Solving Linear Inequalities over the Space of Convex Sets & its Applications to Cryptography and Hydrodynamics

Is a two-party function, possibly with randomized output, securely computable? We provide a finite procedure to answer this question, thereby settling a foundational, three-decade-old open problem in secure computation and information complexity.Beaver-Chor-Kushilevitz [11], [22], [8] answered this question for deterministic output functions. Basu et al. [3] recently gave a geometric characterization of randomized functions securely computable with bounded communication complexity. Randomized functions can have arbitrarily high communication complexity, even for fixed input-output sets [5]. Without an upper bound on the communication complexity, the decidability of the question of whether a given two-party function with randomized output is securely computable was a formidable challenge.We reduce answering this question to proving specific lamination hulls are semi-algebraic. Lamination hulls are an infinite union of recursively defined sets independently motivated by the hydrodynamics literature. We connect this technical objective to solving a system of linear inequalities over convex sets in high dimensions, where inequalities represent the natural containment relation. We present a Gaussian elimination-inspired algorithm to compute the smallest simultaneous solutions to such systems. After that, using these solutions, we prove that our lamination hulls are semi-algebraic.Our technical solution introduces a novel set operator called positive geometric join. In our application context, it characterizes algebraically well-behaved sets that generalize polytopes, which we call hemihedra. The positive geometric join operator and hemihedral sets should interest the broader mathematics and computer science community. These advancements should help further information complexity investigations more broadly via the recently established connection by Basu et al. [3].