Space-Efficient Lattice-Basis Reduction

Space-Efficient Lattice-Basis Reduction

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.

Birthday methods search for useful collisions among many sampled objects; storing the sample collection creates a memory cost. Time-space tradeoffs measure the respective number of operations and stored words.

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 source algorithms and their baseline time and space bounds are not stated in the supplied question.

Source-record limitation: The birthday-method lattice algorithms and baseline time/space bounds to be preserved are absent.

Statement recorded in the supplied source.

Reduce the essentially time-sized space requirement of birthday-method algorithms for lattice-basis reduction while retaining their improved running time. [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

Reduce the memory needed by the cited lattice algorithms while preserving their improved running times.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000242 (OWR-831-016). 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 Complexity Theory $2006$, Oberwolfach Reports, DOI 10.4171/owr/2005/26, pp. 1431--1504 https://doi.org/10.4171/owr/2005/26.
• S3 Regev LWE survey https://cims.nyu.edu/~regev/papers/qcrypto.pdf.
• 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 30000242 and code OWR-831-016 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.

A limited buffer-detection time--space calculation

The report asks for less space in a generalized-birthday procedure, but its entire baseline algorithm and acceptable time tradeoff are not supplied in the extract. Assume integers $M\ge1$ and $1\le S\le M$. We test one simple potential route: discard all but $S$ stored values while continuing to compare each fresh independent uniform value from a set of size $M$ with the current buffer. The buffer may be selected adaptively from the past, contains at most $S$ values, and a collision is detected only by equality between a fresh value and one of those stored values. No comparisons to discarded values, compressed certificates, recomputation or dependent walk are allowed in this restricted model. Thus maintaining a success probability bounded away from zero in $O(\sqrt M)$ draws in this model requires $S=\Omega(\sqrt M)$. This is not a general collision-finding space lower bound: memory can be encoded differently, work can be repeated, and dependent iteration can change what information is retained. It also does not address the multi-list algebraic structure in generalized birthday algorithms.

Verification and remaining gaps

For $M=4$, store the latest value only, starting with one uniform sample and making two further fresh draws. Exhaustive enumeration of all $4^3$ triples gives detection probability $7/16$ for equality of adjacent samples, versus $5/8$ for any repeated value among all three samples. The exact missed case has first and third values equal but different from the discarded middle value. The proposed simple-discard route loses evidence in precisely this way. No improved implementation, acceptable time--space tradeoff for the source algorithm, or unrestricted lower bound was established.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/26. Relevant passages read: 2005/26 pp. 1490--1491, generalized-birthday reduction and its storage bottleneck.

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 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 batch016/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.