Strongly Fair Validity Sample Clauses

Strongly Fair Validity. In this section, we prove that RBA achieves strongly fair validity. Intuitively, when the network operates synchronously, every honest nodes receives all the initial messages from each other before 2λ. Then, as long as the underlying VRF is secure, the probability of being the leader is approximate the uniform distribution, so RBA achieves strongly fair validity. The result is formalized as the following theorem. Theorem 13 (strongly fair validity). Suppose the network is synchronous and F is a secure VRF. Then, RBA achieves strongly fair validity under the assumption of static adversary.

Strongly Fair Validity. We first show that the probability is exactly lower-bounded by the uniform distribution in the ideal world. Then, we show that RBA works the same as the ideal world except the negligible probability. We define the VRF oracle, consisting of two algorithm: Oprove and Overi. Oprove is defined as: 1) Take as input a seed x and a secret key sk. 2) Return Prove(x, sk). Overi is defined as: 1) Take as input a public key pk, a seed x, a value y and a proof π. 2) Return Veri(pk, x, y, π). We also define the ideal functionality of VRF, consisting of two algorithm: Iprove and Iveri. Iprove is defined as: 1) Takes as input a seed x and a secret key sk. Y . Σ k−1 t − i n − i ( n − t ) n . t Σk ( n − t ). n 2) Check whether Q(x, sk) is defined. If not, choose y ← Thus, in expectation, the number of rounds can be computed Q(x, sk) = (y, π). If Q(x, sk) is defined, prove(x, sk) ≤ return Q(x, sk). Σ . Σ by Xxxxx is defined as: − · t n t 4 t ( ) · (6 + 4i) ≤ 6 + n . 1) Takes as input a public key pk, a seed x, a value y and ≥ ≥