Unconditionally Secure Constant-Rounds Multi-party Computation for Equality, Comparison, Bits and Exponentiation

In this paper we are interested in efficient and secure constant round multi-party protocols which provide unconditional security against so called honest-but-curious adversaries. In particular, we design a novel constant round protocol that converts from shares over Zq to shares over the integers working for all shared inputs from Zq. Furthermore, we present a constant round protocol to securely evaluate a shared input on a public polynomial whose running time is linear in the degree of the polynomial. The proposed solution makes use of Chebyshev Polynomials. We show that the latter two protocols can be used to design efficient constant round protocols for the following natural problems: (i) Equality: Computing shares of the bit indicating if a shared input value equals zero or not. This provides the missing building blocks for many constant round linear algebra protocols from the work of Cramer and Damg̊ard [CD01]. (ii) Comparison: Computing shares of a bit indicating which of two shared inputs is greater. (iii) Bits: Computing shares of the binary representation of a shared input value. (iv) Exponentiation: Computing shares of x mod q given shares of x, a and q. Prior to this paper, for all the above mentioned problems, there were in general no efficient constant round protocols known providing unconditional security.