gpt 6 astra ultra attempt on CS-687

LWE dimension comparison: padding, projection, and the lost secret component

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 lattice generated by independent vectors $b_1,\ldots,b_d$ is $L=\{\sum_i z_i b_i:z_i\in\mathbb Z\}$. SVP seeks a shortest nonzero lattice vector; SIVP seeks $d$ independent short vectors, with approximation measured against the relevant successive minimum.

Learning with errors gives noisy linear equations $b_i=\langle a_i,s\rangle+e_i\pmod q$ for a hidden vector $s$. Ring-LWE uses the analogous equations in a polynomial quotient ring, under specified sample and error distributions.

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 approximation factors, error distribution, modulus, and precise source reductions are not specified by the dimension-only question.

Source-record limitation: The reductions and all their quantitative promises other than a dimension comparison are absent.

Statement recorded in the supplied source.

Can the discrepancy between the lattice dimensions in the quantum and classical reductions from GapSVP to learning with errors be removed or explained, namely dimension n for the quantum reduction versus order square-root-n for the classical reduction? [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

Resolve the dimension loss between the stated quantum and classical lattice-to-learning reductions.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002730 (OWR-13353-018). 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 Combinatorial Optimization $2015$, Oberwolfach Reports, DOI 10.4171/owr/2014/51, pp. 2873--2932 https://doi.org/10.4171/owr/2014/51.
• S3 Regev, lattices and learning with errors https://arxiv.org/abs/2401.03703.
• S4 classical GapSVP-to-LWE analysis https://www.aimsciences.org/article/doi/10.3934/amc.2021004.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

Can the discrepancy between the lattice dimensions in the quantum and classical reductions from GapSVP to learning with errors be removed or explained, namely dimension n for the quantum reduction versus order square-root-n for the classical reduction?

Quantitative source scope

Stehle's report, printed pages 2918--2919, compares a quantum worst-case lattice reduction in dimension $n$ with a classical one of lattice dimension $O(\sqrt n)$ for LWE of dimension $n$. It explicitly includes modulus, noise and approximation promises: the quantum statement assumes prime polynomially bounded $q$, $\alpha q\ge2\sqrt n$, and approximation on the order of $n$ times a polylogarithmic factor divided by $\alpha$. The dimension-only catalogue sentence omits those parameters. We do not claim that the following elementary transformations of LWE samples implement or remove the source's worst-case lattice reduction loss.

Attempt 1: pad dimensions and randomize the secret

Let $q$ be prime, write $\mathbb F_q$ for the field of residues, and suppose samples of dimension $r$ have the form $(a,b)$ with uniform independent $a\in\mathbb F_q^r$ and $b=a^Tt+e$, for a fixed secret $t$ and independent errors of some fixed distribution modulo $q$. For $n\ge r$, append independent uniform $c\in\mathbb F_q^{n-r}$ and choose one shared uniform vector $w\in\mathbb F_q^n$. Output \[ A=(a,c),\qquad B=b+A^Tw. \] Then $A$ is uniform, and $B=A^T((t,0)+w)+e$. The new secret is uniform on $\mathbb F_q^n$, independently of all sample vectors, because translation by the fixed $(t,0)$ preserves uniformity. All samples use the same new secret and unchanged error distribution. A search oracle returning it lets us subtract the known $w$ and recover $t$. For a decision reduction, uniform original $b$ remains uniform after adding $A^Tw$, so uniform input also maps to uniform output. This proves the elementary direction $\mathrm{LWE}_r\le\mathrm{LWE}_n$ under the stated sample conventions. It does not turn hardness of a lower-dimensional worst-case lattice problem into hardness for a higher-dimensional worst-case lattice problem. Reversing that reduction direction would be an unsupported inference.

Attempt 2: project public vectors to fewer coordinates

Fix a full-column-rank matrix $T\in\mathbb F_q^{n\times r}$ with $r<n$, chosen independently of the secret. For original samples $b=a^Ts+e$, keep only $a'=T^Ta$ and $b$. If $s=Tt$, this is an exact $r$-dimensional LWE sample with secret $t$. But if $s\notin\operatorname{col}(T)$, the $r+1$ linear forms in $(T^Ta,a^Ts)$ are independent. The associated linear map has rank $r+1$: a dependence would express $s$ as a linear combination of columns of $T$, or contradict the rank of $T$. Gaussian elimination gives $n-r-1$ free coordinates in each fiber, so each output has exactly $q^{n-r-1}$ preimages. Consequently $a^Ts$ is uniform independently of $a'$. Adding the independent error $e$ leaves it uniform and independent. For each such fixed $s$, all projected samples are therefore independent uniform pairs, regardless of the original noise law.

For a uniform original secret, the event $s\in\operatorname{col}(T)$ has probability $q^{r-n}$. Conditional on it, the corresponding $t$ is uniform. For any number $M$ of samples, the entire projected distribution is thus \[ \mathcal P_M=q^{r-n}\mathcal L_{r,M} +(1-q^{r-n})\mathcal U_M, \] where $\mathcal L_{r,M}$ is the ordinary $r$-dimensional LWE distribution with a shared uniform secret and $\mathcal U_M$ is uniform. For every event $E$, $|\mathcal P_M(E)-\mathcal U_M(E)| =q^{r-n}|\mathcal L_{r,M}(E)-\mathcal U_M(E)|\le q^{r-n}$. Thus the total variation distance, defined as the supremum of this difference over events, is at most $q^{r-n}$. No distinguisher can recover more advantage from these projected samples, even with arbitrarily many of them. For $r$ around $\sqrt n$ this particular map discards almost all distinguishing information. The conclusion concerns a fixed secret-independent linear projection with the original right-hand side retained. It is not a lower bound for general classical reductions, adaptive transformations, or quantum algorithms.

Source check and adversarial audit

Regev's arXiv:2401.03703 abstract was inspected; its comment identifies it as a corrected posting of the 2009 paper, not a new 2024 removal of the quantum step. The Sarkar--Singha publisher page was checked for its classical concrete-analysis topic, without treating its full parameter analysis as verified here. The elementary padding and projection proofs above require no lattice-hardness theorem and claim no novelty. The audit checked that the padding mask is shared across samples, that the projected mixture uses one shared secret rather than independently resampling secrets, and that field rank needs prime $q$. The missing step remains a worst-case-to-average-case reduction with all the source's quantitative promises. No computation or theorem prover was used.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. The coordinating Codex (GPT-6; exact serving revision not exposed) also provided a separate model-assisted written-proof review of this manuscript. This is not independent human verification. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

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% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. No source-comparable classical worst-case lattice reduction or explanation ruling one out is established; the projection obstruction applies only to a specified linear sample transformation.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Oberwolfach Report 51/2014; original source passage and surrounding definitions read in the cached primary report on 11 October 2026. https://doi.org/10.4171/owr/2014/51.
• Oded Regev, On Lattices, Learning with Errors, Random Linear Codes, and Cryptography; abstract and corrected-2009-version comment inspected. https://arxiv.org/abs/2401.03703.
• Palash Sarkar and Subhadip Singha, Classical reduction of gap SVP to LWE: A concrete security analysis; publisher title and metadata inspected, full reduction not audited. https://www.aimsciences.org/article/doi/10.3934/amc.2021004.

Final status

Unresolved in this attempt. No source-comparable classical worst-case lattice reduction or explanation ruling one out is established; the projection obstruction applies only to a specified linear sample transformation.