Information-Theoretic Kuplex

Simplex is usually stated with signed quorums that anyone can forward and verify. Here we describe an information-theoretic version. Parties have authenticated point-to-point channels, but no signatures and no transferable quorums. Each party therefore acts only on quorums of messages it received directly.

The motivation is practical as well as conceptual: avoiding signatures removes signature verification from the critical path. Threshold signatures and multi-signatures can reduce this overhead in signed protocols, but their post-quantum variants currently come with substantial CPU, storage, and message-size costs.

This follows the same information-theoretic direction as Information Theoretic HotStuff (IT-HS), but applies it to the Simplex/Kuplex view-exit structure.

Theorem: IT-Kuplex is a signature-free BFT protocol with a robust good-case latency $3\delta$. At $3f+1$, it has a worst-case view latency of $3\Delta+2\delta$, and at $4f+1$ it improves to $3\Delta+\delta$.

IT-Kuplex can be seen as a variant of the recent Forget-IT protocol. The main comparison points are:

• Both protocols obtain a robust good-case latency $3\delta$ as defined in Forget-IT.
• Forget-IT obtains constant storage. IT-Kuplex is based on Kuplex, a latency-optimized Simplex variant, and can therefore require unbounded skip quorums in extreme executions.
• Forget-IT has a worst-case view latency of $4\Delta+3\delta$, while IT-Kuplex obtains $3\Delta+2\delta$ for $3f+1$ and $3\Delta+\delta$ for $4f+1$. One of the “$\Delta$ improvements” comes from the aged quorum technique that assumes clocks are synchronized.

The view entrance gap, aged quorums, and robust good-case latency

If a party enters a view as soon as it has a quorum of $n-f$ messages, then other parties might receive the $n-2f$ honest messages one delay later, and might enter only one delay after that when all honest messages arrive. This would create a gap of $2\Delta$.

We reduce the gap back to $\Delta$ using the following technique, which relies on parties having synchronized clocks.

Each sender in a quorum includes the time at which it sent its message.

Define a quorum to be aged at time $t$ if all messages in the quorum have timestamps that are not later than $t-\Delta$.

Note that Byzantine parties may put arbitrary timestamps on their own messages.

The view entrance gap of a view is the time between the first and last honest entrances to that view. With aged quorums the gap after GST is just $\Delta$. If an honest party holds an aged quorum at time $t$, then all the $n-2f$ honest parties in that quorum sent their messages no later than $t-\Delta$, hence all honest parties will receive those messages no later than time $t$. Hence all honest parties will send a message that will arrive at all other honest parties by time $t+\Delta$, forcing the gap to be at most $\Delta$.

For robust good-case latency, the leader of view $v+1$ may have an aged quorum and be ready to propose while other parties may enter view $v+1$ up to $\Delta$ later. To keep robust good-case latency at $3\delta$, a valid proposal can make parties send Vote$(v+1,1,x)$ and Final$(v+1,x)$ before they enter view $v+1$.

IT-Kuplex with $3f+1$ Parties

Definitions and Quorums

In this section $n=3f+1$ and $Q=2f+1$.

The messages in view $v$ are:

1. Vote$(v,g,x)$, the grade-$g$ value vote for a non-$\bot$ value $x$, where $g\in{1,2,3}$.
2. Bot$(v,g)$, the grade-$g$ bottom vote, where $g\in{1,2,3}$.
3. Final$(v,x)$, the final message for a non-$\bot$ value $x$.

The quorums in view $v$ are:

1. $\mathrm{Q1}_v(x)$ is $Q$ Vote$(v,1,x)$ messages. It is used to send Final$(v,x)$.
2. $\mathrm{F}_v(x)$ is $Q$ Final$(v,x)$ messages. A party that has it decides $x$.
3. $\mathrm{W1}_v$ is $Q$ messages, each either Vote$(v,1,\cdot)$ or Bot$(v,1)$, in which no non-$\bot$ value appears at least $f+1$ times. After $T_v>2\Delta$, it is used to send Bot$(v,2)$ when final_lock(v) is unset.
4. $\mathrm{Q2}_v(x)$ is $Q$ messages, each either Vote$(v,1,x)$ or Vote$(v,2,x)$. It is used to send Vote$(v,3,x)$.
5. $\mathrm{Q3}_v(x)$ is $Q$ Vote$(v,3,x)$ messages. It is used to leave view $v$.
6. $\mathrm{B2}_v$ is $Q$ Bot$(v,2)$ messages. It is the intermediate bottom quorum used to send Bot$(v,3)$ and clear final_lock(v).
7. $\mathrm{B3}_v$ is $Q$ Bot$(v,3)$ messages. It is the bottom quorum used to leave view $v$ without a value.

An honest party:

• Sends at most one grade-1 message in a view: either Vote$(v,1,x)$ for one value $x$, or Bot$(v,1)$.
• Sends Final$(v,x)$ and sets final_lock(v) = x only if it first sent Vote$(v,1,x)$, has not sent Bot$(v,2)$ or Bot$(v,3)$, and has not sent a grade-1 or grade-2 vote for any $y\neq x$.
• While final_lock(v) = x, it sends no Bot$(v,2)$, no Bot$(v,3)$, and no grade-1 or grade-2 vote for any $y\neq x$.

Notes:

• Unlike grade-1 votes, a party may send grade-2 and grade-3 votes for several values.
• Having final_lock(v) = x and learning that an honest party saw $\mathrm{Q2}_v(y)$ for some $y\neq x$ indicates that $x$ was not decided, and hence allows the party to set final_lock(v) = ⊥.
• For proposals, honest leaders use aged quorums, while recipients validate using unaged quorums.
• A valid proposal for view $v$ can make a party send Vote$(v,1,x)$, send Final$(v,x)$, and decide before it starts T_v. Bot$(v,1)$ and the $\mathrm{W1}_v$ timeout rule still wait for T_v.
• final_lock(v) is initially ⊥.
• Parties start with view=0. By convention, every party initially has aged $\mathrm{B3}_0$.

The $3f+1$ IT-Kuplex Protocol

3f+1 IT-Kuplex with view v and designated sender Leader(v):

1. Upon first having aged Q3_{v-1}(x) or aged B3_{v-1}, if view < v:
    Set view = v
    Start timer T_v
    If this party is Leader(v):
        Send <Propose, v, x, w> using aged proposal quorums

2. Upon receiving the first <Propose, v, x, w> from Leader(v), keep it.
   Upon having a kept proposal:
    If all hold:
        - view < v, or (view = v and T_v <= 2Δ)
        - 0 <= w < v
        - for every w < y < v: B3_y
        - w = 0, or Q3_w(x)
        - no <Vote, v, 1, *> sent
        - no <Bot, v, 1> sent
        - final_lock(v) = x or final_lock(v) = ⊥
    Then Send <Vote, v, 1, x>

3. Upon T_v = 2Δ and no <Vote, v, 1, *> sent:
    Send <Bot, v, 1>

4. Upon Q1_v(x), if all hold:
        - sent <Vote, v, 1, x>
        - no <Final, v, *> sent
        - no <Bot, v, 2> sent
        - no <Bot, v, 3> sent
        - no <Vote, v, *, y> sent for y ≠ x
    Send <Final, v, x>
    Set final_lock(v) = x

5. Upon B2_v or f+1 <Bot, v, 3> messages:
    Set final_lock(v) = ⊥

6. Upon f+1 <Vote, v, *, x>:
    If no <Vote, v, 2, x> sent and
       (final_lock(v) = x or final_lock(v) = ⊥):
    Send <Vote, v, 2, x>

7. Upon any of the following holding:
        - f+1 <Bot, v, *> messages,
        - T_v > 2Δ and W1_v,
    If no <Bot, v, 2> sent and final_lock(v) = ⊥,
    Send <Bot, v, 2>

8. Upon Q2_v(x), or f+1 <Vote, v, 3, x> messages:
    If no <Vote, v, 3, x> sent, Send <Vote, v, 3, x>

9. Upon B2_v, or f+1 <Bot, v, 3> messages:
    If no <Bot, v, 3> sent and final_lock(v) = ⊥,
    Send <Bot, v, 3>

10. Upon F_v(x):
    Decide x

Why the $3f+1$ Protocol Works

Claim 1: Validity

If an honest party has $\mathrm{F}_v(x)$, $\mathrm{Q2}_v(x)$, or $\mathrm{Q3}_v(x)$, then some honest party sent Vote$(v,1,x)$. If the leader is honest, then $x$ is the leader’s proposed value.

Proof. Byzantine parties alone cannot supply an $f+1$ trigger, so every honest grade-2 or grade-3 value vote traces back to an honest grade-1 vote for that value. If the leader is honest, the leader fixes the value of every honest grade-1 vote.

Claim 2: Safety in a view

A final quorum $\mathrm{F}_v(x)$ excludes every conflicting final quorum, value quorum, and bottom quorum in view $v$.

Proof. Let $H_x$ be the honest parties that sent Final$(v,x)$ inside the final quorum. Since the final quorum has size $Q$, $|H_x|\geq f+1$. We first show that no party in $H_x$ ever clears final_lock(v) = x.

Before the first such unlock, parties in $H_x$ send no Bot$(v,2)$ and no Bot$(v,3)$. Thus a $\mathrm{B2}_v$ quorum cannot exist: it requires at least $f+1$ honest Bot$(v,2)$ messages, but there are at most $f$ honest parties outside $H_x$. Now consider the first honest sender of Bot$(v,3)$ before such an unlock. It cannot be triggered by $f+1$ earlier Bot$(v,3)$ messages, since Byzantine parties alone provide only $f$ of them, so it must be triggered by $\mathrm{B2}_v$, impossible. Hence no honest Bot$(v,3)$ is sent before the first unlock, and $f+1$ Bot$(v,3)$ messages cannot be received. The first unlock cannot occur.

Thus every party in $H_x$ keeps its lock forever: it sends no Bot$(v,2)$, no Bot$(v,3)$, and no grade-1 or grade-2 vote for $y\neq x$.

A conflicting $\mathrm{Q2}_v(y)$ quorum intersects $H_x$ in an honest party, impossible by the previous paragraph. For $\mathrm{Q3}_v(y)$, take the first honest sender of Vote$(v,3,y)$. That sender must have received $\mathrm{Q2}_v(y)$, because Byzantine parties alone cannot provide $f+1$ earlier grade-3 votes. But $\mathrm{Q2}_v(y)$ cannot exist.

Similarly, a $\mathrm{B2}_v$ quorum would intersect $H_x$ in an honest party that keeps its final lock, so $\mathrm{B2}_v$ cannot exist. The first honest sender of Bot$(v,3)$ therefore cannot be triggered by $\mathrm{B2}_v$, and cannot be triggered by $f+1$ earlier Bot$(v,3)$ messages without an honest predecessor. Thus $\mathrm{B3}_v$ cannot exist. Same-view final uniqueness follows because two final quorums intersect in an honest party, and an honest party sends at most one Final in a view.

Claim 3: Agreement

No two final quorums, in the same view or in different views, can be for different values.

Proof sketch. Same-view agreement is Claim 2. For cross-view agreement, let $k$ be the first view with a final quorum $\mathrm{F}_k(x)$. Consider the first later honest grade-1 vote for $y\neq x$ in a view $r>k$. It was sent after accepting a proposal $(r,y,w)$. If $w<k$, the proposal requires $\mathrm{B3}_k$, contradicting Claim 2. If $w=k$, it requires $\mathrm{Q3}_k(y)$, again contradicting Claim 2. If $k<w<r$, the proposal requires $\mathrm{Q3}_w(y)$. By Claim 1, this quorum implies that some honest party sent Vote$(w,1,y)$. Since $k<w<r$, this contradicts the choice of $r$ as the first later view with an honest grade-1 vote for $y$. Hence no later conflicting final quorum can exist.

Claim 4: Totality

If an honest party sends Final$(v,x)$, every honest party eventually has $\mathrm{Q3}_v(x)$. If an honest party has $\mathrm{Q3}_v(x)$ or $\mathrm{B3}_v$, then every honest party eventually has the same kind of view-exiting quorum for view $v$.

Proof. If an honest party sends Final$(v,x)$, it had $\mathrm{Q1}_v(x)$, including $f+1$ honest grade-1 votes. Every honest party eventually receives those votes. A conflicting final lock would require a $\mathrm{Q1}_v(y)$ quorum for some $y\neq x$, impossible by quorum intersection. Hence every honest party sends Vote$(v,2,x)$. Then every honest party has $\mathrm{Q2}_v(x)$, sends Vote$(v,3,x)$, and has $\mathrm{Q3}_v(x)$.

Now suppose an honest party has $\mathrm{Q3}_v(x)$ or $\mathrm{B3}_v$. The quorum contains at least $f+1$ honest messages of one form: Vote$(v,3,x)$ in the first case, Bot$(v,3)$ in the second. Every honest party eventually receives them. These messages trigger the same grade-3 message at every honest party; in the Bot$(v,3)$ case they first clear final_lock(v). Hence all honest parties eventually have the same exit quorum.

Claim 5: Aged catch-up sets view=v+1 within $\Delta$

After $\mathsf{GST}+\Delta$, whenever an honest party has an aged $\mathrm{Q3}v(x)$ or $\mathrm{B3}_v$ at time $t$, all honest parties have the corresponding unaged quorum by $t+\delta$ and the corresponding aged quorum by $t+\Delta$. Thus any honest party with view < v+1 can set view=v+1 and start $T{v+1}$ by $t+\Delta$.

Proof. Let $t\geq\mathsf{GST}+\Delta$ be a time at which an honest party has an aged $\mathrm{Q3}_v(x)$. The aged quorum contains $f+1$ honest grade-3 votes sent by time $t-\Delta$. Every honest party receives them by time $t$, sends Vote$(v,3,x)$ by time $t$, and has a quorum of honest grade-3 votes by time $t+\delta$. Hence it has an aged $\mathrm{Q3}_v(x)$ by time $t+\Delta$. An honest leader of view $v+1$ has the unaged quorum by time $t+\delta$.

For $\mathrm{B3}_v$, the aged quorum contains $f+1$ honest Bot$(v,3)$ messages sent by time $t-\Delta$. Every honest party receives them by time $t$, clears final_lock(v), sends Bot$(v,3)$ by time $t$, and has a quorum of honest Bot$(v,3)$ messages by time $t+\delta$. Hence it has an aged $\mathrm{B3}_v$ by time $t+\Delta$, and an honest leader of view $v+1$ has the unaged quorum by time $t+\delta$.

A valid proposal may make a party send Vote$(v+1,1,x)$ and Final$(v+1,x)$ earlier. Those messages do not set view=v+1 or start $T_{v+1}$.

Claim 6: Worst-case view latency is $3\Delta+2\delta$

Every post-GST view has an aged view-exiting quorum by time $t_{\mathsf{start}}+3\Delta+2\delta$, where $t_{\mathsf{start}}$ is the latest time at which an honest party sets view=v and starts $T_v$.

Proof. Let

If an aged view-exiting quorum already exists by time $T+\Delta+\delta$, we are done. If some honest party sets view > v before its grade-1 timeout in view $v$, then it already has an aged view-exiting quorum for $v$ before $t_{\mathsf{start}}+2\Delta$; Claim 5 gives every honest party an aged view-exiting quorum by the claimed bound. So assume no honest party sets view > v before its grade-1 timeout. By time $t_{\mathsf{start}}+2\Delta$, every honest party has sent one grade-1 message, either Vote$(v,1,\cdot)$ or Bot$(v,1)$. By time $T$, every honest party has received all honest grade-1 messages.

If at least $f+1$ honest grade-1 messages are Vote$(v,1,x)$, then $x$ is unique. Every honest party receives those $f+1$ honest votes by time $T$. A conflicting Final would require a $\mathrm{Q1}_v(y)$ quorum for some $y\neq x$, which cannot coexist with those honest votes. Hence every honest party sends Vote$(v,2,x)$ by time $T$. Every honest party has a quorum of honest grade-2 votes by time $T+\delta$, sends Vote$(v,3,x)$, and has an aged $\mathrm{Q3}_v(x)$ by time $T+\Delta+\delta$.

Otherwise, the honest grade-1 messages form $\mathrm{W1}_v$ at every honest party by time $T$. No final lock can exist, so every honest party sends Bot$(v,2)$. Every honest party has a quorum of honest Bot$(v,2)$ messages by time $T+\delta$, sends Bot$(v,3)$, and has an aged $\mathrm{B3}_v$ by time $T+\Delta+\delta$.

Since $T+\Delta+\delta=t_{\mathsf{start}}+3\Delta+2\delta$, the claim follows.

Claim 7: Robust good-case latency is $3\delta$

If an honest leader sets view=v after GST at time $t$ and sends its proposal, then all honest parties decide by $t+3\delta$.

Proof. The leader’s proposal quorums are aged at the leader by time $t$. By Claim 5, every honest party has those quorums unaged by time $t+\delta$. The proposal also arrives by time $t+\delta$.

Thus every honest party can validate the proposal by time $t+\delta$. If it has view < v, the proposal can make it send Vote$(v,1,x)$ before $T_v$ starts. If it has view=v, then Claim 5 gives timer value at most $\Delta+\delta\leq 2\Delta$. In either case, the grade-1 voting rule passes. After sending Vote$(v,1,x)$, an honest party can still send Final$(v,x)$ after the cutoff, provided it has not sent Bot$(v,2)$, Bot$(v,3)$, or a grade-1 or grade-2 vote for $y\neq x$. Every $Q$-sized grade-1 quorum contains at least $Q-f=f+1$ honest messages, and in this execution all honest grade-1 messages are Vote$(v,1,x)$. Thus $\mathrm{W1}_v$ cannot exist, so the $\mathrm{W1}_v$ rule cannot make it send Bot$(v,2)$.

Thus the proof does not wait for every party to start T_v: honest parties send Vote$(v,1,x)$ by $t+\delta$, send Final$(v,x)$ by $t+2\delta$, and decide by $t+3\delta$.

The honest-leader execution inside the view does not wait for a timeout or an aged quorum. The timeout forces a view-exiting quorum in the worst case.

Why Grade 2 Is Not Enough for $3f+1$

Leaving on aged $\mathrm{Q2}$ would fail because $\mathrm{Q2}$ does not have totality. For example, one honest party may send Final$(v,x)$ and become locked against Vote$(v,2,y)$ for $y\neq x$, while another honest party has $\mathrm{Q2}_v(y)$. The other honest parties may not have $\mathrm{Q2}_v(y)$.

Grade 3 adds one message delay. On the bottom path, $\mathrm{B2}_v$ contains only $f+1$ honest Bot$(v,2)$ messages, which is not enough to unlock honest parties. The grade-3 step turns this into $f+1$ honest Bot$(v,3)$ messages, which are a proof that unlocking is safe.

This extra grade gives worst-case view latency $3\Delta+2\delta$ at $3f+1$. At $4f+1$, a $Q$-quorum contains $M=2f+1$ honest grade-2 messages, enough to prove that any conflicting final lock cannot still lead to a decision, and hence that unlocking is safe. Thus the protocol can leave on aged $\mathrm{V2}$ or $\mathrm{B2}$ without a grade-3 round, removing one $\delta$ term.

IT-Kuplex with $4f+1$ Parties

Now assume $n=4f+1$, $Q=3f+1$, and $M=2f+1$. Then

Also, every $Q$-quorum contains at least $Q-f=M$ honest messages. These are the quorum facts that let the protocol remove the grade-3 echo from the $3f+1$ protocol.

Protocol Delta

Use the $3f+1$ protocol with these changes.

1. Delete grade 3: no Vote$(v,3,x)$, no Bot$(v,3)$, no $\mathrm{Q3}$, and no $\mathrm{B3}$.
2. The view-exiting and proposal quorums are now $\mathrm{V2}_v(x)$, a $Q$-quorum of Vote$(v,2,x)$ messages, and $\mathrm{B2}_v$, a $Q$-quorum of Bot$(v,2)$ messages. The initial aged quorum is $\mathrm{B2}_0$.
3. The proposal rule uses $\mathrm{V2}_w(x)$ and $\mathrm{B2}_y$ instead of $\mathrm{Q3}_w(x)$ and $\mathrm{B3}_y$; it has no final_lock(v) = x or final_lock(v) = ⊥ guard.
4. Let $\mathrm{M1}_v(x)$ be $M$ Vote$(v,1,x)$ messages, and let $\mathrm{W1}_v$ be $Q$ grade-1 messages in which no value appears at least $M$ times.
5. Let $\mathrm{U2}_v(x)$ be $M$ messages, each either Vote$(v,2,y)$ for some $y\neq x$, or Bot$(v,2)$. A party with final_lock(v) = x clears the lock on $\mathrm{U2}_v(x)$.
6. final_lock(v) is unset until the party sends Final$(v,x)$, and is then set to $x$. While final_lock(v) = x, the party sends no Bot$(v,2)$ and no Vote$(v,2,y)$ for $y\neq x$; Vote$(v,2,x)$ remains allowed.
7. A party sends Vote$(v,2,x)$ on $\mathrm{M1}_v(x)$ or on $f+1$ Vote$(v,2,x)$ messages, if it is unlocked or locked on $x$.
8. A party sends Bot$(v,2)$ on $f+1$ Bot$(v,1)$ messages, on timeout with $\mathrm{W1}_v$, or on $f+1$ Bot$(v,2)$ messages, if it is unlocked.
9. A party decides on $\mathrm{F}_v(x)$ as before. After setting view > v, it still runs the unlock, grade-2, and decision rules for view $v$.

A valid proposal can make a party send Vote$(v,1,x)$ and Final$(v,x)$ before it starts T_v.

Proof Delta

Value origin is unchanged. A final quorum contains an honest Final$(v,x)$ sender, and every honest Vote$(v,2,x)$ traces back to an honest Vote$(v,1,x)$.

For same-view safety, suppose $\mathrm{F}_v(x)$ exists. Let $H_x$ be the honest parties that sent Final$(v,x)$; then $|H_x|\geq M$. Before the first unlock by a party in $H_x$, a $\mathrm{U2}_v(x)$ proof can contain honest messages only from outside $H_x$, and there are at most $f$ such parties. Together with the $f$ Byzantine parties this is fewer than $M$ messages, so the first unlock cannot occur. Thus the parties in $H_x$ never send Bot$(v,2)$ or Vote$(v,2,y)$ for $y\neq x$, which excludes every conflicting $\mathrm{V2}_v(y)$ and $\mathrm{B2}_v$.

Cross-view agreement is the same minimal-later-view argument as before, with $\mathrm{V2}$ replacing $\mathrm{Q3}$ and $\mathrm{B2}$ replacing $\mathrm{B3}$.

Totality is where $4f+1$ saves a grade. If an honest party sends Final$(v,x)$, then its $\mathrm{Q1}_v(x)$ contains $M$ honest Vote$(v,1,x)$ messages; every honest party eventually receives them. A conflicting lock would require $\mathrm{Q1}_v(y)$ for some $y\neq x$, impossible by intersection, so every honest party sends Vote$(v,2,x)$. If an honest party has $\mathrm{V2}_v(x)$ or $\mathrm{B2}_v$, that quorum contains $M$ honest grade-2 messages. Those messages unlock any conflicting honest lock and make every honest party send the matching grade-2 message.

Consider any time $t$ satisfying

An aged $\mathrm{V2}v(x)$ or $\mathrm{B2}_v$ at time $t$ contains $M$ honest grade-2 messages sent by time $t-\Delta$. Every honest party receives them by time $t$, unlocks if needed, and sends the matching grade-2 message by time $t$. Hence every honest party has the same unaged exit quorum by $t+\delta$ and the same aged exit quorum by $t+\Delta$. Future-view Vote and Final messages may arrive earlier, but they do not set view=v+1 or start $T{v+1}$.

For worst-case latency, let $t_{\mathsf{start}}$ be the latest time an honest party sets view=v and starts $T_v$. If an aged exit quorum already exists, the aged catch-up argument applies. Otherwise, by $t_{\mathsf{start}}+2\Delta+\delta$ every honest party has all honest grade-1 messages. If $M$ honest grade-1 messages are Vote$(v,1,x)$, every honest party sends Vote$(v,2,x)$. Otherwise every honest party has $\mathrm{W1}_v$ and no final lock can exist, so every honest party sends Bot$(v,2)$. The exit quorum is aged one $\Delta$ later, so the worst-case view latency is $3\Delta+\delta$.

The honest-leader good case is unchanged: the proposal arrives and validates by $t+\delta$, honest parties send Vote$(v,1,x)$ by $t+\delta$, send Final$(v,x)$ by $t+2\delta$, and decide by $t+3\delta$.

Relation to Simplex

These protocols fit the decomposition in Deconstructing Simplex and the broader Simplex chapter: the outer proposal rule is unchanged, and each view runs an information-theoretic IT-Kuplex instance instead of signed Graded Broadcast. The only interface change is that quorums are directly received, so the $3f+1$ value and bottom quorums are $\mathrm{Q3}$ and $\mathrm{B3}$, while the $4f+1$ quorums are $\mathrm{V2}$ and $\mathrm{B2}$. Parties advance only on aged value or bottom quorums; a future-view proposal can be buffered and then validated once the required unaged quorums arrive.

Notes

• Information Theoretic HotStuff (IT-HS) is an earlier signature-free treatment of a HotStuff-style protocol. IT-HS replaces signed certificates with authenticated message passing and boosting. IT-Kuplex uses the same information-theoretic lens, but starts from Simplex/Kuplex and uses aged quorums to keep the post-GST view entrance gap to one $\Delta$.

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