The empirical conditional probability of correct plaintext recovery given span membership was one throughout the experiment.
  • 5.3 Why is the native recovery threshold sharp?: 56 of 60 public setups had native sampled-orbit rank r_pk = 4n, while four had codimension two.Mean ranks were 79.8 of 80, 91.9 of 92, and 127.9 of 128 for n = 20, 23, 32.
  • 5.3 Why is the native recovery threshold sharp?: At m = r_pk − 1, learned spans contained 98.74–99.22% of exact rank but covered only 4.94–5.37% of future samples; at m = r_pk, coverage rose to 94.98–95.44%.The native threshold reflects the sampler’s comparatively spread-out probability mass, where one missing direction leaves substantial future mass uncovered.
  • 5.4 Is full-span recovery fundamentally necessary?: At m = r_pk, the strong hot-pool sampler learned only 5.1–7.1% of exact rank while covering 97.88–98.03% of future samples.The moderate sampler likewise covered 78.8–80.1% while learning 32.5–39.6% of exact rank; both are synthetic stress tests.

6 Discussion

The discussion reframes attack relevance around future-ciphertext coverage rather than recovery of the secret action or full algebraic orbit. It also limits the claims to exact linear transport and distinguishes sampled-span optimality from universal cryptanalytic lower bounds.

  • 6 Discussion: Security evaluation should consider whether public operations expose a fixed exact linear transport on future ciphertexts, not only whether hidden generators or the full orbit are recoverable.The attack can recover the secret action’s effect without reconstructing the action itself.
  • 6 Discussion: The attack requires public paired samples, a fixed low-dimensional exact linear transport, and a payload recoverable from the transported value.Breaking at least one of these ingredients is identified as necessary to prevent the present attack.
  • 6 Discussion: Future-ciphertext probability mass covered by the sampled span is the relevant cryptanalytic quantity, and high coverage can occur with only a small fraction of full algebraic rank recovered.Thus hiding generators or making full-span reconstruction difficult is not sufficient by itself.
  • Limitations and Scope: The generic analysis does not directly cover nonlinear, essentially hidden-randomness-dependent, or approximately linear transport relations.The stated scope is exact linear key transport satisfying a fixed relation Y_ρ = A X_ρ.
  • Limitations and Scope: The sampled-span optimality result characterizes distribution-free coverage complexity, not a lower bound against every cryptanalytic algorithm.Additional algebraic information, protocol structure, side information, or representations may enable fewer samples.
  • Limitations and Scope: Synthetic non-uniform samplers validate the rank-versus-coverage distinction but are not claimed to model the native randomness distribution of the 2024 scheme.The concrete experiments target attack-mechanism and distributional validation rather than replacement of the mathematical proofs.

7 Related Work

Related work establishes both linear-decomposition cryptanalysis and statistical subspace learning as antecedents. This paper connects them by analyzing arbitrary public-sampling distributions, paired observations under an unknown linear map, future-span coverage, and formal PKE insecurity.

  • Linear-algebraic and linear-decomposition cryptanalysis: Algebraic cryptanalysis has shown that recovering a nominal secret action, such as conjugacy, may be unnecessary for attacking group-based protocols.Linear-algebraic and linear-decomposition methods provide polynomial-time attacks in several algebraic settings.
  • Linear-algebraic and linear-decomposition cryptanalysis: Random public samples have previously replaced explicitly constructed bases in linear cryptanalysis, including the Modified Matrix Modular Cryptosystem.The paper distinguishes its contribution from random spanning and coefficient transfer themselves.
  • Linear-algebraic and linear-decomposition cryptanalysis: Prior work on semidirect-product and matrix-action proposals illustrates recurring vulnerabilities when hidden operations remain accessible through low-dimensional linear representations.The cited line includes efficient cryptanalysis of MAKE and conditions for linear-algebraic attacks.
  • Twisted and skew group-ring constructions: The 2024 twisted–skew construction may be nonassociative, so earlier generic two-sided-multiplication theorems do not apply verbatim.The paper reports no prior explicit attack on the exact CTSP or Algorithms 6–8, while avoiding claims that sampler-only access is necessary.
  • Statistical learning of spans and subspaces: Statistical subspace-learning and stable-compression results provide related learning tools but do not address paired observations, coefficient transfer, or formal PKE security.Conversely, classical linear-decomposition attacks do not characterize distribution-free future-ciphertext coverage.

8 Conclusion

The paper reframes linear-decomposition cryptanalysis around future-ciphertext coverage using public samples, rather than recovery of explicit generators or the full algebraic span. It derives a generic IND–CPA impossibility result and demonstrates the attack on twisted–skew group-ring PKE.

  • The framework uses only the public sampling-and-evaluation interface and targets probability mass of future ciphertexts rather than the complete algebraic span.This shifts the cryptanalytic objective away from reconstructing the full relevant action space.
  • Finite public samples can provide strong guarantees for independently generated future targets under arbitrary encryption-induced distributions.The analysis characterizes the optimal sampled-span complexity and combines coverage with coefficient transfer.
  • Publicly samplable exact linear key transport of polynomial effective dimension cannot provide IND–CPA security when the transported value determines the decryption-critical payload.
  • The twisted–skew group-ring PKE admits a sampler-only attack that yields plaintext recovery and an IND–CPA break despite non-associative multiplication.The construction’s correctness relation induces fixed linear transport when multiplication order is preserved.
  • The conclusions are scoped to exact public linear transport, while noisy, approximate, or randomness-dependent transport remains future work.The evaluation uses a published construction and locally generated keys, randomness, and ciphertexts.
  • Experiments confirm the transport identity and end-to-end attack while showing that probability-mass coverage can outpace algebraic rank recovery.The artifact supports exact arithmetic, attack stages, raw outputs, and regenerated figures.

C Proofs for Preliminaries and Attack Setup

The proofs establish coefficient transfer, distribution-free coverage of sampled spans, and matching sample-complexity bounds. They use stable sample compression for the upper bound and rare-type distributions for lower bounds.

  • Preliminaries and attack setup: The coefficient-transfer identity recovers the transported value whenever a fresh target lies in the span of sampled public vectors.Gaussian elimination supplies the decomposition coefficients, which transfer through the fixed linear map.
  • Sampled-span coverage: The coverage upper bound follows by reducing sampled-span prediction to a realizable stable sample-compression scheme.The compression size is tied to the sampled-orbit dimension and yields the stated distribution-free guarantee.
  • Preliminaries and attack setup: The sampled-span classifier retains a basis of the sample span and reconstructs a zero-error, stable classifier.Stability follows because removing a non-basis sample leaves the span unchanged.
  • Sample-complexity lower bounds: Rare-type distributions establish dimension-dependent lower bounds because unseen independent rare vectors remain outside the observed sample span.With probability at least 1/2, more than half of the rare basis vectors are unseen.
  • Sample-complexity lower bounds: No distribution-free confidence guarantee is possible below the stated linear-in-r/ε threshold when δ<1/2.The lower-bound argument uses a two-dimensional distribution with a rare vector.

D.3 Proof of the IND-CPA No-Go Theorem for Recoverable Payloads

The no-go proof constructs an IND–CPA adversary that samples public pairs, tests whether the challenge target lies in their span, and transfers coefficients to recover the decryption-critical value.

  • The adversary samples public paired examples before receiving the IND–CPA challenge and uses polynomial-time linear algebra.The paired sampler exposes samples of the fixed secret linear transport.
  • If the challenge target lies in the sampled span, Gaussian elimination and coefficient transfer recover its exact transported value.The public recovery algorithm then obtains the recoverable payload.
  • The adversary guesses correctly with certainty on simultaneous span-coverage and payload-recovery events, and otherwise guesses randomly.This directly converts coverage and payload recovery into IND–CPA advantage.
  • The adversary is PPT with O(R(λ)) public samples, and perfect payload recovery gives IND–CPA advantage at least 1/4.More generally, negligible payload-recovery error still leaves non-negligible advantage.
  • Twisted–skew linearity: In the twisted–skew setting, fixed-parenthesization multiplication is additive and F_q-linear in each input without requiring associativity.The resulting map A_sk is F_q-linear and transports sampled inputs to the corresponding outputs.

E.2 Proof of the CTSP Corollary

The CTSP corollary fixes one public key, generates independent public protocol samples, and applies the sampled-span attack to recover the CTSP target with polynomial-time Gaussian elimination.

  • Fixing the first CTSP public key defines a secret linear map A_pk1(x)=(a_1x)bγ_1.
  • Independent secret-key samples generate public vectors that share the same distribution as the CTSP target conditioned on the fixed public key.The sampled vectors, together with the target public key, are i.i.d. from μ_pk1.
  • When the target lies in the sampled span, Gaussian elimination supplies coefficients and Lemma 2.1 recovers the CTSP target.
  • The ring dimension satisfies dim_Fq R=2|G|=4^n, so choosing m=8n−1 gives coverage lower bound at least 1/2.All samples and evaluations are public, and the remaining computation is polynomial-time Gaussian elimination.

E.3 Proof of the Sampler-Only Break Corollary

The proof constructs a sampler-only attack from independent public encryptions of zero, estimates the CTSP target, and recovers the challenge plaintext when the sampled-span event occurs. This yields constant IND–CPA advantage using polynomial-time linear algebra and 8n−1 public encryptions.

  • Independent public encryptions of zero provide CTSP training pairs for the challenge target.
  • The estimator recovers the CTSP target with probability at least 1−r_pk/(m+1).
  • Once the target estimate is correct, the adversary exactly recovers the challenge plaintext.
  • On the sampled-span event, plaintext recovery reveals the IND–CPA challenge bit; otherwise, the adversary guesses uniformly.
  • The attack achieves at least 1/4 IND–CPA advantage using 8n−1 public encryptions and polynomial-time linear algebra in a 4n-dimensional F_q representation.
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