Reusable Garbled RAM under Weaker Assumptions

Reusable Garbled RAM under Weaker Assumptions

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

Garbled RAM encodes a RAM computation to hide data while enabling evaluation. Reusability allows the encoding to serve repeated computations; persistent memory preserves encoded state across them.

A cryptographic security requirement compares an efficient adversary's advantage with a negligible function of the security parameter. A negligible function is smaller than every inverse polynomial eventually; computational and information-theoretic security are different requirements.

Statement recorded in the supplied source.

Can reusable garbled RAM, including schemes with persistent memory, be constructed under assumptions weaker than indistinguishability obfuscation, functional encryption for circuits, or the stronger obfuscation assumptions used by the known constructions? [S2]

Scope qualification.

The relevant reuse, memory-access leakage, and security requirements are those of the source.

Source status.

The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Construct reusable stateful garbled RAM under weaker assumptions while preserving the required efficiency and security.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002628 (OWR-13105-007). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cryptography $2015$, Oberwolfach Reports, DOI 10.4171/owr/2014/35, pp. 1933--1990 https://doi.org/10.4171/owr/2014/35.
• S3 FOCS 2014 reusable garbled RAM https://ieee-focs.org/FOCS-2014-Papers/6517a404.pdf.
• S4 Garbled RAM from one-way functions https://eprint.iacr.org/2014/941.pdf. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002628 and source code OWR-13105-007 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not 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.

Reusing a mask leaks a persistent-state transition

The source asks for reusable garbled RAM and persistent databases under weaker assumptions. We test a restricted proposed implementation: encrypt one persistent bit by XOR with the same one-time pad at every update. This is not claimed to be a full garbled-RAM construction. Start with memory $m_0=0$, choose a uniform secret bit $K$, and expose $c_0=m_0\oplus K$. Compare two one-write histories, both accessing the same address and returning the same public output 0: one writes $m_1=0$, the other writes $m_1=1$. Expose $c_1=m_1\oplus K$. The first observation blocks a direct reuse-of-one-time-randomness route even when the address pattern and public output reveal no difference. The second identifies a local repair but does not make it a reusable garbling: maintaining pad updates, permitting authorized computation, enforcing state consistency, and controlling access-pattern leakage are separate obligations.

Verification and remaining gaps

All two reused-pad outcomes and all four fresh-pad outcomes were enumerated for both histories. This is an exact information leak in an auxiliary XOR-masking model, not an attack on the source's obfuscation-based constructions. No reusable garbled-RAM scheme, persistent-state security proof, or reduction to weaker cryptographic assumptions is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2014/35. Relevant passages read: 2014/35 pp. 1977--1978, reusable garbled RAM, persistent databases and assumption-strength questions.

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.