The idea is to have the window shift by a fraction of its size. We call the resulting window as the -shifting -window, where is the window length and is the length by which it shifts (cf. Denition 1). With careful calibration of and , we can ensure that the window captures the critical window.

This intuition is formally proven later in Theorem 2. Below, we formally describe the chain selection rules. Our new chain selection rule, formally specied as algorithm maxvalid-sc() (see Figure 7), surgically adapts the chain selection rule of Ouroboros Genesis, namely maxvalid-bg() (see Figure 20), but by replacing the long-range fork rule with a new one. We will continue all the discussions by referring to the 21

We will note that the only information about the underlying blockchains needed in any part of the consensus mechanism is a xed number of blocks in the immediate history. Therefore, the consensus verication is succinct. Here, Cloc is the local chain, N = {C1, . . . , C } is the list of chains to choose from. The function isShortRange(C, C (cid:48)) outputs whether or not the chains fork in the short range or not. The function getMinDen(C) outputs the minimum of all the window densities observed thus far in C; it is formally dened in Figure 16. Algorithm maxvalid-sc(Cloc, N = {C1, . . . , C }, ) // Compare Cloc with each candidate chain in N 1. Set Cmax Cloc 2. for = 1, . . . , do if isShortRange(C, Cmax) then // Short-range fork if |C | > |Cmax| then Set Cmax C end if else //Long-range fork if getMinDen(C) > getMinDen(Cmax) then Set Cmax C end if end if end for 3. return Cmax Figure 7: The new chain selection rule.

7.2 Background We introduce many relevant concepts from Ouroboros Genesis which remain the same in our setting too. We begin by recalling a summary of the Ouroboros family of protocols. The Ouroboros family of consensus protocols. Kiayias et al. proposed the rst proof-of-stake (PoS) protocol Ouroboros with rigorous security 22 guarantees. However, the adversarial model only considered synchronous networks. Here, the protocol execution is divided into time units called slots and groups of slots form epochs.