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In a (t,n) threshold scheme any t or more shares can reconstruct the secret s, but less than t shares reveal no information about s. However, an unauthenticated adversary can pretend to be the shareholder at the reconstruction stage. If there were more than t honest shareholders, the unauthenticated adversary without valid share can obtain the secret. To deal with this type of attacks, a model of (t,m,n) group oriented secret sharing (GOSS) scheme was proposed by Miao et al. in 2015. Here the group oriented property means that if m>t parties try to reconstruct the secret, they should all have the authentic shares in advance. It was claimed by Miao et al. that the group oriented property in their GOSS schemes holds in the information-theoretic sense. In this paper, we revisit two instantiations of (t,m,n) group oriented secret sharing schemes and show that these constructions cannot provide the so-called “group oriented property”. Specifically, we develop concrete attacks which allow an unauthenticated adversary with no valid share to participate in the reconstruction phase and obtain the secret provided that there are at least t honest shares presented at the reconstruction phase.

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In this paper, we present a practical and provably secure two-pass authenticated key exchange protocol over ideal lattices, which is conceptually simple and has similarities to the Diffie-Hellman based protocols such as HMQV (CRYPTO 2005) and OAKE (CCS 2013). Our method does not involve other cryptographic primitives—in particular, it does not use signatures—which simplifies the protocol and enables us to base the security directly on the hardness of the ring learning with errors problem. The security is proven in the Bellare-Rogaway model with weak perfect forward secrecy in the random oracle model. We also give a one-pass variant of our two-pass protocol, which might be appealing in specific applications. Several concrete choices of parameters are provided, and a proof-of-concept implementation shows that our protocols are indeed practical.

We use the signal function from RLWE key exchange to derive an efficient zero knowledge authentication protocol to validate an RLWE key p=as+e with secret s and error e in the Random Oracle Model (ROM). With this protocol, a verifier can validate that a key presented to him by a prover P is of the form p=as+e with s,e small and that the prover knows s. We accompany the description of the protocol with proof to show that it has negligible soundness and completeness error. The soundness of our protocol relies directly on the hardness of the RLWE problem. The protocol is applicable for both LWE and RLWE but we focus on the RLWE based protocol for efficiency and practicality. We also present a variant of the main protocol with a commitment scheme to avoid using the ROM.