In this paper we study cryptographic finite abelian groups of unknown order and hardness assumptions in these groups. Abelian groups necessitate multiple group generators, which may be chosen at random. We formalize this setting and hardness assumptions therein. Furthermore, we generalize the algebraic group model and strong algebraic group model from cyclic groups to arbitrary finite abelian groups of unknown order. Building on these formalizations, we present techniques to deal with this new setting, and prove new reductions. These results are relevant for class groups of imaginary quadratic number fields and time-lock cryptography build upon them.

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Lecture Notes in Computer Science , Subseries Security and Cryptology
Secure scalable policy-enforced distributed data processing
ASIACRYPT - International Conference on the Theory and Application of Cryptology and Information Security

van Baarsen, A., & Stevens, M. (2021). On time-lock cryptographic assumptions in abelian hidden-order groups. In Advances in Cryptology - ASIACRYPT (pp. 367–397). doi:10.1007/978-3-030-92075-3_13