residue-apn: coefficient choice in Galois-ring lifts of APN permutations (paper, Lean 4 formalization, blueprint, finite computations)
Year: 2026
Type: software
Abstract: For odd m >= 5, q = 2^m and every odd r with 3 <= r < m and gcd(r, m) = 1, the reduced polynomial P_{m,r} = red((T_r^3)^(2^(m-2))), T_r = sum_{i<r} X^(2^i), has coefficients in F_2 and exactly 3r - 2 terms, is an APN permutation of F_q, and its formal derivative 1 + X^(2^(r-2)) + X^(2^(m-2)) has no zero on F_q and is two-to-one. Its ramification as a map of the projective line is determined for all admissible (m, r): 2^(m-r) finite ramification points, all with index and different exponent 2^(r-2), over the single finite branch value 1. The 0/1 lift satisfies an identity in Z[X] that evaluates it over every Galois ring GR(2^k, m) with m - 1 squarings and one general multiplication. For k >= 2 and every coefficient lift, the differential rows are described in the unit directions outside the kernel of the linear part of the derivative and in the top ideal, and a two-word, two-layer transition is excluded at the residue level. Lean 4 (v4.31.0-rc1, Mathlib d568c8c0; axioms a subset of propext, Classical.choice, Quot.sound; no sorry): Lemma 2.2 (polynomials over F_2 with quadratic support), Theorem 3.1 (all four items, all admissible m, r), the identity of Proposition 5.1, Theorem 4.1 items (1)-(3) and root multiplicities, Proposition 6.1 and Theorem 6.2 for an abstract finite ring with a surjection onto F_q (the Galois-ring instance is in the paper), Proposition 7.1 after the normalisation. Math-source digest c1844f5479439c4edffbfb4c3c5a7427f13c21b45bb7eef3edc75076889ffdda. Not formalized: Lemma 3.2, Corollary 3.3, Remark 4.2, the operation count, the ramification reading, the Galois-ring construction and cardinalities, cited bounds. Licences: the files of this record are distributed under two licences, each for a part of the files; they are not alternatives for the same file. MIT: code, machine-readable, configuration and auxiliary files (Python, shell, Lean sources and configuration, JSON, TSV, DOT, CFF, logs, records and checksums). CC BY 4.0: prose by the author: the paper sources and PDFs, the blueprint text and PDF, and the Markdown documents (the licence texts LICENSE and LICENSE-CC-BY-4.0.txt keep their own terms). The licence of every file is listed in release/DISTRIBUTION_FILES.tsv. Status: Version 1.0.0, deposited under the identifiers given in this record (software version DOI 10.5281/zenodo.23232181; paper version DOI 10.5281/zenodo.23232183).
Author: Fukui, Hiroki
Institution: Kyoto University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220894920