Degradable Agreement in the Presence of Byzantine Faults

Consider a system consisting of a sender that wants to send a value to certain receivers. Byzantine agreement protocols [7, 6, 2] have been proposed to achieve this in the presence of arbitrary failures. The imposed requirement typically is that the fault-free receivers must all agree on the same value [7, 6]. (Dolev [2] analyzes a seemingly weaker form of agreement). It has been shown that such an agreement is impossible if a third or more of the nodes are faulty [7, 6, 2]. We propose an agreement protocol that achieves Lamport's Byzantine agreement [7] up to a certain number of faults and a degraded form of agreement with a higher number of faults. Essentially, the degraded form of agreement allows the fault-free receivers to agree on at most two di erent values one of which is necessarily the default value. The default value is distinguishable from all other values. The proposed approach is named \degradable agreement". Speci cally, m=u-degradable agreement is de ned using two parameters, m and u, and the following four conditions. (The term node refers to the sender and the receivers). (1) If the sender is fault-free and at most m nodes are faulty, then all the fault-free nodes must agree on the sender's value. (2) If the sender is faulty, and the number of faulty nodes is at most m, then all the fault-free nodes must agree on an identical value. (3) If the sender is fault-free, and the number of faulty nodes is more than m but at most u, then the fault-free nodes may be partitioned into at most two classes. The fault-free nodes in one of the classes must agree on the sender's value, and the fault-free nodes in the other class must all agree on the default value. (4) If the sender is faulty, and the number of faulty nodes is more than m but at most u, then the fault-free nodes may be partitioned into at most two classes. The fault-free nodes in one of the classes must agree on the default value, and the fault-free nodes in the other class must all agree on an identical value. It is shown that at least 2m+u+1 nodes and network connectivity of at least m+u+1 are necessary to achieve m=u-degradable agreement. An m=u-degradable agreement algorithm is presented for more than 2m+u nodes. Conditions (3) and (4) imply that, up to u faults, at least m + 1 fault-free nodes are guaranteed to agree on the same value.