A Multi-dimensional Yao ’ s Millionaire Protocol

Yao introduced the “millionaire problem”, in which two parties want to determine who is richer without disclosing anything else about their wealth. This problem deals with single comparison situation; however, in many applications, one often encounters situations where one wants to make multiple comparisons in an “all-or-nothing” fashion: Alice has an n-dimensional vector A = (a 1; : : : ; an), and Bob has another n-dimensional vector B = (b1; : : : ; bn). Alice wants to know whether A dominatesB, i.e. whetherfor all i = 1; : : : ; n, ai > bi. If 9i such thatai < bi, then both parties should learn nothing about the other party’s information, including any partial information, such as the relationship between any a i, bi pair, for i = 1; : : : ; n. This problem cannot be solved by just using the solution to Yao’s Millionaire Problemn times, once for each dimension: That would inappropriately reveal the relative ordering of individual ai; bi pairs in the case where the answer to the domination question is “no”. We propose a novel and efficient solution to this multi-dimensional Yao’s Millionaire Problem. The communication complexity of our scheme is linear in the number of bits needed to represent Alice’s and bob’s vectors.