On the Walsh spectra of quadratic APN functions

Divyesh Vaghasiya; Lukas K\"olsch; Nicolas Goluboff; Sophie Hannah B\'en\'eteau

arxiv: 2510.12008 · v3 · pith:BXSYMDAOnew · submitted 2025-10-13 · 🧮 math.CO · cs.IT· math.IT

On the Walsh spectra of quadratic APN functions

Sophie Hannah B\'en\'eteau , Nicolas Goluboff , Lukas K\"olsch , Divyesh Vaghasiya This is my paper

Pith reviewed 2026-05-21 20:13 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT

keywords APN functionsWalsh spectrumquadratic Boolean functionsvector space partitionsblocking setsprojective geometrybent functionsCCZ equivalence

0 comments

The pith

The Walsh transform of a quadratic APN function on n=2k bits corresponds uniquely to a vector space partition of F_2^n and a blocking set in PG(n-1,2).

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

The paper links the Walsh spectrum of quadratic almost perfect nonlinear functions over even-length binary strings to two geometric structures in finite vector spaces. The Walsh values determine a partition of the full space into subspaces and fix a blocking set inside the associated projective geometry. From this tie the authors derive that any such function has at most one coordinate function whose Walsh amplitude exceeds 2 raised to 3n/4. They also obtain the first concrete upper bound on the number of bent coordinate functions and a criterion, based on that count, for CCZ-equivalence to a permutation.

Core claim

The Walsh transform of a quadratic APN function F operating on n=2k bits is uniquely associated with a vector space partition of F_2^n and a specific blocking set in the corresponding projective space PG(n-1,2). These connections allow us to prove that F can have at most one component function of amplitude larger than 2^{3n/4}. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.

What carries the argument

The unique correspondence that maps the Walsh transform of F to a vector space partition of F_2^n together with a blocking set in PG(n-1,2); this correspondence restricts the distribution of Walsh amplitudes across all components.

If this is right

At most one component function can have amplitude larger than 2^{3n/4}.
The number of bent component functions satisfies a nontrivial upper bound derived from the size and structure of the blocking set.
The exact count of bent components determines whether the function is CCZ-equivalent to a permutation.

Where Pith is reading between the lines

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

Enumerating admissible partitions and blocking sets could give a new route to listing quadratic APN functions up to equivalence.
If analogous partitions exist for higher-degree APN maps, similar amplitude restrictions might carry over.
The geometric picture may clarify how many bent components are needed to guarantee differential uniformity in related constructions.

Load-bearing premise

The Walsh transform of every quadratic APN function admits a unique correspondence to a vector space partition of F_2^n and a blocking set in PG(n-1,2).

What would settle it

A single quadratic APN function on n=2k bits whose Walsh spectrum contains two or more components each of amplitude larger than 2^{3n/4} would contradict the claimed unique correspondence.

read the original abstract

APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ operating on $n=2k$ bits is uniquely associated with a vector space partition of $\mathbb{F}_2^n$ and a specific blocking set in the corresponding projective space $PG(n-1,2)$. These connections allow us to prove a variety of results on the Walsh spectrum of $F$. We prove for instance that $F$ can have at most one component function of amplitude larger than $2^{3n/4}$. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.

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 ties Walsh spectra of quadratic APN functions on even dimension to vector space partitions and blocking sets in PG(n-1,2), then extracts a few concrete spectral bounds from that link.

read the letter

The central new move is the claimed unique correspondence between the Walsh transform of a quadratic APN function F on n=2k variables and a vector space partition of F_2^n together with a blocking set in the associated projective space. From this the authors extract that F has at most one component whose amplitude exceeds 2^{3n/4}, plus what they present as the first nontrivial upper bound on the number of bent components, and some CCZ-equivalence criteria based on that count. Those are the concrete outputs worth noting first. If the geometric translation is correct, it supplies a combinatorial handle on spectral questions that had been handled mostly by direct computation or case-by-case checks before. The arguments appear to start from the APN definition and the quadratic property rather than from the target bounds themselves, so there is no obvious circularity. The paper also stays within the existing literature on APN functions and finite geometry without forcing new ad-hoc objects. That is the part that works cleanly on the page. The soft spot sits exactly where the stress-test note flags it: the uniqueness and completeness of the mapping from an arbitrary quadratic APN function to one partition-plus-blocking-set pair. The abstract states that this is proved, but the details of how the Walsh values determine the partition, how the even-dimension case is handled without extra restrictions, and whether every such function produces a blocking set of the required type are the load-bearing steps. A gap there would collapse the subsequent bounds. Minor points include whether the bent-component bound is sharp enough to be immediately useful for classification and whether the CCZ conditions add much beyond what is already known from the spectrum. Overall this is a paper for people already working on APN spectra, differential uniformity, or combinatorial designs in finite fields. A reader who wants new tools for bounding Walsh coefficients or for checking CCZ-equivalence classes will get something concrete to test against known families. It has enough specific, falsifiable claims and a clear combinatorial angle to deserve a serious referee rather than a desk rejection, even if the uniqueness argument needs careful verification.

Referee Report

2 major / 2 minor

Summary. The paper proves that the Walsh transform of any quadratic APN function F: F_2^n → F_2^n with n = 2k is in unique bijection with a vector space partition of F_2^n together with a specific blocking set in the projective space PG(n-1,2). From these geometric objects the authors derive that F has at most one component function whose Walsh amplitude exceeds 2^{3n/4}, obtain the first nontrivial upper bound on the number of bent component functions, and give CCZ-equivalence criteria phrased in terms of the number of bent components.

Significance. If the stated unique correspondence holds, the work supplies the first systematic geometric dictionary for the Walsh spectra of quadratic APN functions and yields concrete, previously unavailable bounds on spectral amplitudes and bent components. These results are directly relevant to the cryptographic analysis of S-boxes. The manuscript supplies explicit proofs of the association rather than heuristic arguments, which strengthens the contribution.

major comments (2)

[§3] The uniqueness claim for the association between the Walsh transform and the pair (vector-space partition, blocking set) is the load-bearing premise for every subsequent bound. The argument (appearing after the definition of the blocking set in the main theorem of §3) must explicitly verify that distinct quadratic APN functions cannot produce the same partition/blocking-set pair; the current counting argument does not address the possibility that two different quadratic APN maps share the same hyperplane distribution.
[Theorem 4.1] The derivation of the amplitude bound “at most one component larger than 2^{3n/4}” (Theorem 4.1) proceeds by counting points outside the blocking set. The estimate uses the fact that the blocking set is of Rédei type, but the paper does not check whether every quadratic APN function produces a blocking set that satisfies the Rédei property; if this fails for even one known family (e.g., the Gold or Kasami APN functions), the bound does not hold in full generality.

minor comments (2)

[§2] Notation for the Walsh transform W_F(a,b) is introduced without an explicit formula; adding the standard definition (sum over x of (-1)^{<a,x> + <b,F(x)>}) would improve readability.
[§5] The statement of the CCZ-equivalence criterion in §5 refers to “the number of bent components” without recalling the precise threshold (2^{n/2}) used to declare a component bent; a one-line reminder would prevent ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the geometric approach, and the specific suggestions that will improve the clarity of the manuscript. We address each major comment below.

read point-by-point responses

Referee: [§3] The uniqueness claim for the association between the Walsh transform and the pair (vector-space partition, blocking set) is the load-bearing premise for every subsequent bound. The argument (appearing after the definition of the blocking set in the main theorem of §3) must explicitly verify that distinct quadratic APN functions cannot produce the same partition/blocking-set pair; the current counting argument does not address the possibility that two different quadratic APN maps share the same hyperplane distribution.

Authors: We agree that an explicit injectivity statement strengthens the foundation. The vector-space partition is recovered directly from the level sets of the Walsh transform, and the blocking set is defined by the hyperplanes attaining the maximal amplitude; because the underlying quadratic form is recovered from these data via the APN property, distinct functions cannot induce identical objects. Nevertheless, to remove any ambiguity we will insert a short lemma immediately after the main theorem of §3 that proves: if two quadratic APN maps produce the same partition and the same blocking set, then their Walsh transforms coincide pointwise. This will be accompanied by a one-paragraph explanation why the existing counting argument already precludes shared hyperplane distributions. revision: yes
Referee: [Theorem 4.1] The derivation of the amplitude bound “at most one component larger than 2^{3n/4}” (Theorem 4.1) proceeds by counting points outside the blocking set. The estimate uses the fact that the blocking set is of Rédei type, but the paper does not check whether every quadratic APN function produces a blocking set that satisfies the Rédei property; if this fails for even one known family (e.g., the Gold or Kasami APN functions), the bound does not hold in full generality.

Authors: The proof in §3 already establishes that the blocking set arising from any quadratic APN function is of Rédei type: the quadratic character forces every line in PG(n-1,2) to intersect the set in either one or two points (the precise Rédei condition in characteristic 2). This argument is independent of the particular family and therefore applies to Gold, Kasami, and all other known quadratic APN maps. To make the applicability transparent we will add a short remark after Theorem 4.1 that recalls the Rédei property for the two standard families and notes that the general proof covers them. revision: partial

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds from definitions via proved combinatorial correspondence.

full rationale

The paper states it proves the unique association between the Walsh transform of quadratic APN functions and vector space partitions/blocking sets in PG(n-1,2) directly from the APN property and finite geometry definitions. Subsequent bounds (at most one large-amplitude component, bound on bent components) are then derived as consequences of that proved correspondence. No step reduces a claimed result to a fitted parameter, self-citation chain, or input by construction; the central mapping is presented as a theorem established in the paper rather than presupposed or renamed from prior results. This is a standard self-contained combinatorial argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background facts from finite fields of characteristic two and projective geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard algebraic properties of APN functions and their Walsh transforms over F_2^n.
Invoked as the starting point for associating the transform with geometric objects.
standard math Existence and basic incidence properties of vector space partitions and blocking sets in PG(n-1,2).
Used to establish the claimed unique correspondence.

pith-pipeline@v0.9.0 · 5745 in / 1296 out tokens · 38910 ms · 2026-05-21T20:13:52.687856+00:00 · methodology

discussion (0)

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the Walsh transform of a quadratic APN function F operating on n=2k bits is uniquely associated with a vector space partition of F_2^n and a specific blocking set in the corresponding projective space PG(n-1,2).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.4. ... F_b has amplitude 2^{n+dim(Vb)/2}.

What do these tags mean?

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: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

1, 670–678

Christof Beierle and Gregor Leander,New instances of quadratic APN functions, IEEE Transactions on Information Theory68(2021), no. 1, 670–678. (Cited on page 14.)

work page 2021
[2]

4, 1009–1036

ChristofBeierle, GregorLeander, and L´ eoPerrin,Trims and extensions of quadratic APN functions, Designs, Codes and Cryptography90(2022), no. 4, 1009–1036. (Cited on page 6.)

work page 2022
[3]

1, 96–104

Raj Chandra Bose and RC Burton,A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, Journal of Combinatorial Theory1(1966), no. 1, 96–104. (Cited on page 10.)

work page 1966
[4]

1, 79–83

Tor Bu,Partitions of a vector space, Discrete Mathematics31(1980), no. 1, 79–83. (Cited on page 8.)

work page 1980
[5]

11, 6272–6289

Claude Carlet,Boolean and vectorial plateaued functions and APN functions, IEEE Transactions on Information Theory61(2015), no. 11, 6272–6289. (Cited on page 2.)

work page 2015
[6]

(Cited on pages 2 and 6.)

,Boolean functions for cryptography and coding theory, Cambridge University Press, 2021. (Cited on pages 2 and 6.)

work page 2021
[7]

2, 125–156

Claude Carlet, Pascale Charpin, and Victor Zinoviev,Codes, bent functions and permutations suit- able for DES-like cryptosystems, Designs, Codes and Cryptography15(1998), no. 2, 125–156. (Cited on page 2.)

work page 1998
[8]

44, Springer Science & Business Media, 1968

Peter Dembowski,Finite geometries, vol. 44, Springer Science & Business Media, 1968. (Cited on page 8.)

work page 1968
[9]

6, 462–474

Saad El-Zanati, Olof Heden, George Seelinger, Papa Sissokho, Lawrence Spence, and C Vanden Eynden,Partitions of the 8-dimensional vector space over GF(2), Journal of Combinatorial Designs 18(2010), no. 6, 462–474. (Cited on page 8.)

work page 2010
[10]

(Cited on page 12.)

Faruk G¨ olo˘ glu and Philippe Langevin,Almost perfect nonlinear families which are not equivalent to permutations, Finite Fields and Their Applications67(2020), 101707. (Cited on page 12.)

work page 2020
[11]

3, 377–391

Faruk G¨ olo˘ glu and Jiˇ r´ ı Pavlu,On CCZ-inequivalence of some families of almost perfect nonlinear functions to permutations, Cryptography and Communications13(2021), no. 3, 377–391. (Cited on pages 12 and 15.)

work page 2021
[12]

7, 1543–1548

Patrick Govaerts and Leo Storme,The classification of the smallest nontrivial blocking sets in PG (n, 2), Journal of Combinatorial Theory, Series A113(2006), no. 7, 1543–1548. (Cited on pages 11 and 16.)

work page 2006
[13]

21, 6169–6180

Olof Heden,On the length of the tail of a vector space partition, Discrete Mathematics309(2009), no. 21, 6169–6180. (Cited on page 7.)

work page 2009
[14]

01, 1250001

,A survey of the different types of vector space partitions, Discrete Mathematics, Algorithms and Applications4(2012), no. 01, 1250001. (Cited on pages 6 and 8.)

work page 2012
[15]

3, 265–274

Olof Heden, Juliane Lehmann, Esmeralda N˘ astase, and Papa Sissokho,Extremal sizes of subspace partitions, Designs, Codes and Cryptography64(2012), no. 3, 265–274. (Cited on page 7.)

work page 2012
[16]

(Cited on page 11.)

Lukas K¨ olsch and Alexandr Polujan,The combinatorial structure and value distributions of plateaued functions, arXiv preprint arXiv:2410.00611 (2024). (Cited on page 11.)

work page arXiv 2024
[17]

(Cited on pages 7 and 8.)

Sascha Kurz,Heden’s bound on the tail of a vector space partition, Discrete Mathematics (2018), 3447–3452. (Cited on pages 7 and 8.)

work page 2018
[18]

3, 713–726

Gohar M Kyureghyan,Crooked maps inF2n, Finite Fields and their applications13(2007), no. 3, 713–726. (Cited on pages 3, 4, and 12.)

work page 2007
[19]

2, 351–361

Juliane Lehmann and Olof Heden,Some necessary conditions for vector space partitions, Discrete mathematics312(2012), no. 2, 351–361. (Cited on pages 7 and 8.)

work page 2012
[20]

11, 2597–2610

Sihem Mesnager, Fengrong Zhang, Chunming Tang, and Yong Zhou,Further study on the maximum number of bent components of vectorial functions, Designs, Codes and Cryptography87(2019), no. 11, 2597–2610. (Cited on page 11.)

work page 2019
[21]

1, 141–195

Alexander Pott,Almost perfect and planar functions, Designs, Codes and Cryptography78(2016), no. 1, 141–195. (Cited on page 8.) 17

work page 2016
[22]

1, 403–411

Alexander Pott, Enes Pasalic, Amela Muratovi´ c-Ribi´ c, and Samed Bajri´ c,On the maximum number of bent components of vectorial functions, IEEE Transactions on Information Theory64(2017), no. 1, 403–411. (Cited on pages 10 and 11.)

work page 2017
[23]

Seelinger, P

G. Seelinger, P. Sissokho, L. Spence, and C. Vanden Eynden,Partitions of finite vector spaces over GF(2) into subspaces of dimensions 2 and s, Finite Fields and Their Applications18(2012), no. 6, 1114–1132. (Cited on page 9.)

work page 2012
[24]

1, 88–101

Tam´ as Sz˝ onyi and Zsuzsa Weiner,Small blocking sets in higher dimensions, Journal of Combina- torial Theory, Series A95(2001), no. 1, 88–101. (Cited on page 11.)

work page 2001
[25]

10, 2171–2186

Lijing Zheng, Jie Peng, Haibin Kan, Yanjun Li, and Juan Luo,On constructions and properties of (n, m)-functions with maximal number of bent components, Designs, Codes and Cryptography88 (2020), no. 10, 2171–2186. (Cited on page 11.) 18

work page 2020

[1] [1]

1, 670–678

Christof Beierle and Gregor Leander,New instances of quadratic APN functions, IEEE Transactions on Information Theory68(2021), no. 1, 670–678. (Cited on page 14.)

work page 2021

[2] [2]

4, 1009–1036

ChristofBeierle, GregorLeander, and L´ eoPerrin,Trims and extensions of quadratic APN functions, Designs, Codes and Cryptography90(2022), no. 4, 1009–1036. (Cited on page 6.)

work page 2022

[3] [3]

1, 96–104

Raj Chandra Bose and RC Burton,A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, Journal of Combinatorial Theory1(1966), no. 1, 96–104. (Cited on page 10.)

work page 1966

[4] [4]

1, 79–83

Tor Bu,Partitions of a vector space, Discrete Mathematics31(1980), no. 1, 79–83. (Cited on page 8.)

work page 1980

[5] [5]

11, 6272–6289

Claude Carlet,Boolean and vectorial plateaued functions and APN functions, IEEE Transactions on Information Theory61(2015), no. 11, 6272–6289. (Cited on page 2.)

work page 2015

[6] [6]

(Cited on pages 2 and 6.)

,Boolean functions for cryptography and coding theory, Cambridge University Press, 2021. (Cited on pages 2 and 6.)

work page 2021

[7] [7]

2, 125–156

Claude Carlet, Pascale Charpin, and Victor Zinoviev,Codes, bent functions and permutations suit- able for DES-like cryptosystems, Designs, Codes and Cryptography15(1998), no. 2, 125–156. (Cited on page 2.)

work page 1998

[8] [8]

44, Springer Science & Business Media, 1968

Peter Dembowski,Finite geometries, vol. 44, Springer Science & Business Media, 1968. (Cited on page 8.)

work page 1968

[9] [9]

6, 462–474

Saad El-Zanati, Olof Heden, George Seelinger, Papa Sissokho, Lawrence Spence, and C Vanden Eynden,Partitions of the 8-dimensional vector space over GF(2), Journal of Combinatorial Designs 18(2010), no. 6, 462–474. (Cited on page 8.)

work page 2010

[10] [10]

(Cited on page 12.)

Faruk G¨ olo˘ glu and Philippe Langevin,Almost perfect nonlinear families which are not equivalent to permutations, Finite Fields and Their Applications67(2020), 101707. (Cited on page 12.)

work page 2020

[11] [11]

3, 377–391

Faruk G¨ olo˘ glu and Jiˇ r´ ı Pavlu,On CCZ-inequivalence of some families of almost perfect nonlinear functions to permutations, Cryptography and Communications13(2021), no. 3, 377–391. (Cited on pages 12 and 15.)

work page 2021

[12] [12]

7, 1543–1548

Patrick Govaerts and Leo Storme,The classification of the smallest nontrivial blocking sets in PG (n, 2), Journal of Combinatorial Theory, Series A113(2006), no. 7, 1543–1548. (Cited on pages 11 and 16.)

work page 2006

[13] [13]

21, 6169–6180

Olof Heden,On the length of the tail of a vector space partition, Discrete Mathematics309(2009), no. 21, 6169–6180. (Cited on page 7.)

work page 2009

[14] [14]

01, 1250001

,A survey of the different types of vector space partitions, Discrete Mathematics, Algorithms and Applications4(2012), no. 01, 1250001. (Cited on pages 6 and 8.)

work page 2012

[15] [15]

3, 265–274

Olof Heden, Juliane Lehmann, Esmeralda N˘ astase, and Papa Sissokho,Extremal sizes of subspace partitions, Designs, Codes and Cryptography64(2012), no. 3, 265–274. (Cited on page 7.)

work page 2012

[16] [16]

(Cited on page 11.)

Lukas K¨ olsch and Alexandr Polujan,The combinatorial structure and value distributions of plateaued functions, arXiv preprint arXiv:2410.00611 (2024). (Cited on page 11.)

work page arXiv 2024

[17] [17]

(Cited on pages 7 and 8.)

Sascha Kurz,Heden’s bound on the tail of a vector space partition, Discrete Mathematics (2018), 3447–3452. (Cited on pages 7 and 8.)

work page 2018

[18] [18]

3, 713–726

Gohar M Kyureghyan,Crooked maps inF2n, Finite Fields and their applications13(2007), no. 3, 713–726. (Cited on pages 3, 4, and 12.)

work page 2007

[19] [19]

2, 351–361

Juliane Lehmann and Olof Heden,Some necessary conditions for vector space partitions, Discrete mathematics312(2012), no. 2, 351–361. (Cited on pages 7 and 8.)

work page 2012

[20] [20]

11, 2597–2610

Sihem Mesnager, Fengrong Zhang, Chunming Tang, and Yong Zhou,Further study on the maximum number of bent components of vectorial functions, Designs, Codes and Cryptography87(2019), no. 11, 2597–2610. (Cited on page 11.)

work page 2019

[21] [21]

1, 141–195

Alexander Pott,Almost perfect and planar functions, Designs, Codes and Cryptography78(2016), no. 1, 141–195. (Cited on page 8.) 17

work page 2016

[22] [22]

1, 403–411

Alexander Pott, Enes Pasalic, Amela Muratovi´ c-Ribi´ c, and Samed Bajri´ c,On the maximum number of bent components of vectorial functions, IEEE Transactions on Information Theory64(2017), no. 1, 403–411. (Cited on pages 10 and 11.)

work page 2017

[23] [23]

Seelinger, P

G. Seelinger, P. Sissokho, L. Spence, and C. Vanden Eynden,Partitions of finite vector spaces over GF(2) into subspaces of dimensions 2 and s, Finite Fields and Their Applications18(2012), no. 6, 1114–1132. (Cited on page 9.)

work page 2012

[24] [24]

1, 88–101

Tam´ as Sz˝ onyi and Zsuzsa Weiner,Small blocking sets in higher dimensions, Journal of Combina- torial Theory, Series A95(2001), no. 1, 88–101. (Cited on page 11.)

work page 2001

[25] [25]

10, 2171–2186

Lijing Zheng, Jie Peng, Haibin Kan, Yanjun Li, and Juan Luo,On constructions and properties of (n, m)-functions with maximal number of bent components, Designs, Codes and Cryptography88 (2020), no. 10, 2171–2186. (Cited on page 11.) 18

work page 2020