Paper 2026/2404 Original: eprint.iacr.org/2026/2404 ↗

The Geometry of Collusion Leakage in Inner-Product Functional Encryption

Abstract

Inner-product functional encryption (IPFE) allows the holder of a functional key for a vector $y$ to learn $\langle x,y\rangle$ from an encryption of $x$, and nothing more. Keys, however, accumulate: a coalition holding sufficiently many independent keys recovers the plaintext. We model each institution by the span $W_i\leq\F_q^n$ of its authorized key vectors and show that a coalition $I$ determines the plaintext exactly modulo $(\sum_{i\in I}W_i)^\perp$; for a uniform plaintext it learns exactly $\dim\sum_{i\in I}W_i$ field symbols. The resulting leakage function is a representable and entropic polymatroid. We then study authorization spaces of dimension $r$ whose pairwise intersections have dimension at most $h$. We prove that some coalition of size $c\geq2$ always learns at least $\min\{n,2r-h+c-2\}$ dimensions, and we give a Vandermonde-type construction in which every coalition of size $c$ learns exactly this amount, for all $c$ simultaneously. Hence the largest achievable reconstruction threshold is $n-2r+h+2$, which gives an exact tradeoff between functional diversity and collusion resistance. The optimal configuration is a sunflower, an equidistant subspace code and a generalized arc; its orthogonal complements form a generalized dual arc, and for $r=1$ it reduces to Reed--Solomon secret sharing.