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arxiv: 2606.08662 · v1 · pith:5Q6Z4BKLnew · submitted 2026-06-07 · 🧮 math.NT · cs.CR· cs.DS

Uncertainty Principles for the Number Theoretic Transform

Giulio Malavolta , Alon Rosen This is my paper

Pith reviewed 2026-06-27 17:56 UTC · model grok-4.3

classification 🧮 math.NT cs.CRcs.DS
keywords uncertainty principlenumber-theoretic transformsupport sizefinite fieldspolynomial identity testingexponential polynomialssparsity tradeoff
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The pith

For fixed prime q and all but finitely many p ≡ 1 mod q, every nonzero f in F_p^{Z_q} and its NTT satisfy |Supp(f)| + |Supp(hat f)| ≥ q+1.

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

The paper proves that the number-theoretic transform satisfies a sparsity tradeoff analogous to classical uncertainty principles. For any fixed prime q and for almost all primes p that are congruent to 1 modulo q, any nonzero function f from Z_q into the field F_p has the property that the sizes of the supports of f and its transform add to at least q+1. Consequently a k-sparse f forces its transform to have support size at least q-k+1. The authors also establish a probabilistic version of the bound that holds on average over primes p when p is only polynomially larger than q, and they use the bound to construct a black-box identity tester for k-sparse exponential polynomials of bounded degree whose soundness error vanishes.

Core claim

For every fixed prime q and for all but finitely many primes p ≡ 1 (mod q) every nonzero f∈ F_p^{Z_q} and its number-theoretic transform hat f satisfy |Supp(f)| + |Supp(hat f)| ≥ q+1. A probabilistic version of the same bound holds when the size of p is bounded by a fixed power of q.

What carries the argument

The number-theoretic transform of f, obtained by evaluating f against the characters of Z_q using a primitive q-th root of unity in F_p.

If this is right

  • Any k-sparse function has NTT support of size at least q-k+1.
  • There exists a black-box identity test for k-sparse exponential polynomials of degree at most d whose soundness error vanishes as long as q is moderately larger than k.

Where Pith is reading between the lines

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

  • The same support-sum bound may be usable for identity testing of other classes of sparse algebraic objects whose evaluation maps admit similar character decompositions.
  • Because the bound is proved by averaging over primes, explicit constructions of suitable primes p would turn the probabilistic version into a fully deterministic algorithm for the same PIT task.

Load-bearing premise

A primitive q-th root of unity exists inside the field F_p, which holds exactly when p ≡ 1 mod q.

What would settle it

An explicit nonzero function f in F_p^{Z_q} together with a prime p ≡ 1 mod q, both larger than any fixed bound depending on q, such that |Supp(f)| + |Supp(hat f)| is strictly less than q+1.

read the original abstract

Motivated by polynomial identity testing with exponentials (Li and Wu, ITCS'26), we study uncertainty principles for the number-theoretic transform (NTT). We show that the NTT satisfies strong sparsity tradeoffs: For every fixed prime $q$ and for all but finitely many primes $p \equiv 1 \pmod q$ every nonzero $f\in \mathbb F_p^{\mathbb Z_q}$ and its number-theoretic transform $\hat f$ satisfy \[ |\mathrm{Supp}(f)| + |\mathrm{Supp}(\hat f)| \ge q+1. \] Thus, a $k$-sparse function has transform support at least $q-k+1$. As our main technical contribution, we prove a probabilistic version of the above uncertainty principle, averaged over primes $p$, in the regime $p=q^{O(1)}$. As an application, we obtain a black-box identity test for $k$-sparse exponential polynomials of degree at most $d$ with vanishing soundness error, for $q$ moderately larger than $k$.

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 paper proves uncertainty principles for the number-theoretic transform. For fixed prime q and all but finitely many primes p ≡ 1 (mod q), every nonzero f ∈ F_p^{Z_q} satisfies |Supp(f)| + |Supp(ˆf)| ≥ q + 1. It also establishes a probabilistic version of this bound averaged over primes p = q^{O(1)}, and applies the result to obtain a black-box identity test for k-sparse exponential polynomials of degree at most d with vanishing soundness error when q is moderately larger than k.

Significance. The result supplies a clean, parameter-free sparsity tradeoff for the NTT that directly strengthens polynomial identity testing with exponentials. The probabilistic averaging over primes in the bounded-size regime is a useful technical contribution for the PIT application. The direct linear-algebraic nature of the underlying argument (generalized Vandermonde) is a strength.

minor comments (2)
  1. [Abstract / Main Theorem] The deterministic statement is phrased with the qualifier 'all but finitely many primes p'. The generalized Vandermonde argument (distinct nonzero roots in F_p for p ≡ 1 mod q) shows the bound holds for every such p; the manuscript could strengthen the theorem statement without changing its correctness.
  2. [Introduction] The citation 'Li and Wu, ITCS'26' refers to a forthcoming paper; a brief note on its status or an arXiv reference would improve completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept. The report accurately captures the main results on uncertainty principles for the NTT and the PIT application.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's core claim is a direct mathematical statement proved via a standard linear-algebra argument (non-vanishing of a generalized Vandermonde matrix over F_p for distinct roots of unity). This relies only on the definition of the NTT (a standard number-theoretic fact) and basic field properties, with no fitted parameters, self-referential definitions, or load-bearing self-citations. The deterministic bound and its probabilistic averaged version are derived independently without reducing to prior results by the same authors or renaming known patterns. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard facts about the existence of roots of unity in finite fields when p ≡ 1 mod q; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Existence of a primitive q-th root of unity in F_p whenever p ≡ 1 mod q
    Required to define the NTT; standard fact from elementary number theory.

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