Condense to Conduct and Conduct to Condense
Tomasz Kazana
TL;DR
This work pioneers explicit low-conductance permutations by connecting CondDeg_α to Multi-Source-Somewhere-Condenser structures, establishing a near-complete information-theoretic characterization of conductance for permutations. It provides an efficient construction achieving CondDeg_α(π) ≤ $w - \left\lfloor\tfrac{w}{3}\right\rfloor c$ for prime $n$ and $α$ within a specified range, with $c = Θ(ε)$ and $n_0 = Θ(1/ε^2)$, and proves a matching Converse Theorem showing the equivalence with condenser properties. The technical core combines 3-source condenser ideas (via constructions like $\pi(a,b,c)=(a,b,c+ab)$) with a hierarchical extension to arbitrary $w$, including detailed partitioning arguments and entropy analyses. The results illuminate a deep link between permutation conductance and condensers, offering a rigorous framework for designing explicit primitives with provable randomness properties useful to cryptography and related information-theoretic applications.
Abstract
In this paper, we present the first explicit examples of low-conductance permutations. The notion of conductance of permutations was introduced by Dodis et al. in "Indifferentiability of Confusion-Diffusion Networks", where the search for low-conductance permutations was first initiated and motivated. As part of our contribution, we not only provide these examples, but also offer a general characterization of the problem: we show that low-conductance permutations are equivalent to permutations possessing the information-theoretic properties of Multi-Source-Somewhere-Condensers, a specific variant of somewhere condensers.