Alphabeta MathHow 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.
Extended-Euclidean modular inverse with bounded bit cost
Definition
Input an integer and the least residue . Set and . For , divide
and update to the least residue of modulo . Stop at the first index for which . Return the least residue if , and return NO_INVERSE otherwise. For , the first division has zero divisor under this indexing; handle it separately by returning NO_INVERSE.
The algorithm uses binary integer arithmetic. Its correctness and bit bound for are proved in Correctness and bit cost of the modular inverse recurrence.
Depends on
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$
- Division with remainder in $\mathbb{Z}$: for $a \in \mathbb{Z}$ and $b > 0$ there are unique $q, r \in \mathbb{Z}$ with $a = qb + r$ and $0 \le r < b$
- The Euclidean algorithm: for $a \in \mathbb{Z}$ and $b > 0$ the remainder descent from $(a,b)$ terminates, and the last nonzero remainder is $\gcd(a,b)$
- For $n\ge1$, $[a]_n$ is a unit if and only if $\gcd(a,n)=1$
Used by
Dependency tree · two levels
35 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)
- D. Proos and C. Zalka, Shor's discrete logarithm quantum algorithm for elliptic curves (standard reference, not scraped)