One-round secure comparison of integers

We consider the problem of securely evaluating the Greater Than (GT) predicate and its extension – transferring one of two secrets, depending on the result of comparison. We generalize our solutions and show how to securely decide membership in the union of a set of intervals. We then consider the related problem of comparing two encrypted numbers. We show how to efficiently apply our solutions to practical settings, such as auctions with the semi-honest auctioneer, proxy selling, etc. All of our protocols are one round