Title: From Sigma-Protocols to Threshold Signatures and Secure Federated Learning

This family of schemes came to interest during the new NIST call for standardisation of signature designs, thanks to the trust of the community on their (quantum) resistance and the very malleable underlying structure. The core idea is to consider a zero knowledge proof for the knowledge of a secret group element $g$ such that $x’ = g \star x$ with $x,x’$ public, then render it to a signature (non-interactive protocol) via the classical Fiat-Shamir transform. Several schemes have been proposed with different instantiations of these assumptions: for example, LESS and MEDS use code equivalence (on different metrics).

The mathematical structure of the group action can be exploited to optimize the schemes and to define a threshold version of the signature for any group action, moreover for the ones induced by linear code equivalence it is possible to exploit Abelian subgroups of the general linear group to achieve better parameters.