gpt 6 astra ultra attempt on CS-519

Non-Hopfian maps: public lifts invert words and finite images erase kernels

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 group is nonabelian if some elements do not commute. It is Hopfian if every surjective endomorphism is injective; a non-Hopfian group admits a surjective endomorphism with nontrivial kernel. Using an infinite group computationally requires finite encodings of elements and explicit operations or decision problems.

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 agenda does not specify a group, endomorphism, security assumption, or attack task. The existence of a noninjective epimorphism alone is not treated as a security guarantee. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

Applications of infinite non-abelian groups to cryptography

Several cryptanalytic methods against group-based cryptography suggest possible use cases for infinite non-abelian groups; for example, non-Hopfian groups. We would like to develop tools to understand how these infinite objects can fit into a cryptographic context, and whether their use can be used to address some of the cryptanalytic methods deployed against group-based cryptography. [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.

Short English statement

The source explores whether infinite nonabelian groups could support new cryptographic constructions.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001322 (AIM-CRYPTOGRAPHY-0009). 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 AIM Workshop Problem Lists, source linked by catalogue record AIM-CRYPTOGRAPHY-0009. http://aimpl.org/postquantgroup/3/.

Verbatim supplied catalogue question

Applications of infinite non-abelian groups to cryptography

Several cryptanalytic methods against group-based cryptography suggest possible use cases for infinite non-abelian groups; for example, non-Hopfian groups. We would like to develop tools to understand how these infinite objects can fit into a cryptographic context, and whether their use can be used to address some of the cryptanalytic methods deployed against group-based cryptography.

Attempt and results

The source proposes infinite nonabelian platforms without a particular scheme. We test the tempting implication that a surjective noninjective endomorphism should be hard to invert. Two exact obstructions below explain why that implication needs additional distributions and encoding constraints.

Public generator lifts give an any-preimage algorithm.

Let $G=\langle a_1,\ldots,a_d\rangle$ and let $\phi:G\to G$ be surjective. Assume public words $w_i$ satisfy $\phi(w_i)=a_i$. For an input word $u=a_{i_1}^{\epsilon_1}\cdots a_{i_L}^{\epsilon_L}$, $\epsilon_j\in\{+1,-1\}$, replace each letter by $w_{i_j}^{\epsilon_j}$. The resulting word $v$ satisfies $\phi(v)=u$ by the homomorphism property. If $C=\max_i|w_i|$, its length is at most $CL$, and simple concatenation writes it using at most a constant times that many symbols. Different words representing the same group element may yield different preimages, which is allowed for an any-preimage relation. This does not recover a particular sampled preimage, obey a promised short-output distribution, or discover the lifts when they are unavailable. Surjectivity alone ensures lifts exist, not that their lengths are small or that one can find them efficiently.

Every finite homomorphic observation kills an iterated kernel.

Assume now $G$ is finitely generated and $\phi$ is a surjective endomorphism. For any finite group $Q$, the set $\operatorname{Hom}(G,Q)$ is finite: a homomorphism is determined by images of $d$ generators, so there are at most $|Q|^d$ choices. Precomposition $f\mapsto f\circ\phi$ is injective because $\phi$ is surjective. An injective map of a finite set is surjective. Thus every $f:G\to Q$ has the form $h\circ\phi$, and $f$ kills $\ker\phi$. The same argument applied to the surjective map $\phi^t$ shows that $f$ kills $\ker\phi^t$ for every $t\ge1$. Consequently distinct elements differing by an element of any iterated kernel have identical values under every finite-group homomorphism. If a target group $H$ is residually finite, meaning each nonidentity element has a nonidentity image in some finite group, any homomorphism $G\to H$ also kills these kernels: otherwise composing with a separating finite quotient contradicts the preceding result. In particular a finitely generated residually finite group cannot have a surjective noninjective endomorphism. This is a standard argument, rederived rather than treated as evidence of cryptographic hardness. It limits faithful finite homomorphic encodings; it does not say every finite byte serialization is a homomorphism.

Finite generation is genuinely needed for the finite-image claim.

Let $G=\bigoplus_{j\ge0} S_3$ be the group of finite-support sequences of permutations, with coordinatewise multiplication. It is infinite and nonabelian. The left shift $\phi(g_0,g_1,\ldots)=(g_1,g_2,\ldots)$ is a surjective endomorphism with nontrivial kernel the first $S_3$ factor; inserting an identity at the front is an explicit efficient right inverse in the finite-list encoding. The first-coordinate projection to $S_3$ does not kill that kernel. The group is not finitely generated: any finite set of sequences has support in a finite set of coordinates, as does the subgroup it generates. Thus this example both defeats non-Hopfianity-as-inversion-hardness and checks why the finite-generation hypothesis cannot be omitted above.

Context, adversarial audit, and gap

The official AIM workshop description was inspected; AIMPL retrieval failed and the exact agenda is preserved from the supplied snapshot. The supplied research summary mentions the first two mechanisms, so no novelty is claimed. The audit checked public lift availability, output length, finite versus infinite Hom sets, and the distinction between finite homomorphic observations and arbitrary encodings. No protocol, input distribution, or security reduction has been constructed, and no computation was run.

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. 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. The lift and finite-image obstructions do not build or analyze a specified secure infinite-group protocol; non-Hopfianity alone supplies neither one-wayness nor an efficient cryptographic representation.

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.

• Official AIM workshop description, Post-quantum group-based cryptography, 29 April--3 May 2024; provenance and scope inspected on 11 October 2026. AIMPL pages failed retrieval, so the exact problem wording remains supplied-snapshot material. https://aimath.org/workshops/upcoming/postquantgroup/.

Final status

Unresolved in this attempt. The lift and finite-image obstructions do not build or analyze a specified secure infinite-group protocol; non-Hopfianity alone supplies neither one-wayness nor an efficient cryptographic representation.