A k-of-N threshold signature scheme is that for a group with N members, the number of participants to generate a valid threshold signature is k or more than k. It is considered to be a variant of multi-signature and is widely applied nowadays. For many threshold signature schemes, trusted third parties(TTP) are essential in the construction. However, a dealer(which is a TTP) is not permitted to exist in some particular conditions. We thus focus on proposing the dealer-less version of a threshold signature scheme.
In the field of supersingular isogeny, Luca De Feo et al. sketched a threshold HHS signature scheme, which can be adapted into CSI-FiSh and its ancestors(CSIDH, SeaSign, basic version of CSI-FiSh). Therefore, one is able to construct threshold signature schemes based on isogeny with the help of dealers.
In this paper, we introduce three dealer-less threshold signature schemes based on the basic version of CSI-FiSh, CSI-FiSh and CSI-SharK, respectively. After that, we illustrate their correctness and security. Finally, we compare the theoretical efficiencies with the ones proposed by Luca De Feo et al.
Keywords: Hard Homogeneous Spaces(HHS) ; Threshold signature; Joint Random-Secret Shar- ing(JRSS) ; isogeny; CSI-FiSh; CSI-SharK.