Security consequences of Weil descent for the class-group realization

Definitions and mathematical statement

For a finite cyclic group $G=\langle g\rangle$, the discrete-logarithm problem is to recover $x$ modulo $|G|$ from $g$ and $h=g^x$. In a finite-field system $G$ is typically a specified subgroup of a finite field's multiplicative group. The source asks whether Weil descent compromises a proposed realization of such systems in a class group.

The realization itself is not included: no algebraic object defining the class group, map relating the finite-field subgroup to it, parameter family, subgroup order, or public encoding is supplied. These are essential to a statement comparing discrete-logarithm difficulty. Denote the missing construction by $\mathcal R$; the target is to determine whether a Weil-descent method yields a relevant algorithmic advantage against the discrete-logarithm instances produced by $\mathcal R$.

For context, Weil restriction for a finite field extension $K/k$ can represent $K$-rational points of a suitable algebraic object by $k$-rational points of an object of larger dimension. This geometric reformulation alone does not specify an attack complexity. A precise security question must fix field characteristic and extension degree, the construction and descent, instance distribution, complexity model, and benchmark algorithms; it must also distinguish rigorous bounds, heuristic running times, and concrete parameter security. No universal insecurity statement for class groups or finite-field systems is substituted.

The supplied summary reports Weil descent as a documented attack avenue, especially relevant to some small-characteristic or composite-degree constructions, while saying the specific proposed realization was not conclusively assessed. No exact construction or attack reference is supplied. The 2006 workshop problem and later partial-progress assessment are source attributions pending current primary-literature review.

Short English statement

Does Weil descent make discrete logarithms easier in the workshop's proposed class-group realization of a finite-field system? The realization, parameters and complexity criterion are absent, so the security comparison cannot be made precise from this excerpt alone.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800005$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06. Attributed to Gerhard Frey.
• IPAM, Workshop on Number Theory and Cryptography---Open Problems (2006), attachment 6219, PDF page 18, Problem 4: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule. Supplied PDF/legacy-PPT visual audit accessed 2026-07-29.
• The provenance reports inspecting 19 surviving attachments and excluding repeated animation/build material and solved definitions. No separate realization specification or Weil-descent complexity publication is supplied in this record.