International Association
for Cryptologic Research

Secret sharing schemes split a secret into multiple shares that are usually distributed to distinct participants with the goal that only authorized subsets of participants can recover it. We show that SETUP (Secretly Embedded Trapdoor with Universal Protection) attack can be embedded in 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 knowledge can be enough to reveal the secret. We exemplify the proposed attack against Shamir\'s threshold scheme, as being the most well-known and used secret sharing scheme. Finally, we consider some prevention techniques against the attack.