Minimal Assumptions for Additive Randomized Encodings

Minimal Assumptions for Additive Randomized Encodings

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.

An additive randomized encoding locally maps each party's input into group elements whose aggregate permits decoding the desired output while hiding additional input information. Computational privacy is indistinguishability from a simulator based on the prescribed output.

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 function family, corruption/leakage model, randomness setup, and meaning of fully secure are not specified.

Source-record limitation: The function family, setup, leakage/corruption model, and full security requirement are not supplied.

Statement recorded in the supplied source.

Determine the weakest computational assumptions under which efficient, fully secure additive randomized encodings can be constructed. [S2]

Source status.

The supplied catalogue labels this source record partially solved; 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 2026-08-22.

Short English statement

Determine the weakest assumptions needed for efficient secure additive randomized encodings.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006183 (OWR-14299084-005). 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 Cryptography $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/3, pp. 141--174 https://doi.org/10.4171/owr/2025/3.
• S3 Bitansky, Erabelli, and Garg, ARE from PKE https://eprint.iacr.org/2025/104.pdf.
• S4 Halevi, Ishai, Kushilevitz, and Rabin, ARE https://eprint.iacr.org/2023/870.pdf.
• S5 OWR 2025/3 Cryptography report https://ems.press/content/serial-article-files/51349.
• 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 30006183 and code OWR-14299084-005 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.

The source and the checked primary Bitansky--Erabelli--Garg abstract already report ARE constructions from public-key encryption. That is an attributed upper assumption, not a proof of minimality. This attempt asks whether computational hardness could rescue a much simpler model in which every local encoding is deterministic after public setup and there are no hidden coins.

A quantitative public-setup obstruction

Let $R$ be public random setup and let efficient deterministic maps $a_x(R),b_y(R)$ take values in an efficiently represented Abelian group. The visible sum is $S_{xy}(R)=a_x(R)+b_y(R)$, and the decoder sees $(R,S)$. Assume for each input the probability of wrong decoding of AND is at most $\delta$. Suppose each of the two efficiently computable comparison tests \[ (R,S)\longmapsto\mathbf1\{S\ne a_0(R)+b_0(R)\} \] has distinguishing advantage at most $\varepsilon$ between the input pairs $(0,0),(0,1)$ and $(0,0),(1,0)$, respectively. Full computational privacy would in particular bound these tests. Thus negligible decoding error and negligible computational privacy error cannot coexist in this all-public-coins deterministic-local model, regardless of a hypothesized hard problem: the distinguishing test is an elementary efficient equality comparison. The attempted shortcut of replacing hidden randomized encodings by deterministic public transforms fails before hardness assumptions become relevant.

Verification and remaining gaps

Exhaustive finite-group checks verify the cancellation implication for all four local outputs and the error bound on illustrative public setups. Hidden independent or correlated randomness, secret setup and the actual PKE construction lie outside this obstruction. No necessary cryptographic assumption for those constructions, or equivalence establishing a weakest assumption, is proved here.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2025/3. Relevant passages read: OWR 2025/3, printed page151, Bitansky ARE Part2.

https://eprint.iacr.org/2025/104. Read primary abstract: ARE from public-key encryption and one-sided lifting; full construction/security proof not audited.

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, read and reviewed the written mathematical arguments, independently reran the finite checks, 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.

A separate OpenAI gpt-6-astra, reasoning ultra, invocation also read this written argument and independently performed the finite checks specified in the local peer-review audit. The exact serving revision is not exposed. This is model peer review, not independent human verification.

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 batch026/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.