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

Byunguk Kim; Minwoo Ko; Namhun Koo; Soonhak Kwon

arxiv: 2605.23224 · v1 · pith:7EUZB5JLnew · submitted 2026-05-22 · 💻 cs.IT · cs.CR· math.IT

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

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

Pith reviewed 2026-05-25 03:16 UTC · model grok-4.3

classification 💻 cs.IT cs.CRmath.IT

keywords APN exponentsboomerang uniformitybinomial functionscharacteristic 3finite fieldsdifferential uniformitypower functionscryptographic S-boxes

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The pith

Two classes of binomials over finite fields of characteristic 3 achieve boomerang uniformity zero and local perfect nonlinearity.

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

The paper first parametrizes the APN exponents in characteristic 3 that arise from the Zha-Wang construction and verifies numerically that this list accounts for all known APN power functions through dimension 13. It then uses this list to construct and prove two families of binomials F_r(x) = x^r (1 + χ(x)) that are locally perfect nonlinear and attain the smallest possible boomerang uniformity value of zero. A third binomial with exponent 3^n - 3 is shown to be locally almost perfect nonlinear with boomerang uniformity exactly one when n is odd and at least five, and its complete boomerang spectrum is obtained by evaluating the relevant character sums. These explicit families supply concrete examples of functions whose differential and boomerang properties meet the strongest known bounds in characteristic 3.

Core claim

The central claim is that two explicit classes of binomials built from APN exponents produced by the Zha-Wang parametrization are locally perfect nonlinear and possess boomerang uniformity zero, while the binomial with exponent r = 3^n - 3 is locally almost perfect nonlinear with boomerang uniformity one for every odd n at least five and has its full boomerang spectrum determined by character-sum evaluation.

What carries the argument

The binomial functions F_r(x) = x^r (1 + χ(x)) over F_{3^n} constructed from APN exponents of the Zha-Wang form, together with the evaluation of character sums that determine the boomerang spectrum.

If this is right

The two proved classes give infinite families of functions attaining the absolute minimum boomerang uniformity in characteristic 3.
The binomial with exponent 3^n - 3 supplies an explicit example whose boomerang spectrum is completely known for all odd n at least five.
Numerical verification up to n = 13 indicates that the Zha-Wang list already captures every previously recorded APN power function in this setting.
The proofs extend earlier binomial constructions with low boomerang uniformity from other characteristics to the case of characteristic 3.

Where Pith is reading between the lines

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

If a complete classification of APN exponents in characteristic 3 ever appears, the same binomial construction would immediately produce additional families with boomerang uniformity zero.
The pattern observed for odd n suggests that direct computation of the boomerang uniformity for the exponent 3^n - 3 at the next few odd values of n could serve as a quick consistency check.
Analogous binomial constructions might be attempted in characteristic 2 or 5 once comparable parametrizations of APN exponents become available.

Load-bearing premise

The Zha-Wang parametrization supplies every APN exponent that produces the claimed binomial families.

What would settle it

An explicit APN exponent for some n greater than 13 that lies outside the Zha-Wang list and yields a binomial with boomerang uniformity strictly larger than zero would falsify the claimed generality of the two zero-uniformity classes.

read the original abstract

Recent studies on binomials of the form $F_r(x) = x^r(1 + \chi(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in characteristic $3$, where instances of extremely low boomerang uniformity-namely $0$ or $1$-seem to arise more frequently than in other characteristics. First, we provide a systematic analysis of Almost Perfect Nonlinear (APN) power functions in characteristic $3$. We present an explicit parametrization of APN exponents arising from the construction of Zha and Wang and demonstrate through numerical results for $n \le 13$ that this generalized framework accounts for several previously known and sporadic APN instances. Building on this classification, we identify and rigorously prove two classes of binomials $F_r$ that are locally-PN and possess the minimum possible boomerang uniformity of $0$. These classes involve exponents derived from the aforementioned APN construction and the differentially 4-uniform exponent $r = 2 \cdot 3^{\frac{n-1}{2}} + 1$. Furthermore, we analyze the binomial $F_r$ with $r = 3^n - 3$, proving that it is locally-APN with boomerang uniformity $1$ when $n\ge 5$ is odd, and completely determine its boomerang spectrum through the evaluation of character sums. Our results clarify and extend existing studies on the cryptographic properties of binomials, providing a systematic characterization of several classes of binomials with very low boomerang uniformity in characteristic $3$.

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.

Desk Editor's Note private letter to a colleague

The paper parametrizes APN exponents in char 3 from Zha-Wang and proves boomerang uniformity 0 for two binomial families plus the full spectrum for a third.

read the letter

The core result is an explicit parametrization of APN exponents in characteristic 3 drawn from the Zha-Wang construction, followed by proofs that two families of binomials built on those exponents achieve boomerang uniformity 0, and a complete determination of the boomerang spectrum for the binomial with exponent 3^n-3 when n is odd and at least 5. The authors also note that the same parametrization recovers known APN instances up to n=13 in their numerical checks. What the paper does cleanly is separate the parametrization step from the uniformity proofs. The character-sum arguments for uniformity 0 and for the spectrum stand on their own; the numerics serve only as a coverage check rather than a premise. That separation keeps the claims direct. The local-PN property for the two classes and the local-APN property for the third case follow from the same evaluations. The work stays inside the Zha-Wang framework and inside characteristic 3, which limits how far the statements reach but also keeps the proofs manageable. No evidence appears of post-hoc parameter fitting or circular use of the data in the uniformity claims. The reliance on one external construction is acknowledged rather than hidden. This is a narrow but technically precise piece for people who already work on APN power functions and boomerang uniformity in finite fields. The proofs are standard for the area yet applied to new explicit families, so the paper deserves a serious referee.

Referee Report

0 major / 1 minor

Summary. The paper provides a systematic analysis of APN power functions in characteristic 3 by giving an explicit parametrization of APN exponents from the Zha-Wang construction, supported by numerical checks for n ≤ 13. It proves that two classes of binomials F_r are locally-PN with boomerang uniformity 0, using exponents from the APN construction and r = 2 · 3^{(n-1)/2} + 1. For the binomial with r = 3^n - 3 when n is odd and ≥5, it shows local-APN property with boomerang uniformity 1 and fully determines the boomerang spectrum using character sum evaluations.

Significance. If the results hold, the paper makes a significant contribution to the study of cryptographic properties of functions over finite fields of characteristic 3. The explicit parametrization and rigorous proofs of minimal boomerang uniformity for specific binomials, along with the complete boomerang spectrum determination, provide valuable insights for S-box design. The use of character-sum techniques for the spectrum result is a strength, offering a parameter-free derivation in that part.

minor comments (1)

[Abstract] The symbol χ(x) is introduced without prior definition or reference; a brief clarification would improve readability for readers unfamiliar with the context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation, the recognition of the paper's contributions to APN exponents and boomerang uniformity in characteristic 3, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives its main results on boomerang uniformity and spectra via explicit parametrization of APN exponents taken from the external Zha-Wang construction together with direct character-sum evaluations. Numerical checks up to n=13 serve only to confirm coverage of known cases and are not premises of the uniformity proofs. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via citation appear in the derivation chain. The central theorems remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard finite-field arithmetic and character-sum identities; no free parameters or new entities are introduced.

axioms (2)

standard math Finite fields of characteristic 3 satisfy the usual field axioms and Frobenius automorphism properties
Invoked throughout the definitions of the functions F_r and the differential/boomerang uniformity measures.
domain assumption Character sums over finite fields can be evaluated to obtain exact boomerang spectra
Used to completely determine the boomerang spectrum of the binomial with exponent 3^n-3.

pith-pipeline@v0.9.0 · 5852 in / 1417 out tokens · 34209 ms · 2026-05-25T03:16:54.475387+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 1 internal anchor

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[1] [1]

Blondeau, A

C. Blondeau, A. Canteaut and P. Charpin, “Differential Properties ofx7→x 2t−1,”IEEE Trans. Inf. Theory, vol. 57, no. 12, pp.8127-8137, Dec. 2011. DOI : 10.1109/TIT.2011.2169129

work page doi:10.1109/tit.2011.2169129 2011

[2] [2]

Arithmetization-oriented APN permutations,

L. Budaghyan and M. Pal, “Arithmetization-oriented APN permutations,”Des. Codes Cryp- togr., vol. 93, no. 4, pp.1067-1088, Apr. 2025. DOI : 10.1007/s10623-024-01487-7

work page doi:10.1007/s10623-024-01487-7 2025

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Boomerang connectivity table: A new cryptanalysis tool,

C. Cid, T. Huang, T. Peyrin, Y. Sasaki and L. Song, “Boomerang connectivity table: A new cryptanalysis tool,” EUROCRYPT 2018, Lect. Notes Comput. Sci. 10821, pp.683–714, Mar

work page 2018

[4] [4]

DOI : 10.1007/978-3-319-78375-8 22

work page doi:10.1007/978-3-319-78375-8

[5] [5]

Cyclotomy, Higher Congruences, and Waring’s Problem,

L. E. Dickson, “Cyclotomy, Higher Congruences, and Waring’s Problem,”Amer. J. Math., vol. 57, no. 2, pp. 391-424, Apr. 1935. DOI : 10.2307/2371217

work page doi:10.2307/2371217 1935

[6] [6]

Ternarym-Sequences with Three- Valued Cross-Correlation Function: New Decimations of Welch and Niho Type,

H. Dobbertin, T. Helleseth, V. Kumar, and H. Martinsen, “Ternarym-Sequences with Three- Valued Cross-Correlation Function: New Decimations of Welch and Niho Type,”IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1473-1481, May 2001. DOI : 10.1109/18.923728

work page doi:10.1109/18.923728 2001

[7] [7]

New Families of Almost Perfect Nonlinear Power Map- pings,

T. Helleseth, C. Rong, and D. Sandberg, “New Families of Almost Perfect Nonlinear Power Map- pings,”IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 475-485, Mar. 1999. DOI : 10.1109/18.748997

work page doi:10.1109/18.748997 1999

[8] [8]

The differential spectrum and boomerang spectrum of a class of locally-APN functions,

Z. Hu, N. Li, L. Xu, X. Zeng and X. Tang, “The differential spectrum and boomerang spectrum of a class of locally-APN functions,”Des. Codes Cryptogr., vol. 91, no. 5, pp.1695-1711, 2023. DOI : 10.1007/s10623-022-01161-w

work page doi:10.1007/s10623-022-01161-w 2023

[9] [9]

On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic,

N. Koo and S. Kwon, “On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic,”IEEE Trans. Inf. Theory, vol. 72, no. 3, pp.1928–1942, Mar. 2026. DOI : 10.1109/TIT.2026.3657603

work page doi:10.1109/tit.2026.3657603 1928

[10] [10]

Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic

N. Koo, S. Kwon, M. Ko, and B. Kim, “Locally-APN Binomials with Low Boomerang Unifor- mity in Odd Characteristic,” under revision forFinite Fields and Their Applications, preprint available at https://arxiv.org/abs/2512.17603

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

New families of APN functions in characteristic 3 or 5,

E. Leducq, “New families of APN functions in characteristic 3 or 5,” inArithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, vol. 574, pp.115-123, AMS, 2012. DOI : 10.1090/conm/574/11419

work page doi:10.1090/conm/574/11419 2012

[12] [12]

New results about the boomerang uniformity of permuta- tion polynomials,

K. Li, L. Qu, B. Sun, and C. Li, “New results about the boomerang uniformity of permuta- tion polynomials,”IEEE Trans. Inf. Theory, vol. 65, no. 11, pp.7542–7553, Nov. 2019. DOI : 10.1109/TIT.2019.2918531

work page doi:10.1109/tit.2019.2918531 2019

[13] [13]

Lidl and H

R. Lidl and H. Niederreiter,Finite fields. Cambridge university press, 1997

work page 1997

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A further study on the Ness-Helleseth function,

C. Lyu, X. Wang, and D. Zheng, “A further study on the Ness-Helleseth function,”Finite Fields Appl.vol. 98, Sep. 2024, Art. no. 102453. DOI : 10.1016/j.ffa.2024.102453

work page doi:10.1016/j.ffa.2024.102453 2024

[15] [15]

The Differential and Boomerang Properties of a Class of Bi- nomials,

S. Mesnager and H. Wu, “The Differential and Boomerang Properties of a Class of Bi- nomials,”IEEE Trans. Inf. Theory, vol. 71, no. 6, pp. 4854-4871, Jun. 2025. DOI : 10.1109/TIT.2025.3550851 29

work page doi:10.1109/tit.2025.3550851 2025

[16] [16]

A New Family of Ternary Almost Perfect Nonlinear Mappings,

G. J. Ness and T. Helleseth, “A New Family of Ternary Almost Perfect Nonlinear Mappings,” IEEE Trans. Inf. Theory, vol. 53, no. 7, pp.2581-2586, Jul. 2007. DOI : 10.1109/TIT.2007.899508

work page doi:10.1109/tit.2007.899508 2007

[17] [17]

Differentially uniform mappings for cryptography,

K. Nyberg, “Differentially uniform mappings for cryptography,” EUROCRYPT ’93,Lect. Notes Comput. Sci.vol. 765, pp. 55-64, 1994. DOI : 10.1007/3-540-48285-7 6

work page doi:10.1007/3-540-48285-7 1994

[18] [18]

A note on the differential spectrum of the Ness-Helleseth function,

K. Ren, M. Xiong, and H. Yan, “A note on the differential spectrum of the Ness-Helleseth function,”Adv. Math. Commun., vol. 22, pp. 106-131, Jun. 2026. DOI : 10.3934/amc.2026002

work page doi:10.3934/amc.2026002 2026

[19] [19]

A direct method for calculating the differential spectrum of an APN power mapping,

Y. Xia, F. Bao, S. Chen and T. Helleseth, “A direct method for calculating the differential spectrum of an APN power mapping,”Cryptogr. Commun., vol. 18, no. 1, pp.27-44, Jan. 2026. DOI : 10.1007/s12095-024-00764-5

work page doi:10.1007/s12095-024-00764-5 2026

[20] [20]

More Differential Properties of the Ness- Helleseth Function,

Y. Xia, F. Bao, S. Chen, C. Li and T. Helleseth, “More Differential Properties of the Ness- Helleseth Function,”IEEE Trans. Inf. Theory, vol. 70, no. 8, pp.6076-6090, Aug. 2024. DOI : 10.1109/TIT.2024.3408882

work page doi:10.1109/tit.2024.3408882 2024

[21] [21]

Further investigation on differential properties of the generalized Ness-Helleseth function,

Y. Xia, C. Li, F. Bao, S. Chen and T. Helleseth, “Further investigation on differential properties of the generalized Ness-Helleseth function,”Des. Codes Cryptogr., vol. 93, no. 6, pp.1549-1573, Jun. 2025. DOI : 10.1007/s10623-024-01525-4

work page doi:10.1007/s10623-024-01525-4 2025

[22] [22]

The differential spectrum of a ternary power mapping,

Y. Xia, X. Zhang, C. Lin and T. Helleseth, “The differential spectrum of a ternary power mapping,”Finite Fields Appl.vol. 64, 2020, Art. no. 101660. DOI : 10.1016/j.ffa.2020.101660

work page doi:10.1016/j.ffa.2020.101660 2020

[23] [23]

On the Inequivalence of Ness-Helleseth APN Functions,

X. Zeng, L. Hu, Y. Yang and W. Jiang, “On the Inequivalence of Ness-Helleseth APN Functions,” IACR Cryptol. ePrint Arch., 2007. 2007/379, 2007

work page 2007

[24] [24]

Power functions with low uniformity on odd characteristic finite fields,

Z. Zha and X. Wang, “Power functions with low uniformity on odd characteristic finite fields,” Sci. China Math., vol. 53, no. 8, pp.1931-1940, Aug. 2010. DOI : 10.1007/s11425-010-3149-x. 30

work page doi:10.1007/s11425-010-3149-x 1931