SETUP in secret sharing schemes using random values

Secret sharing schemes divide a secret among multiple participants so that only authorized subsets of parties can reconstruct it. We show that SETUP (Secretly Embedded Trapdoor with Universal Protection) attack can be embedded in secret sharing schemes that employ enough randomness to give the attacker an overwhelming advantage to access the secret. In case of ideal schemes, a coalition of a few participants (within at least one is the attacker) can succeed the attack, while in case of non-ideal schemes the attacker’s knowledge can be enough to reveal the secret. We exemplify the attack against Shamir’s threshold scheme, which is the most well-known and used secret sharing scheme. Finally, we consider some prevention techniques against the proposed attack.