A lossy identification scheme using the subgroup decision assumption

Lossy identification schemes are used to construct tightly secure signature schemes via the Fiat-Shamir heuristic in the random oracle model. Several lossy identification schemes are instantiated by using the short discrete logarithm assumption, the ring-LWE assumption and the subset sum assumption, respectively. For assumptions concerning the integer factoring, Abdalla, Ben Hamouda and Pointcheval [3] recently presented lossy identification schemes based on the φ-hiding assumption, the QR assumption and the DCR assumption, respectively. In this paper, we propose new instantiations of lossy identification schemes. We first construct a variant of the Schnorr's identification scheme, and show its lossiness under the subgroup decision assumption. We also construct a lossy identification scheme which is based on the DCR assumption. Our DCRbased scheme has an advantage relative to the ABP's DCR-based scheme since our scheme needs no modular exponentiation in the response phase. Therefore our scheme is suitable when it is transformed to an online/offline signature.