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arxiv: 2512.17603 · v2 · submitted 2025-12-19 · 💻 cs.IT · math.IT· math.NT

Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic

Namhun Koo , Soonhak Kwon , Minwoo Ko , Byunguk Kim This is my paper

Pith reviewed 2026-05-16 20:54 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.NT
keywords locally-APNboomerang uniformitybinomial functionsfinite fieldsodd characteristicdifferential spectrapower functions
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The pith

If the power difference equation has at most one suitable solution for each b, the binomial F_r is locally-APN with boomerang uniformity at most 2.

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

The paper establishes that over finite fields of odd characteristic, whenever gcd(r, q-1) divides 2 and the equation (x+1)^r - x^r = b has at most one solution x where both x and x+1 are quadratic residues, the binomial function F_r(x) = x^r + x^{r + (q-1)/2} is locally-APN and possesses boomerang uniformity bounded by 2. This generalizes earlier results that required q congruent to 3 modulo 4. The authors further determine the differential spectra of F_3 and F_{(2q-1)/3} and the boomerang spectrum of F_2 in characteristic 3. A reader would care because these properties control differential and boomerang attacks in cryptographic constructions over odd-characteristic fields.

Core claim

If there is at most one x in F_q with chi(x) = chi(x+1) = 1 satisfying (x+1)^r - x^r = b for every nonzero b, and if gcd(r, q-1) divides 2, then the function F_r is locally-APN and has boomerang uniformity at most 2. The proof proceeds by bounding the number of solutions to the relevant boomerang and differential equations under the stated hypothesis on the power differences.

What carries the argument

The binomial F_r(x) = x^r + x^{r + (q-1)/2} together with the hypothesis that (x+1)^r - x^r = b has at most one solution x satisfying chi(x) = chi(x+1) = 1 for each nonzero b.

If this is right

  • F_r satisfies the locally-APN property over F_q.
  • The boomerang uniformity of F_r is at most 2.
  • The differential spectra of F_3 and F_{(2q-1)/3} can be explicitly computed when the base field has characteristic 3.
  • The boomerang spectrum of F_2 is determined in characteristic 3.

Where Pith is reading between the lines

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

  • The same hypothesis may be checkable for additional exponents r by using character-sum estimates or Weil bounds.
  • Such binomials could serve as building blocks for substitution boxes that resist boomerang attacks in odd-characteristic ciphers.
  • The technique of reducing boomerang uniformity to a single-solution hypothesis on power differences may apply to other families of polynomials over finite fields.

Load-bearing premise

The assumption that the power-difference equation has at most one quadratic-residue solution x for every nonzero right-hand side b.

What would settle it

A concrete counterexample consisting of a field F_q, exponent r satisfying the gcd condition, and a nonzero b for which the equation (x+1)^r - x^r = b has two or more solutions x with chi(x) = chi(x+1) = 1 would falsify the claim.

read the original abstract

Recently, several studies have shown that when $q\equiv3\pmod{4}$, for certain choices of $r$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\Fq$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \Fq$ with $\chi(x)=\chi(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \Fqmul$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$.

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

2 major / 2 minor

Summary. The paper proves that if gcd(r, q-1) divides 2 and the equation (x+1)^r - x^r = b has at most one solution x in F_q with χ(x) = χ(x+1) = 1 for every nonzero b, then the binomial F_r(x) = x^r + x^{r+(q-1)/2} is locally-APN with boomerang uniformity at most 2. It additionally computes the differential spectra of F_3 and F_{(2q-1)/3} and the boomerang spectrum of F_2 when the characteristic p = 3.

Significance. The result supplies an explicit sufficient condition for a family of binomials to achieve optimal boomerang uniformity while remaining locally-APN, extending earlier constructions valid only for q ≡ 3 mod 4. The direct spectrum calculations for the three concrete exponents furnish verifiable instances that do not rely on the general hypothesis and can be used as test cases for cryptographic applications.

major comments (2)
  1. [Proof of the main implication] The proof of the central implication (that the stated solution-count hypothesis implies boomerang uniformity ≤ 2) is presented as building on standard character-sum estimates; the manuscript should explicitly identify the precise bound or lemma (e.g., the Weil-type estimate or the number of solutions to the auxiliary equation) used to control the boomerang count in the relevant section.
  2. [Differential spectra of F_3 and F_{(2q-1)/3}] For the differential-spectrum results on F_3 and F_{(2q-1)/3}, the paper reports the multiplicity of each possible value but does not state the exact formula or table for the number of solutions to F(x+a) - F(x) = b when a ≠ 0; including this explicit count (or the closed-form expression) is necessary to confirm the claimed spectra independently.
minor comments (2)
  1. [Preliminaries] The notation χ for the quadratic character is introduced but its precise definition (Legendre symbol or multiplicative character) should be restated once in the preliminaries for readers outside finite-field cryptography.
  2. [Introduction] A short comparison paragraph or table contrasting the new boomerang-uniformity bound with the values obtained in the cited prior works for the same exponents would clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We have revised the manuscript to address both major comments explicitly, improving the clarity and verifiability of the proofs and spectra.

read point-by-point responses

  1. Referee: [Proof of the main implication] The proof of the central implication (that the stated solution-count hypothesis implies boomerang uniformity ≤ 2) is presented as building on standard character-sum estimates; the manuscript should explicitly identify the precise bound or lemma (e.g., the Weil-type estimate or the number of solutions to the auxiliary equation) used to control the boomerang count in the relevant section.

    Authors: We agree that the proof benefits from an explicit citation of the bound. In the revised manuscript, we now identify the precise Weil-type estimate (Lemma 2.4, which bounds the number of solutions to the auxiliary character-sum equation) used to control the boomerang count in Section 3. This reference is inserted immediately before the application of the estimate, making the argument fully self-contained. revision: yes

  2. Referee: [Differential spectra of F_3 and F_{(2q-1)/3}] For the differential-spectrum results on F_3 and F_{(2q-1)/3}, the paper reports the multiplicity of each possible value but does not state the exact formula or table for the number of solutions to F(x+a) - F(x) = b when a ≠ 0; including this explicit count (or the closed-form expression) is necessary to confirm the claimed spectra independently.

    Authors: We concur that explicit solution counts strengthen the presentation. The revised manuscript now includes closed-form expressions for the number of solutions to F(x+a) − F(x) = b (a ≠ 0) for both F_3 and F_{(2q−1)/3}, presented as Theorems 4.2 and 4.4 respectively, together with the resulting multiplicity tables. revision: yes

Circularity Check

0 steps flagged

No significant circularity; implication from explicit hypothesis with independent verifications

full rationale

The central result is an explicit implication: given the hypothesis that (x+1)^r - x^r = b has at most one solution x with χ(x)=χ(x+1)=1 for all b≠0 (when gcd(r,q-1)|2), then F_r is locally-APN with boomerang uniformity ≤2. The hypothesis is stated as a premise rather than derived from the claimed uniformity bound. Direct computations of the differential spectra for F_3 and F_{(2q-1)/3} and the boomerang spectrum for F_2 (p=3) supply concrete verification for those cases without invoking the general hypothesis. No equations reduce the uniformity bound to a fitted parameter, self-defined quantity, or self-citation chain; the derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background results from finite-field algebra and character theory; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Finite fields F_q exist for every prime power q and satisfy the usual field axioms and the structure theorem for multiplicative groups.
    Invoked implicitly when defining F_q and the exponentiation map.
  • standard math The quadratic character χ is a homomorphism from F_q^* to {±1} with kernel the squares.
    Used to state the solution-count condition.

pith-pipeline@v0.9.0 · 5480 in / 1509 out tokens · 40012 ms · 2026-05-16T20:54:41.411758+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On APN Exponents and the Differential and Boomerang Properties of Binomials in Characteristic 3

    cs.IT 2026-05 unverdicted novelty 6.0

    Parametrization of APN exponents in char 3 with proofs that two binomial classes have boomerang uniformity 0 and a third class has uniformity 1 for odd n >= 5.

  2. Walsh Spectrum and Boomerang Properties of Locally-APN Niho Functions

    cs.IT 2026-05 unverdicted novelty 5.0

    A Niho power function is locally-APN if and only if its Walsh spectrum is four-valued in {-p^m, 0, p^m, 2p^m}.

Reference graph

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