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arxiv: 2605.01786 · v1 · submitted 2026-05-03 · 💻 cs.IT · math.IT

Walsh Spectrum and Boomerang Properties of Locally-APN Niho Functions

Yuehui Cui , Jinquan Luo , Can Xiang This is my paper

Pith reviewed 2026-05-09 16:37 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Niho functionslocally-APNWalsh spectrumcyclic codesboomerang propertiesdifferential spectrumpower functionsfinite fields
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The pith

Niho power functions are locally-APN exactly when their Walsh spectrum takes the four values -p^m, 0, p^m and 2p^m.

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

The paper establishes an if-and-only-if link between the locally-APN property for Niho-type power functions and a specific four-valued Walsh spectrum. A function satisfies the local-APN condition precisely when its Walsh transform hits only those four numbers, which in turn forces the related cyclic codes to have exactly four nonzero weights of the stated form. The authors then derive the differential spectrum, the Feistel Boomerang Connectivity Table, and the second-order zero differential spectra for all such functions. A sympathetic reader cares because locally-APN functions can resist differential attacks better than functions with the same differential uniformity, and the spectral test offers a concrete way to identify them.

Core claim

We show that a Niho type power function is locally-APN if and only if its Walsh spectrum takes four values in {-p^m, 0, p^m, 2p^m}. Equivalently, the associated cyclic codes have four nonzero weights: p^{m-1}(p-1)(p^m + k) for k = 0, 1, -1, -2. We also study properties of Niho type locally-APN power functions, including their differential spectrum, Walsh spectrum, Feistel Boomerang Connectivity Table and second-order zero differential spectra.

What carries the argument

The equivalence that ties the locally-APN property of a Niho power function directly to its Walsh spectrum having exactly the four values {-p^m, 0, p^m, 2p^m}, which also fixes the weight distribution of the associated cyclic codes.

If this is right

  • The associated cyclic codes have exactly the four nonzero weights p^{m-1}(p-1)(p^m + k) for k = 0, 1, -1, -2.
  • The differential spectrum of these locally-APN Niho functions is completely determined by the four-valued Walsh spectrum.
  • The Feistel Boomerang Connectivity Table and second-order zero differential spectra take explicit forms that follow from the same spectral condition.

Where Pith is reading between the lines

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

  • The characterization supplies a practical test that could be used to search for additional locally-APN power functions by inspecting only their Walsh transforms.
  • The link between local APN and four-valued spectra may suggest similar equivalences for other classes of functions or for boomerang uniformity measures.
  • The weight formulas for the cyclic codes open the possibility of deriving new bounds or constructions in coding theory that rely on the same Niho exponents.

Load-bearing premise

The functions are Niho-type power functions over finite fields of odd prime characteristic, and the standard definitions of locally-APN, Walsh spectrum, and differential uniformity apply without further restrictions.

What would settle it

Exhibit a concrete Niho power function over an odd-characteristic field whose Walsh spectrum contains a value outside {-p^m, 0, p^m, 2p^m} yet the function is still locally-APN, or one whose spectrum is exactly those four values but the function fails to be locally-APN.

read the original abstract

Recently, the Walsh spectrum and boomerang properties of special power functions have aroused widespread research interest, owing to their important applications in cryptography and information security. In particular, locally-APN functions may offer superior resistance against differential cryptanalysis compared to other functions of equivalent differential uniformity. Up till now only a small number of locally-APN functions have been studied. In this paper, we show that a Niho type power function is locally-APN if and only if its Walsh spectrum takes four values in \(\{-p^m, 0, p^m, 2p^m\}\). Equivalently, the associated cyclic codes have four nonzero weights: $p^{m-1}(p-1)(p^m + k)$ for $k = 0, 1, -1, -2$. Moreover, we also study properties of Niho type locally-APN power functions, including their differential spectrum, Walsh spectrum, Feistel Boomerang Connectivity Table(FBCT for short) and second-order zero differential spectra.

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

0 major / 2 minor

Summary. The manuscript establishes an if-and-only-if characterization for Niho-type power functions over finite fields F_{p^{2m}} with p odd: such a function is locally-APN precisely when its Walsh spectrum is supported exactly on the four values {-p^m, 0, p^m, 2p^m}. Equivalently, the associated cyclic codes have exactly four nonzero weights of the form p^{m-1}(p-1)(p^m + k) for k = 0, 1, -1, -2. Both directions are obtained via direct evaluation of character sums for monomials with Niho exponents, followed by the standard translation to code weights using MacWilliams identities. The paper additionally determines the differential spectrum, Walsh spectrum, Feistel Boomerang Connectivity Table, and second-order zero differential spectra of the locally-APN Niho functions.

Significance. If the characterization holds, the result supplies a complete spectral criterion for local-APN among Niho power functions, which is useful for constructing functions with strong resistance to differential cryptanalysis. The proofs rely on standard definitions and direct character-sum techniques without ad-hoc restrictions on p or m, and the additional spectral and boomerang analyses provide concrete cryptographic profiles. This contributes to the small collection of known locally-APN functions and links the property to code-weight distributions in a falsifiable way.

minor comments (2)
  1. [Abstract] Abstract, line 3: the phrasing 'a Niho type power function is locally-APN if and only if' is slightly ambiguous; it should be clarified that the equivalence holds among Niho-type power functions (as made precise in the body).
  2. The manuscript would benefit from a short table or example computation for small m (e.g., m=1 or m=2) that explicitly lists the four Walsh values and the corresponding code weights to illustrate the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its contributions. The report recommends minor revision but lists no specific major comments. We appreciate the accurate summary of the if-and-only-if characterization and the additional spectral analyses provided in the paper.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central result is an if-and-only-if equivalence between the locally-APN property and a four-valued Walsh spectrum for Niho-type power functions, established via direct evaluation of character sums over finite fields of odd characteristic and the standard MacWilliams translation to code weights. The argument invokes only the usual definitions of Walsh transform, differential uniformity, and Niho exponents; no parameters are fitted to data, no self-citations form a load-bearing chain, and no quantity is defined in terms of itself or renamed as a prediction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the stated claim alone; no explicit free parameters, new entities, or non-standard axioms are visible.

axioms (1)
  • standard math Standard properties of finite fields of odd characteristic and the definitions of Niho exponents, Walsh transform, and locally-APN hold.
    Invoked implicitly by the statement of the theorem.

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Reference graph

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