Randomized Secure Two-Party Computation for Modular Conversion, Zero Test, Comparison, MOD and Exponentiation

When secure arithmetic is required, computation based on secure multiplication (MULT) is much more efficient than computation based on secure boolean circuits. However, a typical application can also require other building blocks, such as comparison, exponentiation and the modulo (MOD) operation. Secure solutions for these functions proposed in the literature rely on bit-decomposition or other bit-oriented methods, which require O(`) MULTs for `-bit inputs. In the absence of a known bit-length independent solution, the complexity of the whole computation is often dominated by these non-arithmetic functions. To resolve the above problem, we start with a general modular conversion, which converts secret shares over distinct moduli. For this, we proposed a probabilistically correct protocol for this with a complexity that is independent of `. Then, we show that when these non-arithmetic functions are based on secure modular conversions, they can be computed in constant rounds and O(k) MULTs, where k is a parameter for an error rate of 2−Ω(k). To promote our protocols to be actively secure, we apply O(k) basic zero-knowledge proofs, which cost at most O(k) exponentiation computation, O(1) rounds and O(k(` + κ)) communication bits, where κ is the security parameter used in the commitment scheme.