Efficient Lossy Trapdoor Functions Based on Subgroup Membership Assumptions

We propose a generic construction of lossy trapdoor function from the subgroup membership assumption. We present three concrete constructions based on the k-DCR assumption over $\mathbb{Z}^*_{N^2}$ , the extended p-subgroup assumption over $\mathbb{Z}^*_{N^2}$ , and the decisional RSA subgroup membership assumption over $\mathbb{Z}^*_{N}$ . Our constructions are more efficient than the previous construction from the DCR assumption over $\mathbb{Z}^*_{N^{s}}$ (s ≤ 3).