ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A composite-order cyclic discrete-log Fourier sample

Example

Work in the multiplicative group $F_{31}^{\times}$ and let $g = 9$ . Its powers satisfy

9^{2} \equiv 19, 9^{3} \equiv 16, 9^{4} \equiv 20, 9^{5} \equiv 25, 9^{15} \equiv 1 (m o d 31) .

Since the proper divisors of $15$ are $1, 3, 5$ and $9, 16, 25$ are all different from $1$ modulo $31$ , $g$ has exact order $q = 15$ . Set $x = 20 = g^{4}$ , so the discrete logarithm is $a = 4$ .

The cyclic DLP subgroup is $H = ⟨ (- 4, 1) ⟩ \leq Z_{15}^{2}$ . The Fourier label $(c, d) = (2, 8)$ lies in its annihilator because $d \equiv a c \equiv 4 \cdot 2 \equiv 8 (m o d 15)$ . Here $\gcd (2, 15) = 1$ and $2^{- 1} \equiv 8 (m o d 15)$ , so the sample recovers

d c^{- 1} \equiv 8 \cdot 8 \equiv 4 (m o d 15) .

Verification is direct: $9^{4} \equiv 20 = x (m o d 31)$ . Under the exact uniform annihilator sample law, this particular pair has probability $1 / 15$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms (17 April 2025) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)