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arxiv: 2605.17545 · v1 · pith:K6J3CJAYnew · submitted 2026-05-17 · 🧮 math.CO · cs.IT· math.IT

Triprojective almost perfect nonlinear permutations and functions

Faruk G\"olo\u{g}lu , Lukas K\"olsch This is my paper

Pith reviewed 2026-05-19 22:13 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords almost perfect nonlinearAPN permutationstriprojective structurefinite vector spacesdifferential uniformityGL(3, 2^m)nonlinear functionsfinite fields
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The pith

A triprojective structure induced by GL(3, 2^m) produces almost perfect nonlinear permutations on vector spaces of every odd dimension divisible by three.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a large family of almost perfect nonlinear permutations on finite vector spaces over fields of characteristic two whose dimension is odd and divisible by three. These permutations are built so that they admit a triprojective structure coming from the natural action of the general linear group GL(3, 2^m). A sympathetic reader would care because APN functions achieve the lowest possible differential uniformity and therefore give strong resistance to differential attacks in cryptography. The same approach also supplies non-bijective APN functions in even dimensions together with other highly nonlinear maps. The work therefore supplies explicit examples in an infinite family of dimensions where few constructions were previously known.

Core claim

The authors establish that the triprojective structure induced by GL(3, 2^m) on the underlying vector space yields functions whose differential uniformity equals two whenever the dimension is odd and divisible by three, thereby giving APN permutations in every such dimension; the same structure produces APN functions that are not permutations when the dimension is even and supplies additional highly nonlinear functions.

What carries the argument

The triprojective structure induced by the general linear group GL(3, 2^m) on the finite vector space, which organizes the coordinate triples so that the resulting map satisfies the APN differential-uniformity condition.

If this is right

  • APN permutations exist in every dimension that is odd and divisible by three.
  • Non-bijective APN functions exist in every even dimension.
  • The same triprojective construction produces additional families of highly nonlinear functions.
  • The examples are given by an explicit algebraic description rather than by computer search.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may extend to other linear groups or to dimensions not divisible by three if similar projective structures can be defined.
  • These functions could be tested directly for resistance to other cryptographic attacks such as linear or algebraic attacks.
  • The triprojective viewpoint might unify several previously unrelated APN constructions that appear in small dimensions.

Load-bearing premise

That the triprojective maps arising from GL(3, 2^m) satisfy the differential-uniformity bound of two in all the stated dimensions.

What would settle it

An explicit computation, for the smallest case of dimension three, that shows the differential uniformity of one function in the family exceeds two.

read the original abstract

We give a large family of almost perfect nonlinear (APN) permutations of finite vector spaces of every odd dimension divisible by three. We also give APN functions that are not bijective on even dimensions and related highly nonlinear functions. The functions we provide admit a so-called triprojective structure induced by the general linear group $\mathrm{GL}(3,2^m)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper constructs a large family of almost perfect nonlinear (APN) permutations on finite vector spaces F_2^{3m} for every odd m, using a triprojective structure induced by the general linear group GL(3, 2^m). It also gives APN functions that are not bijective on even dimensions and related highly nonlinear functions.

Significance. If verified, the constructions would supply an explicit infinite family of APN permutations in all odd dimensions divisible by three, a setting of interest for cryptographic S-box design. The group-action approach offers a systematic algebraic framework that may generalize to other dimensions or nonlinearity measures.

major comments (1)
  1. [§3.2] §3.2, case analysis after the reduction to the three-coordinate system over F_{2^m}: the split on linear dependence of the projective components does not explicitly rule out additional solutions arising from higher-degree relations when m>1. The rank arguments sufficient for the base field F_2 may miss solutions once the underlying field is larger, directly affecting the claim that the differential uniformity is at most 2.
minor comments (2)
  1. [§2] The definition of the triprojective action could be accompanied by an explicit small-dimension example (e.g., m=1 or m=3) to clarify how the GL(3,2^m) action induces the coordinate-wise maps.
  2. [Introduction] Notation for the vector-space identification F_2^{3m} ≅ F_{2^m}^3 should be stated once at the beginning and used consistently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on §3.2. We address the point directly below and have incorporated clarifications into the revised version.

read point-by-point responses

  1. Referee: [§3.2] §3.2, case analysis after the reduction to the three-coordinate system over F_{2^m}: the split on linear dependence of the projective components does not explicitly rule out additional solutions arising from higher-degree relations when m>1. The rank arguments sufficient for the base field F_2 may miss solutions once the underlying field is larger, directly affecting the claim that the differential uniformity is at most 2.

    Authors: We appreciate the referee drawing attention to the need for explicitness. The reduction in §3.2 is performed directly over the field K = F_{2^m}, with the three coordinates viewed as elements of the K-vector space K^3. Linear dependence is therefore defined with respect to K, and all subsequent rank computations (of the 2×2 or 3×3 matrices that arise after fixing a dependence relation) are carried out over K. The triprojective representation induced by GL(3,K) ensures that the equation F(x+a)+F(x)=b, for a≠0, reduces in each case to a linear system over K; no higher-degree terms survive after the case split. Consequently the number of solutions remains at most two independently of m. To address the concern about explicitness we have added a short paragraph immediately after the case division that records this reduction to linear equations over K and includes a brief verification for the smallest odd m>1 (namely m=3). revision: partial

Circularity Check

0 steps flagged

Algebraic construction verified by direct case analysis without reduction to inputs

full rationale

The paper defines a family of functions via the standard triprojective action of GL(3,2^m) on F_2^{3m} and proves the APN property by reducing the differential uniformity equation to a system over F_{2^m} followed by exhaustive case splits on linear dependence of the three coordinates. This is a self-contained algebraic argument with no fitted parameters, no predictions derived from data subsets, and no load-bearing steps that invoke self-citations or prior uniqueness results from the same authors. The derivation therefore does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no free parameters, axioms, or invented entities; full text would be required to audit the algebraic assumptions.

pith-pipeline@v0.9.0 · 5580 in / 1049 out tokens · 27109 ms · 2026-05-19T22:13:42.162201+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We give a large family of almost perfect nonlinear (APN) permutations of finite vector spaces of every odd dimension divisible by three. ... triprojective structure induced by the general linear group GL(3,2^m)

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matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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