Paper 2024/1452 Original: eprint.iacr.org/2024/1452 ↗

On the Complexity of Cryptographic Groups and Generic Group Models

Abstract

Ever since the seminal work of Diffie and Hellman, cryptographic (cyclic) groups have served as a fundamental building block for constructing cryptographic schemes and protocols. The security of these constructions can often be based on the hardness of (cyclic) group-based computational assumptions. Then, the generic group model (GGM) has been studied as an idealized model (Shoup, EuroCrypt 1997), which justifies the hardness of many (cyclic) group-based assumptions and shows the limits of some group-based cryptosystems. We stress that, the importance of the length of group encoding, either in a concrete group-based construction or assumption, or in the GGM, has not been studied. In this work, we initiate a systematic study on the complexity of cryptographic groups and generic group models, varying in different lengths of group encodings, and demonstrate evidences that ``the length matters''. More concretely, we have the following results: -- We show that there is no black-box/relativizing reduction from the CDH-secure groups (i.e., over such groups, the computational Diffie-Hellman assumption holds) with shorter encodings, to the CDH-secure groups with longer encodings, within the same security parameter. More specifically, given any arbitrary longer CDH-secure group, it is impossible to generically shorten the group encoding and obtain a shorter CDH-secure group within the same group order. -- We show that there is a strict hierarchy of the GGMs with different lengths of encodings. That is, in the framework of indifferentiability, the shorter GGM is strictly stronger than the longer ones, even in the presence of computationally bounded adversaries.