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We construct efficient ring signatures (RS) from isogeny and lattice assumptions. Our ring signatures are based on a logarithmic OR proof for group actions. We instantiate this group action by either the CSIDH group action or an MLWE-based group action to obtain our isogeny-based or lattice-based RS scheme, respectively. Even though the OR proof has a binary challenge space and therefore requires a number of repetitions which is linear in the security parameter, the sizes of our ring signatures are small and scale better with the ring size N than previously known post-quantum ring signatures. We also construct linkable ring signatures (LRS) that are almost as efficient as the non-linkable variants. The isogeny-based scheme produces signatures whose size is an order of magnitude smaller than all previously known logarithmic post-quantum ring signatures, but it is relatively slow (e.g. 5.5 KB signatures and 79 s signing time for rings with 8 members). In comparison, the latticebased construction is much faster, but has larger signatures (e.g. 30 KB signatures and 90 ms signing time for the same ring size). For small ring sizes our lattice-based ring signatures are slightly larger than state-ofthe-art schemes, but they are smaller for ring sizes larger than N ≈ 1024.

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