Threshold signature scheme with threshold verification based on multivariate linear polynomial

Secret sharing schemes are multi-party protocols related to key establishment. They also facilitate distributed trust or shared control for critical activities (e.g., signing corporate cheques and opening bank vaults), by gating the critical action on cooperation from t(t ∈ Z ⁺) of n(n ∈ Z ⁺) users. A (t, n) threshold scheme (t < n) is a method by which a trusted party computes secret shares Γ _i (1 ⩽ i ⩽ n) from an initial secret Γ ₀ and securely distributes Γ _i to user. Any t or more users who pool their shares may easily recover Γ ₀, but any group knowing only t−1 or fewer shares may not. By the ElGamal public key cryptophytes and the Schnorr’s signature scheme, this paper proposes a new (t, n) threshold signature scheme with (k,m) (k,m ∈ Z ⁺) threshold verification based on the multivariate linear polynomial.