Variance: Secure Two-Party Protocol for Solving Yao\u27s Millionaires\u27 Problem in Bitcoin

Secure multiparty protocols are useful tools for parties wishing to jointly compute a function while keeping their input data secret. The millionaires’ problem is the first secure two-party computation problem, where the goal is to securely compare two private numbers without a trusted third-party. There have been several solutions to the problem, including Yao’s protocol [Yao, 1982] and Mix and Match [Jakobsson and Juels, 2000]. However, Yao’s Protocol is not secure in the malicious model and Mix and Match unnecessarily releases theoretically breakable encryptions of information about the data that is not needed for the comparison. In addition, neither protocol has any verification of the validity of the inputs before they are used. In this thesis, we introduce Variance, a privacy-preserving two-party protocol for solving the Yao’s millionaires’ problem in a Bitcoin setting, in which each party controls several Bitcoin accounts (public Bitcoin addresses) and they want to find out who owns more bitcoins without revealing (1) how many accounts they own and the balance of each account, (2) the addresses associated with their accounts, and (3) their total wealth of bitcoins while assuring the other party that they are not claiming more bitcoin than they possess. We utilize commitments, encryptions, zero knowledge proofs, and homomorphisms as the major computational tools to provide a solution to the problem, and subsequently prove that the solution is secure against active adversaries in the malicious model

The Logic of Counting Query Answers

We consider the problem of counting the number of answers to a first-order formula on a finite structure. We present and study an extension of first-order logic in which algorithms for this counting problem can be naturally and conveniently expressed, in senses that are made precise and that are motivated by the wish to understand tractable cases of the counting problem

Approximating Local Homology from Samples

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are robust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratification learning, feasible in high dimensions.Comment: 23 pages, 14 figure