Nonexistence of generalized bent functions and the quadratic norm form equations
Chang Lv, Yuqing Zhu
TL;DR
This work proves broad nonexistence results for generalized bent functions of type $[n,2p^e]$ by translating the problem into the nonexistence of integral points on quadratic norm form equations over cyclotomic subfields. The authors establish a general criterion: if $E$ is a complex subfield of a cyclotomic field and $F=E\cap\mathbb R$, then for primes $p>(4q^n)^k$ the norm equation $N_{E/F}(\alpha)=q^n$ has no solution in $\mathfrak o_E$, with a descent mechanism to smaller fields underpinning the proof. Applying this to GBFs with $t=2p^e$ yields a universal nonexistence when $p$ is sufficiently large relative to $n$ and $f=\mathrm{ord}_p(2)$ (threshold $p>2^{2\mathcal B(l)+nl}$, $l=\frac{2(p-1)}{(3-(-1)^f)f}$); under ERH there are infinitely many such primes for fixed $n$. Moreover, the paper conducts GRH-based computations to establish nonexistence for relatively small $p$ by testing norm-form solvability via explicit class-group computations, providing concrete examples and an algorithmic framework for further search. The results advance understanding of GBFs by tying their existence to deep number-theoretic constraints in cyclotomic fields, with potential implications for cryptographic constructions relying on GBFs.
Abstract
We present a new result on the nonexistence of generalized bent functions (GBFs)from (Z/tZ)^n to Z/tZ (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no integral points, we obtain a universalresult on the nonexistence of GBFs with type [n,2p^e] when p and n satisfy a certain inequality, and by computational methods with a widely accepted hypothesis, Generalized Riemann Hypothesis, we also achieve some results on the nonexistence of GBFs for relatively small p.