Cryptographic Replacement of Pseudorandom Steganographic Functions

Cryptographic Replacement of Pseudorandom Steganographic Functions

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A pseudorandom function family is efficiently computable with a secret key and computationally indistinguishable from a uniformly random function under the specified oracle access. Steganography hides the presence of a message within an ordinary communication channel.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The proposed replacement class and security experiment are not specified in the supplied sentence; the source terminology is retained.

Source-record limitation: The cryptographic replacement class and security notion are not specified.

Statement recorded in the supplied source.

Can the pseudorandom functions used in the steganographic constructions be replaced by cryptographic functions? [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Determine whether the cited steganographic construction can use the proposed alternative cryptographic functions.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002327 (OWR-12481-018). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Combinatorics and Probability $2014$, Oberwolfach Reports, DOI 10.4171/owr/2013/18, pp. 1087--1152 https://doi.org/10.4171/owr/2013/18.
• S3 Liskiewicz--Reischuk--Woelfel, security levels in steganography https://doi.org/10.1016/j.tcs.2017.06.007.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30002327 and code OWR-12481-018 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

When an oracle hybrid is actually valid

The source's grey-box steganographic models distinguish what the channel chooser, sender and warden know. Its phrase about replacing pseudorandom functions does not itself specify a security game for a replacement family. We test the familiar oracle-hybrid route with all its simulation assumptions exposed. Let $E^O$ be a probabilistic polynomial-time oracle experiment, including every participant and all information supplied to them, making polynomially many oracle queries. Suppose the probability that its output bit is 1 differs by at most $\eta$ from a reference cover experiment when $O$ is a uniformly random function. Suppose further that the function family $F_K$ is secure against every such oracle distinguisher with advantage at most $\epsilon$. The important failed step is treating unavailable auxiliary information as though the reduction could generate it from oracle access. For example, if the experiment also receives the secret key $K$, it can query a fixed input $x$ and test whether the answer equals its locally computed $F_K(x)$. In the keyed-function world this test always accepts. In a random-function world independent of $K$, with $\ell$-bit outputs, it accepts with probability $2^{-\ell}$. The advantage is therefore $1-2^{-\ell}$. This is not a break of standard secret-key pseudorandom-function security: that security never gives the distinguisher $K$. It shows why the reduction must account for the source model's information distribution, including coalitions.

Verification and remaining gaps

For $\ell=3$ all eight random output strings were enumerated, giving one match and advantage $7/8$ for any fixed expected string. This checks the information obstruction, not a cryptographic security assumption. No new stegosystem is constructed, no pseudorandom family is replaced by a provably weaker one, and no ambiguity in the original replacement question is silently resolved. The proved statement is a conditional standard hybrid lemma whose prerequisites must be supplied by a precise source-specific model.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2013/18. Relevant passages read: 2013/18 pp. 1129--1130, grey-box channel models, coalition information and pseudorandom-function discussion.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch019/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.