Indistinguishability of Sum of Permutations: A Fourier Analytic Route to Classical and Quantum Security

We study classical and quantum indistinguishability of sums of independent random permutations and related transformations from permutations to functions. Let $G$ be a finite abelian group of order $N$, and let $π^k_+(x)=π_1(x)+\cdots+π_k(x)$ for $k\geq2$ independent uniform random permutations of $