{"id":"8a4e2151-c8bd-4759-b4c1-43ba41b50b6e","arxiv_id":"2506.04383","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"HFKR hashes noisy affine-map trajectories over Z^2 to derive keys, and the paper reports fractal dimension and hash diffusion metrics, but it does not prove the claimed post-quantum security.","lead":"This paper proposes HFKR, a method that hashes random-looking paths generated by noisy affine maps over a grid to produce cryptographic keys. The authors say the paths are fractal and that this gives post-quantum security, but no formal security proof is provided and the main experimental comparison is flawed.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central HFKR security claim is unsupported: no reduction links SPIP hardness to key recovery, and the paper supplies no min-entropy bound for hashed trajectories; the empirical fractal and diffusion results do not fill that gap.","rationale":"The reader's weakest assumption identifies the same two coupled gaps: no reduction from SPIP to HFKR key recovery, and no demonstrated min-entropy for the hashed trajectory. My stress-test confirms that both are load-bearing. The paper's own conclusion defers the reduction, and the experimental section never measures key-distribution min-entropy. The BLAKE3 comparison is additionally confounded by an unstated output length, but that is secondary; even a perfect diffusion experiment would not establish key-recovery hardness. A computational min-entropy check on the actual key-generation algorithm would settle the more basic question of whether the key space is large enough to protect against guessing. If the min-entropy is low, the scheme fails regardless of SPIP. If it is high, the missing reduction still leaves the central claim unproven. Either way, the current manuscript does not support the claimed post-quantum security, so the REJECT verdict is appropriate.","tokens_in":6561,"tokens_out":5426,"duration_ms":60849,"concrete_test":"Using the published GitHub code and the exact parameter ranges used in Section 4 (x0, affine coefficient bounds, bmin/bmax, epsilon, and n in {128, 500, 2000, 5000}), run at least 10^6 independent executions of Algorithm 1, hash each trajectory with SHA3-512, and estimate the number of distinct keys and their min-entropy. If the distinct-key count is far below 2^512 or a targeted search over the bounded noise and affine choices recovers a non-negligible fraction of keys, then the path-entropy assumption fails and the central security claim collapses. This check directly settles whether the hash input actually has the cryptographic min-entropy the paper assumes.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is that HFKR post-quantum security arises from the synergy between symbolic fractality and hash-based entropy amplification. That claim requires two conditions: (1) recovering the key k = H(x0 || x1 || ... || xn) in Eq. (2) is computationally hard, and (2) the distribution of k has enough min-entropy to prevent brute-force guessing. Neither condition is established. Section 5 explicitly defers 'formalizing hardness reductions that link SPI style symbolic inversion with hash-based obfuscation' to future work, so the only asserted bridge from the SPIP hardness result [15] to HFKR key-recovery hardness is missing. This is not a cosmetic omission: an adversary seeking k need not invert the symbolic path at all, so SPIP hardness alone does not imply key-recovery hardness. The paper also never estimates the min-entropy of the keys produced by Algorithm 1 over its parameter and noise space. The Section 4 entropy analysis measures only the difference between perturbed and unperturbed hash outputs, not the actual key distribution. The reported fractal dimension near 1.06 is consistent with a nearly one-dimensional, low-complexity trajectory, which could imply low path entropy rather than high entropy. Thus the central security claim rests on an unverified entropy assumption and an absent reduction, not on the empirical results presented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Hashed Fractal Key Recovery (HFKR) problem, a proposed non-algebraic cryptographic construction. Algorithm 1 generates a symbolic trajectory over Z^2 via noisy contractive affine maps, and Eq. (2) derives a cryptographic key as k = H(x0 || ... || xn). The paper claims that HFKR achieves post-quantum security from the 'synergy' of symbolic fractality and hash-based entropy amplification. The empirical component measures the fractal dimension of generated trajectories (stabilizing near 1.06) and compares the avalanche/diffusion properties of SHA3-512, SHAKE256, and BLAKE3 across 250 perturbation trials, reporting Hamming distances, bit-flip rates, entropy deviations, and chi-square uniformity tests.","tokens_in":6884,"tokens_out":2983,"duration_ms":41632,"significance":"If the central claim were established, a non-algebraic, structure-free key-recovery-hard primitive would be a notable contribution to post-quantum cryptography, especially given the scarcity of constructions that avoid lattice and number-theoretic assumptions. The paper has clear strengths: the construction is precisely specified, the empirical methodology is reproducible (the code is publicly linked), and the diffusion metrics are concrete and falsifiable. However, the significance is sharply limited because the paper does not provide a security model, a reduction from any established hard problem to key recovery, or an entropy bound for the produced keys. As it stands, the paper supplies empirical evidence about trajectory geometry and hash avalanche behavior, but not a cryptographic hardness result.","major_comments":[{"comment":"The paper's central security claim is unsupported by any reduction. Section 5 explicitly defers 'formalizing hardness reductions that link SPI style symbolic inversion with hash-based obfuscation under standard adversarial models' to future work. This is not a cosmetic gap: an adversary seeking k = H(x0 || ... || xn) need not recover the symbolic path x0,...,xn at all, so the cited #P-hardness/PSPACE-hardness of SPIP [15] does not, by itself, imply key-recovery hardness. The manuscript needs either a reduction from key recovery to SPIP, or a direct lower-bound argument for the HFKR problem; neither appears.","section":"Section 5, Eq. (2)"},{"comment":"The BLAKE3 comparison is invalid because the output lengths are not matched. BLAKE3's default output is 256 bits, while SHA3-512 and SHAKE256 outputs are 512 bits in this experiment. For a 256-bit random output, the expected Hamming distance between two independent samples is 128 bits; the reported mean of 128.21 for BLAKE3 is therefore exactly what ideal diffusion would produce, not 'weaker diffusion.' The conclusion that BLAKE3 exhibits 'notably weaker diffusion' is an artifact of comparing raw bit counts across different digest sizes. The comparison should use a normalized metric (e.g., Hamming distance divided by output length) or equal-length outputs.","section":"Section 4, Table 1 and Figure 2"},{"comment":"The entropy analysis does not measure the entropy of the key distribution. The reported 'entropy difference ΔH' is computed between perturbed and unperturbed hash outputs, which is an avalanche measure, not an estimate of the min-entropy of k over the parameter and noise space of Algorithm 1. The paper never bounds the min-entropy of the concatenated trajectory or of the resulting key, so there is no basis for claiming resistance to brute-force or guessing attacks. The observed fractal dimension stabilizing near 1.06 is, if anything, a warning sign: a nearly one-dimensional object has low effective dimensionality, which could imply low path entropy rather than high entropy.","section":"Section 4, entropy analysis"},{"comment":"The paper calls HFKR a 'novel cryptographic hardness assumption' but never defines the assumption formally: there is no adversarial success experiment, no advantage function, no security parameter, and no statement of what is conjectured to be hard. In particular, the phrase 'post-quantum security' is used throughout without a quantum adversary model (e.g., QROM or quantum circuit model). As a result, the paper's abstract and conclusion go beyond what the evidence supports: the empirical results are consistent with standard hash-function avalanche properties and do not establish a new hardness assumption.","section":"Section 3, Algorithm 1 and surrounding text"}],"minor_comments":[{"comment":"The citation to Barnsley [16] appears mismatched: reference [16] is listed as an arXiv preprint with no author name, while the text attributes the fractal-symbolic connection to Barnsley and Kitchens. Please correct the reference list or the in-text attribution.","section":"References [16] and [17]"},{"comment":"Figure captions do not specify the sample size, the perturbation offsets, or the units on the axes. The text mentions 250 trials and 5 perturbation offsets, but the captions should state this directly for readability.","section":"Section 4, captions of Figures 2-4"},{"comment":"There is a typographical error: 'SPI style symbolic inversion' should be 'SPIP-style symbolic inversion.'","section":"Section 5"},{"comment":"The claim that the trajectory is 'non-invertible due to stochastic perturbations and rounding' is asserted without proof. Rounding is indeed non-injective, but the statement would benefit from a precise argument or a reference, especially since invertibility is not equivalent to hardness of inversion.","section":"Section 3, Eq. (1)"}],"recommendation":"reject","confidential_remarks":"The paper's only hardness foundation is the author's own prior work [15], and no independent verification or proof sketch of that result is included. Since the current manuscript relies entirely on [15] for its claimed SPIP hardness, and since Section 5 explicitly defers the reduction to HFKR, the central contribution is a collection of empirical observations rather than a cryptographic result. The BLAKE3 comparison error is also a serious correctness issue in the experimental section. I would recommend rejection, though the underlying idea of studying hash-perturbed symbolic trajectories may be worth revisiting if the author later supplies a formal hardness reduction and an entropy bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper introduces the \"Hashed Fractal Key Recovery problem\" — basically the author's earlier Symbolic Path Inversion Problem (SPIP) followed by a hash. The genuinely new part is the packaging: a key-generation algorithm built on symbolic walks with a hash on top, plus some empirical diffusion measurements. The central claim, that HFKR is a post-quantum-secure non-algebraic primitive, is not established anywhere in the manuscript. The paper itself says in Section 5 that reductions linking SPIP inversion to hash-based recovery are future work. That is not a cosmetic gap: an adversary wanting the key doesn't have to invert the symbolic path at all. SPIP hardness alone, even if true, doesn't transfer through a hash.\n\nWhat the paper does well: it is clearly written, the algorithm is specified, the code is on GitHub, and the author openly flags the missing reductions. The avalanche experiments on SHA3-512 and SHAKE256 are standard but honestly reported. There is no sign of fabrication; the numbers are plausible.\n\nThe soft spots, in proportion:\n\n1. The BLAKE3 comparison is invalid. BLAKE3's default output is 256 bits; SHA3-512 outputs 512 bits. The paper reports BLAKE3's mean Hamming distance of ~128 as \"weaker diffusion,\" but 128/256 is exactly the expected avalanche rate. The output lengths are never disclosed, so the comparison is apples-to-oranges. That is a real, identifiable error.\n\n2. No min-entropy argument for the hashed keys. Section 4 measures the difference between perturbed and unperturbed hashes (avalanche), not the distribution of keys produced by Algorithm 1 over the parameter and noise space. The fractal dimension stabilizing near 1.06 is close to a space-filling curve in 2D, but that says little about the entropy of the concatenated path; a low-dimensional trajectory can still have low complexity.\n\n3. The chi-square p-values of 1.0000 are interpretative noise. The author half-acknowledges this in the text, but then uses \"higher chi-square statistic\" to claim SHA3/Shake diffuse better. That's not what the test shows.\n\nOverall: this is a proposal with a named problem and some reproducible measurements, but the load-bearing security claim is an assertion, not a result. The paper doesn't deserve a serious referee on its current form. If the author later supplies a real reduction and a min-entropy bound, the underlying direction might be worth a second look.\n\nRecommendation: desk reject. It is an honest, organized preprint, but the central claim is unsupported and the empirical comparison is flawed. Cite it only as an example of why avalanche tests alone don't establish key-recovery hardness.","headline":"A named primitive that is really SPIP-plus-hashing, with the post-quantum security claim resting on an unproved reduction and a botched BLAKE3 comparison; the reproducible experiments don't rescue the central claim.","tokens_in":7369,"tokens_out":3004,"would_cite":false,"duration_ms":27492,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes deriving post-quantum keys by hashing chaotic symbolic walks, replacing algebraic hardness assumptions with symbolic complexity plus hash diffusion.","keywords":["Hashed Fractal Key Recovery","Symbolic Path Inversion Problem","post-quantum cryptography","symbolic dynamics","hash-based obfuscation","fractal dimension","chaos theory","non-algebraic hardness assumptions"],"falsifier":"Enumerate all $m^n$ symbolic paths for a small trajectory length $n$, hash each one, and measure the min-entropy of the resulting key distribution; if it falls well below 256 bits, HFKR cannot provide the claimed security regardless of SPIP's hardness.","tokens_in":6359,"feed_emoji":"🔑","tokens_out":11356,"duration_ms":95626,"temperature":0.7,"pith_summary":"The paper is trying to establish that a cryptographic key can be generated by hashing a chaotic symbolic walk, with no factorization, discrete-logarithm, or lattice assumption underneath. It builds on the Symbolic Path Inversion Problem (SPIP), whose hardness it takes as given, and adds a hash layer: the concatenated walk is run through SHA3-512, SHAKE256, or BLAKE3 to obtain a fixed-size key. Over 250 perturbation trials, the walks show a box-counting dimension that stabilizes near 1.06, and SHA3-512 and SHAKE256 convert a small path change into about 255 flipped bits out of 512, which the paper interprets as hash-based amplification of symbolic divergence. If the scheme is sound, it would supply a lightweight, algebra-free route to key generation that is meant to hold up against quantum attackers.","feed_headline":"Chaos plus hashing aims to replace algebraic post-quantum keys","feed_subtitle":"If the claim holds, keys could come from chaos rather than number theory, but the formal hardness proof is still pending.","key_machinery":"The mechanism is the symbolic trajectory $P = \\{x_0, x_1, \\ldots, x_n\\}$ generated by the recursion $x_{i+1} = \\lfloor A_i x_i + b_i + \\delta_i \\rfloor$, where $A_i$ is a contractive linear map, $b_i$ a bounded translation, and $\\delta_i$ bounded noise; the path is concatenated into a message $M = x_0 \\| x_1 \\| \\cdots \\| x_n$ and hashed to a key $k = H(M)$. The trajectory's role is to supply combinatorial and fractal entropy, while $H$ acts as an entropy amplifier and one-way compressor. The box-counting dimension, which stabilizes near 1.06 in the experiments, is the paper's quantitative handle on the fractal character of the symbolic paths.","core_discovery":"On the paper's own terms, the discovery is that symbolic chaos and hashing form a usable non-algebraic primitive. Contractive affine maps over $\\mathbb{Z}^2$, perturbed by bounded noise and rounded down, generate paths whose combinatorial growth and fractal structure can be measured, and a cryptographic hash compresses those paths into keys while amplifying any change in the path. The author claims that this synergy is what gives HFKR its post-quantum character: SPIP supplies intractability, and the hash supplies one-wayness and the avalanche effect, so seeing only the key should not reveal the path. The empirical results are read as confirming the entropy foundation, with the box-counting dimension converging to roughly 1.06 and SHA3-512 and SHAKE256 reaching mean Hamming distances near 255 bits, while BLAKE3 shows weaker diffusion. The paper does not claim to prove the full reduction from SPIP hardness to hashed key recovery; that proof is explicitly left for future work.","pith_inferences":["A natural stress test the paper leaves implicit is to enumerate all symbolic paths for small $n$ and compute the true min-entropy of $H(M)$; if that entropy falls below the key length, the advertised security bound would not hold.","Because the implementation samples fresh affine maps at every step as a stochastic extension of SPIP, a complete treatment would need a reduction showing that this extension preserves the hardness of the original SPIP problem.","The pairing of p-values near 1.0000 with BLAKE3's lower mean Hamming distance suggests that uniform bit-flip statistics and strong diffusion are separable properties; a minimum-Hamming-distance metric over many perturbations would make the distinction explicit."],"forward_implications":["If HFKR is sound for the proposed parameters, post-quantum key generation would no longer be tied to algebraic problems such as lattice inversion, opening a distinct design space for key generation.","Hash selection becomes part of the security argument: SHA3-512 and SHAKE256 preserve symbolic divergence well, while BLAKE3's weaker diffusion makes it a poor choice for this construction.","The stabilized fractal dimension near 1.06 gives a quantitative target for choosing trajectory length in future instantiations of the scheme.","The construction's simplicity supports the paper's claim that it can run in constrained environments, provided the missing SPIP-to-HFKR reduction is eventually supplied."],"supporting_citations":[{"why":"Supplies the SPIP intractability claim (#P-hard and PSPACE-hard) that HFKR inherits as its security foundation.","marker":"[15]"},{"why":"Supports the paper's assertion that symbolic substitution systems can produce fractal measures and dimensions.","marker":"[16]"},{"why":"Provides the symbolic-dynamics background for treating the trajectory space as a discrete fractal tree.","marker":"[17]"},{"why":"An earlier fractal-and-chaos cryptosystem that still rests on RSA, used by the paper as the contrast class it moves beyond.","marker":"[12]"},{"why":"An earlier fractal public-key system that depends on Diffie-Hellman seeding, illustrating the algebraic scaffolding HFKR aims to remove.","marker":"[14]"},{"why":"The author's prior result on fractal attractors in random iterated function systems, grounding the stochastic trajectory model.","marker":"[11]"}],"fun_headline_variants":["Fractal paths + hashing: a new route to post-quantum keys","Non-algebraic keys from fractal chaos and hashing","SPIP + hash: post-quantum security without algebra","Hashed fractal paths aim for quantum-resistant keys","Fractal chaos + hashing to generate post-quantum keys"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scheme's security rests on SPIP being genuinely intractable for the short, stochastic walks used here and on the hash layer inheriting that hardness; the paper defers the proof of the inheritance step.","fun_headline_variants_meta":{"raw":{"variants":["Fractal paths + hashing: a new route to post-quantum keys","Non-algebraic keys from fractal chaos and hashing","SPIP + hash: post-quantum security without algebra","Hashed fractal paths aim for quantum-resistant keys","Fractal chaos + hashing to generate post-quantum keys"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000519,"raw_usage":{"total_tokens":2555,"prompt_tokens":1026,"completion_tokens":1529,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":1440}},"tokens_in":642,"tokens_out":1529,"duration_ms":10232,"temperature":1.0,"reasoning_tokens":1440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:43:13.519670+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all $m^n$ symbolic paths for a small trajectory length $n$, hash each one, and measure the min-entropy of the resulting key distribution; if it falls well below 256 bits, HFKR cannot provide the claimed security regardless of SPIP's hardness.","supporting_citations":[{"cited_title":"On the Intractability of Chaotic Symbolic Walks: Toward a Non-Algebraic Post-Quantum Hardness Assumption","cited_arxiv_id":"2505.22644","evidence_quote":"Supplies the SPIP intractability claim (#P-hard and PSPACE-hard) that HFKR inherits as its security foundation."},{"cited_title":"Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems","cited_arxiv_id":"1308.3819","evidence_quote":"Supports the paper's assertion that symbolic substitution systems can produce fractal measures and dimensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the symbolic-dynamics background for treating the trajectory space as a discrete fractal tree."},{"cited_title":"Fractal-based hybrid cryptosys- tem: Enhancing image encryption with rsa, homomorphic encryption, and chaotic maps,","cited_arxiv_id":null,"evidence_quote":"An earlier fractal-and-chaos cryptosystem that still rests on RSA, used by the paper as the contrast class it moves beyond."},{"cited_title":"Effi- ciency analysis for public key systems based on fractal functions,","cited_arxiv_id":null,"evidence_quote":"An earlier fractal public-key system that depends on Diffie-Hellman seeding, illustrating the algebraic scaffolding HFKR aims to remove."}],"review_version":1}