Proof of the Goresky Klapper Conjecture on Decimations of L -sequences
Todd Cochrane, Sergei Vladimirovich Konyagin · SIAM Journal on Discrete Mathematics · 2011
Let p be an odd prime and $\mathbb E=\{2,4, \dots, p-1\}$ the set of nonzero even residues in $\mathbb Z_p = \mathbb Z/(p)$. We prove that for $p>13$, if the mapping $x \to Ax^k$ is a permutation of $\mathbb Z_p$, but not the identity mapping, then the mapping is not a permutation of $\mathbb E$. This establishes a conjecture of Goresky and Klapper stating that any two distinct decimations of a binary $\ell$-sequence are cyclically distinct.