Efficient multiplicative-to-additive function from Joye-Libert cryptosystem and its application to threshold ECDSA

Threshold ECDSA receives interest lately due to its widespread adoption in blockchain applications. A common building block of all leading constructions involves a secure conversion of multiplicative shares into additive ones, which is called the multiplicative-to-additive (MtA) function. MtA dominates the overall complexity of all existing threshold ECDSA constructions. Specifically, O(n²) invocations of MtA are required in the case of n active signers. Hence, improvement of MtA leads directly to significant improvements for all state-of-the-art threshold ECDSA schemes. In this paper, we design a novel MtA by revisiting the Joye-Libert (JL) cryptosystem. Specifically, we revisit JL encryption and propose a JL-based commitment, then give efficient zero-knowledge proofs for JL cryptosystem which are the first to have standard soundness. Our new MtA offers the best time-space complexity trade-off among all existing MtA constructions. It outperforms state-of-the-art constructions from Paillier by a factor of 1.85 to 2 in bandwidth and 1.2 to 1.7 in computation. It is 7× faster than those based on Castagnos-Laguillaumie encryption only at the cost of 2× more bandwidth. While our MtA is slower than OT-based constructions, it saves 18.7× in bandwidth requirement. In addition, we also design a batch version of MtA to further reduce the amortised time and space cost by another 25%.