% Network and Distributed Systems Security (NDSS) Symposium 2021\\
% 21-24 February 2021\\
% ISBN 1-891562-66-5\\
% https://dx.doi.org/10.14722/ndss.2021.23xxx\\
% www.ndss-symposium.org
%}
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\begin{abstract}
We present an actively secure threshold scheme in the setting of Hard Homogeneous Spaces (HHS) which allows fine-grained access structures. More precisely, we elevate a
%given
passively secure isogeny-based threshold scheme to an actively secure setting. We prove the active security and simulatability of our advanced schemes.
By characterising the necessary properties, we open our schemes to a significantly wider field of applicable secret sharing schemes. Furthermore, we show that Shamir's scheme has our generalised properties, and thereby our approach truly represents a less restrictive generalisation.
%In their recent work \cite{FeoM20}, DeFeo and Meyer introduced a method for executing isogeny based public key encryptions on a private key, that had been stored in Shamir's information theoretical secret sharing scheme \cite{Shamir:1979:HSS}. While their approach enables a distributed en- and decryption for a threshold scheme, it is rather restrictive in that the access structure is limited to sets of a minimum size and lacks granularity. Furthermore their approach is only passively secure, i.e., a misbehaving adversary can not be detected.
%In this work we elevate \cite{PKC:DeFMey20}'s scheme to an actively secure setting, that is misbehaving player's are detected while the protocol is being executed. We prove the active security and simulatability of our advanced schemes. Furthermore we expand the range of secret sharing schemes able to support DeFeo's and Meyer's approach by defining some characterising properties of suitable schemes. For schemes fulfilling said properties the same security guarantees hold in regards to \cite{PKC:DeFMey20}'s approach. Furthermore we show that Shamir's scheme has our generalised properties, and thereby our approach truly is a generalisation of \cite{PKC:DeFMey20}.
\begin{comment}In their recent work \cite{PKC:DeFMey20}, De Feo and Meyer introduced a method for executing isogeny-based public key encryption on a private key, that had been stored in Shamir's information theoretical secret sharing scheme \cite{Shamir:1979:HSS}. While their approach enables a distributed en- and decryption for a threshold scheme, it is rather restrictive in that the access structure is limited to sets of a minimum size and lacks granularity. Furthermore, their approach is only passively secure, i.e., a misbehaving adversary can not be detected.
In this work we elevate \cite{PKC:DeFMey20}'s scheme to an actively secure setting, that is misbehaving players are detected while the protocol is being executed. We prove the active security and simulatability of our advanced schemes. Furthermore, we expand the range of secret sharing schemes able to support De Feo's and Meyer's approach by defining some characterising properties of suitable schemes. For schemes fulfilling said properties the same security guarantees hold in regard to \cite{PKC:DeFMey20}'s approach. Furthermore, we show that Shamir's scheme has our generalised properties, and thereby our approach truly is a generalisation of \cite{PKC:DeFMey20}.
\end{comment}
\keywords{post-quantum cryptography
\and isogeny-based cryptography
\and threshold cryptography
}
\end{abstract}
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