For many years, analysis in distributed programs, particularly in Byzantine consensus and state machine replication (SMR), has targeted on two fundamental targets: consistency and liveness. Consistency means all nodes agree on the identical sequence of transactions, whereas liveness ensures the system continues so as to add new ones. Nonetheless, these properties don’t cease dangerous actors from altering the order of transactions after they’re acquired.
In public blockchains, that hole in conventional consensus ensures has change into a significant issue. Validators, block builders or sequencers can exploit their privileged position in block ordering for monetary achieve, a apply often called maximal extractable worth (MEV). This manipulation consists of worthwhile frontrunning, backrunning and sandwiching of transactions. As a result of transaction execution order determines validity or profitability in DeFi functions, the integrity of transaction ordering is significant for sustaining equity and belief.
To deal with this vital safety hole, transaction order-fairness has been proposed as a 3rd important consensus property. Honest-ordering protocols be certain that the ultimate order of transactions will depend on exterior, goal components, resembling arrival instances (or receiving order) and is proof against adversarial reordering. By limiting how a lot energy a block proposer has to reorder transactions, these protocols transfer blockchains nearer to being clear, predictable, and MEV-resistant.
The Condorcet paradox and impossibility of perfect equity
Essentially the most intuitive and strongest notion of equity is Obtain-Order-Equity (ROF). Informally outlined as “first acquired, first output,” ROF dictates that if a ample variety of transactions (tx) arrive at a majority of nodes sooner than one other transaction (tx′), then the system is required to order tx earlier than tx′ for execution.
Nonetheless, attaining this universally accepted “order equity” is basically inconceivable until it’s assumed that every one nodes can talk instantaneously (i.e., working straight away synchronous exterior community). This impossibility outcome stems from a shocking connection to social selection idea, particularly the Condorcet paradox.
The Condorcet paradox illustrates how, even when each particular person node maintains a transitive inside ordering of transactions, the collective choice throughout the system can lead to what are often called non-transitive cycles. For instance, it’s potential {that a} majority of nodes obtain transaction A earlier than B, a majority obtain B earlier than C, and a majority obtain C earlier than A. Therefore, the three majority preferences type a loop (A→B→C→A). Because of this no single, constant ordering of the transactions A, B and C can ever fulfill all majority preferences concurrently.
This paradox demonstrates why the aim of completely attaining Obtain-Order-Equity is inconceivable in asynchronous networks, and even in synchronous networks that share a typical clock if exterior community delays are too lengthy. This impossibility necessitates the adoption of weaker equity definitions, resembling batch order equity.
Hedera Hashgraph and flaw of median timestamping
Hedera, which employs the Hashgraph consensus algorithm, seeks to approximate a robust notion of receive-order equity (ROF). It does this by assigning every transaction a closing timestamp computed because the median of all nodes’ native timestamps for that transaction.
Nonetheless, that is inherently liable to manipulation. A single adversarial node can intentionally distort its native timestamps and invert the ultimate ordering of two transactions, even when all sincere individuals acquired them within the right order.
Contemplate a easy instance with 5 consensus nodes (A, B, C, D and E) the place Node E acts maliciously. Two transactions, tx₁ and tx₂, are broadcast to the community. All sincere nodes obtain tx₁ earlier than tx₂, so the anticipated closing order ought to be tx₁ → tx₂.
On this instance, the adversary assigns tx₁ a later timestamp (3) and tx₂ an earlier one (2) to skew the median.
When the protocol computes the medians:
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For tx₁, the timestamps (1, 1, 4, 4, 3) yield a median of three.
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For tx₂, the timestamps (2, 2, 5, 5, 2) yield a median of two.
As a result of the ultimate timestamp of tx₁ (3) is bigger than that of tx₂ (2), the protocol outputs tx₂ → tx₁, thus reversing the true order noticed by all sincere nodes.
This toy instance demonstrates a vital flaw: The median perform, whereas showing impartial, is paradoxically the precise explanation for unfairness as a result of it may be exploited by even a single dishonest participant to bias the ultimate transaction order.
In consequence, Hashgraph’s often-touted “truthful timestamping” is a surprisingly weak notion of equity. The Hashgraph consensus fails to ensure receive-order equity and as a substitute will depend on a permissioned validator set quite than on cryptographic ensures.
Reaching sensible ensures
Nonetheless, to avoid the theoretical impossibility demonstrated by Condorcet, sensible fair-ordering schemes should chill out the definition of equity ultimately.
The Aequitas protocols launched the criterion of Block-Order-Equity (BOF), or batch-order-fairness. BOF dictates that if sufficiently many nodes obtain a transaction tx earlier than one other transaction tx′, then tx should be delivered in a block earlier than or concurrently tx′, that means no sincere node can ship tx′ in a block after tx. This relaxes the rule from “should be delivered earlier than” (the requirement of ROF) to “should be delivered no later than”.
Contemplate three consensus nodes (A, B and C) and three transactions: tx₁, tx₂, and tx₃. A transaction is taken into account “acquired earlier” if at the very least two of the three nodes (a majority) observe it first.
If we apply majority voting to find out a world order:
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tx₁ → tx₂ (agreed by A and C)
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tx₂ → tx₃ (agreed by A and B)
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tx₃ → tx₁ (agreed by B and C)
These preferences create a loop: tx₁ → tx₂ → tx₃ → tx₁. On this scenario, there’s no single order that may fulfill everybody’s view without delay, which suggests strict ROF is inconceivable to attain.
BOF solves this by grouping all of the conflicting transactions into the identical batch or block as a substitute of forcing one to return earlier than one other. The protocol merely outputs:
Block B₁ = {tx₁, tx₂, tx₃}
Because of this, from the protocol’s perspective, all three transactions are handled as in the event that they occurred on the identical time. Contained in the block, a deterministic tie-breaker (resembling a hash worth) decides the precise order by which they’ll be executed. By doing this, BOF ensures equity for each pair of transactions and retains the ultimate transaction log constant for everybody. Every one is processed no later than the one which precedes it.
This small however necessary adjustment lets the protocol deal with conditions the place transaction orderings battle, by grouping these conflicting transactions into the identical block or batch. Importantly, this doesn’t end in a partial ordering, as each node should nonetheless agree on one single, linear sequence of transactions. The transactions inside every block are nonetheless organized in a hard and fast order for execution. In instances when no such conflicts happen, the protocol nonetheless achieves the stronger ROF property.
Whereas Aequitas efficiently achieved BOF, it confronted vital limitations, notably that it had very excessive communication complexity and will solely assure weak liveness. Weak liveness implies {that a} transaction’s supply is barely assured after the complete Condorcet cycle it is part of is accomplished. This might take an arbitrarily very long time if cycles “chain collectively.”
The Themis protocol was launched to implement the identical robust BOF property, however with improved communication complexity. Themis achieves this utilizing three methods: Batch Unspooling, Deferred Ordering, and Stronger Intra-Batch Ensures.
In its commonplace type, Themis requires every participant to alternate messages with most different nodes within the community. The quantity of communication required will increase with the sq. of the variety of community individuals. Nonetheless, in its optimized model, SNARK-Themis, nodes use succinct cryptographic proofs to confirm equity while not having to speak instantly with each different participant. This reduces the communication load in order that it grows solely linearly, which permits Themis to scale effectively even in giant networks.
Assume 5 nodes (A–E) taking part in consensus obtain three transactions: tx₁, tx₂, and tx₃. Resulting from community latency, their native orders differ:
As in Aequitas, these preferences create a Condorcet cycle. However as a substitute of ready for the complete cycle to be resolved, Themis retains the system transferring utilizing a technique referred to as batch unspooling. It identifies all transactions which are a part of the cycle and teams them into one set, referred to as a strongly linked element (SCC). On this case, all three transactions belong to the identical SCC, which Themis outputs as a batch-in-progress, labeled Batch B₁ = {tx₁, tx₂, tx₃}.
By doing this, Themis permits the community to maintain processing new transactions even whereas the interior order of Batch B₁ remains to be being finalized. This ensures the system stays stay and avoids stalling.
Overview:
The idea of good equity in transaction ordering could seem simple. Whoever’s transaction reaches the community first ought to be processed first. Nonetheless, because the Condorcet paradox demonstrates, this perfect can’t maintain in actual, distributed programs. Completely different nodes see transactions in numerous orders, and when these views battle, no protocol can construct a single, universally “right” sequence with out compromise.
Hedera’s Hashgraph tried to approximate this perfect with median timestamps, however that method depends extra on belief than on proof. A single dishonest participant can distort the median and flip transaction order, revealing that “truthful timestamping” will not be actually truthful.
Protocols like Aequitas and Themis transfer the dialogue ahead by acknowledging what can and can’t be achieved. As an alternative of chasing the inconceivable, they redefine equity in a manner that also preserves order integrity beneath actual community situations. What emerges will not be a rejection of equity, however its evolution. This evolution attracts a transparent line between perceived equity and provable equity. It exhibits that true transaction-order integrity in decentralized programs can’t rely on status, validator belief or permissioned management. It should come from cryptographic verification embedded within the protocol itself.
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