Uncertainty Principles for the Number Theoretic Transform

2026-06-07

AI Summary

This work investigates the uncertainty principle in the Number Theoretic Transform (NTT), establishing that for any nonzero function over a finite field, the sum of the sizes of its support and that of its NTT is at least \( q+1 \). It presents the first deterministic and probabilistic uncertainty principles tailored to the NTT. By integrating tools from algebraic number theory, Fourier analysis, and probabilistic methods, the study characterizes the fundamental limits on the sparsity structure of functions and their NTTs over finite fields, under the condition that the underlying prime \( p = q^{O(1)} \). As an application, the authors derive a zero-error black-box identity testing algorithm for \( k \)-sparse exponential polynomials, achieving perfect reliability when \( q \gtrsim k \), thereby significantly advancing the interplay between sparse recovery and algebraic complexity theory.

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 $k$.

Problem

• uncertainty principle
• number-theoretic transform
• sparsity
• support
• polynomial identity testing

Innovation

• Number-Theoretic Transform
• Uncertainty Principle
• Sparse Polynomials
• Polynomial Identity Testing
• Probabilistic Method