Secure Distributed Modular Exponentiation: Systematic Analysis and New Results

Modular exponentiation is a vital function in public key cryptography. Dozens of protocols, including encryption schemes, signature schemes, pseudorandom functions, and more, perform this operation on a secret base and/or a secret exponent. In a multiparty computation setting, these secret data might be shared over multiple parties who wish to compute this modular exponentiation in a secure and distributed way. However, whereas typical frameworks for secure multiparty computation based on secret sharing provide a basic tool box of secure distributed computation of, most importantly, randomness generation, addition, and multiplication, the status quo of secure distributed modular exponentiation is unsatisfactory. In this work, we provide a complete and comprehensive overview on existing protocols for perfectly secure distributed exponentiation differing depending on whether the inputs and outcomes are public or shared. We perform a detailed complexity computation of the currently existing protocols, observing that earlier authors have overestimated the complexity of their own protocols, and close the remaining open problem: protocols for secure distributed exponentiation with secret base, secret exponent, and public outcome. We prove that the presented protocol is universally composably secure in the presence of malicious adversaries. We finally exemplify the practical relevance of the new protocol by demonstrating how it can be used for pseudorandom generation and signing.