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Subject: Re: [bitcoin-dev] TAPLEAF_UPDATE_VERIFY covenant opcode
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I like this proposal, I think it has interesting use cases! I'm quick to
charitably Matt's comment, "I=E2=80=99ve been saying we need more covenants
research and proposals before we move forward with one", as before we move
forward with *any.* I don't think that these efforts are rival -- different
opcodes for different nodes as they say.

I've previously done some analysis comparing Coin / Payment Pools with CTV
to TapLeafUpdate which make CTV out favorably in terms of chain load and
privacy.

On the "anyone can withdraw themselves in O(1) transactions" front, is that
if you contrast a CTV-style tree, the withdraws are O(log(n)) but E[O(1)]
for all participants, e.g. summing over the entire tree as it splits to
evict a bad actor ends up being O(N) total work over N participants, so you
do have to look at the exact transactions that come out w.r.t. script size
to determine which Payment Pool has overall less chain work to trustlessly
withdraw. This is compounded by the fact that a Taproot for N participants
uses a O(log N) witness.


Let's do out that basic math. First, let's assume we have 30 participants.
The basic script for each node would be:

TLUV: Taproot(Tweaked Key, {<KEY> DUP "" 1 TLUV
    CHECKSIGVERIFY
    IN_OUT_AMOUNT SUB <BAL> GREATERTHANOREQUAL, ...})

Under this, the first withdraw for TLUV would require in witnesses stack:
Assume average amount is 0.005BTC, so we have 4.2 B users =3D 18.9 bits =3D=
3
bytes

1 signature (1+64 bytes) + (1 Script =3D (+ 1 1 32 1 1 1 1 1 1 1 3 1 1) =3D=
 46
bytes) + (1 taproot path =3D 2 + 33 + log2(N)*32)
=3D 146+log2(N)*32.

now, because we delete the key, we need to sum this from N=3D0 to N=3D30:

>>> sum([65+46+35+math.log(N,2)*32 for N in range(1, 31)])
7826.690154943152 bytes of witness data

Each transaction should have 1 input (40 bytes), 2 outputs (2* (34+8) =3D
84), 4 bytes locktime, 4 bytes version, 2 byte witness flag, 1 byte in
counter 1 byte out counter  =3D 136 bytes (we already count witnesses above=
)


136 * 30 + 7827 =3D 11907 bytes to withdraw all trustlessly

Now for CTV:
-CTV: Taproot(MuSigKey(subparties), <H splits pool with radix 4> CTV)

*sidebar: **why radix 4? A while ago, I did the math out and a radix of 4
or 5 was optimal for bare script... assuming this result holds with
taproot.*


balance holders: 0..30
you have a base set of transactions paying out: 0..4 4..8 8..12 12..16
16..20 20..24 24..27 27..30
interior nodes covering: 0..16 16..30
root node covering: 0..30

The witness for each of these looks like:

(Taproot Script =3D 1+1+32+1) + (Taproot Control =3D 33) =3D 68 bytes

A transaction with two outputs should have 1 input (40 bytes), 2 outputs
(2* (34+8) =3D 84), 4 bytes locktime, 4 bytes version, 2 byte witness flag,=
 1
byte in counter 1 byte out counter  =3D 136 bytes + 68 bytes witness =3D 20=
4
A transaction with three outputs should have 1 input (40 bytes), 3 outputs
(3* (34+8) =3D 126), 4 bytes locktime, 4 bytes version, 2 byte witness flag=
,
1 byte in counter 1 byte out counter  =3D 178 bytes + 68 bytes witness =3D =
246
A transaction with 4 outputs should have 1 input (40 bytes), 4 outputs (4*
(34+8) =3D 126), 4 bytes locktime, 4 bytes version, 2 byte witness flag, 1
byte in counter 1 byte out counter  =3D 220 bytes + 68 bytes witness =3D 28=
8

204 + 288*6 + 246*2 =3D 2424 bytes

Therefore the CTV style pool is, in this example, about 5x more efficient
in block space utilization as compared to TLUV at trustlessly withdrawing
all participants. This extra space leaves lots of headroom to e.g.
including things like OP_TRUE anchor outputs (12*10) =3D 120 bytes total fo=
r
CPFP; an optional script path with 2 inputs for a gas-paying input (cost is
around 32 bytes for taproot?). The design also scales beyond 30
participants, where the advantage grows further (iirc, sum i =3D 0 to n log=
 i
is relatively close to n log n).

In the single withdrawal case, the cost to eject a single participant with
CTV is 204+288 =3D 492 bytes, compared to 65+46+35+math.log(30,2)*32+136 =
=3D
439 bytes. The cost to eject a second participant in CTV is much smaller as
it amortizes -- worst case is 288, best case is 0 (already expanded),
whereas in TLUV there is limited amortization so it would be about 438
bytes.

The protocols are identical in the cooperative case.

In terms of privacy, the CTV version is a little bit worse. At every
splitting, radix of the root nodes total value gets broadcast. So to eject
a participant, you end up leaking a bit more information. However, it might
be a reasonable assumption that if one of your counterparties is
uncooperative, they might dox you anyways. CTV trees are also superior
during updates for privacy in the cooperative case. With the TLUV pool, you
must know all tapleafs and the corresponding balances. Whereas in CTV
trees, you only need to know the balances of the nodes above you. E.g., we
can update the balances

from: [[1 Alice, 2 Bob], [3 Carol, 4 Dave]]
to: [[2.5 Alice, 0.5 Bob], [3 Carol, 4 Dave]]

without informing Carol or Dave about the updates in our subtree, just that
our slice of participants signed off on it. So even in the 1 party
uncooperative case, 3/4 the pool has update privacy against them, and the
subtrees that share a branch with the uncooperative also have this privacy
recursively to an extent.

CTV's TXID stability also lets you embed payment channels a bit easier, but
perhaps TLUV would be in an ANYPREVOUT universe w.r.t. channel protocols so
that might matter less across the designs. Nevertheless, I think it's
exciting to think about these payment pools where every node is an
updatable channel.

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<div dir=3D"ltr"><div class=3D"gmail_default" style=3D"font-family:arial,he=
lvetica,sans-serif;font-size:small;color:rgb(0,0,0)">I like this proposal, =
I think it has interesting use cases! I&#39;m quick to charitably Matt&#39;=
s comment, &quot;<span style=3D"font-family:Arial,Helvetica,sans-serif;colo=
r:rgb(34,34,34)">I=E2=80=99ve been saying we need more covenants research a=
nd proposals before we move forward with one&quot;, as before we move forwa=
rd with <i>any.</i>=C2=A0I don&#39;t think that these efforts are rival -- =
different opcodes for different nodes as they say.</span></div><div class=
=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;font-siz=
e:small;color:rgb(0,0,0)"><span style=3D"font-family:Arial,Helvetica,sans-s=
erif;color:rgb(34,34,34)"><br></span></div><div class=3D"gmail_default" sty=
le=3D"font-family:arial,helvetica,sans-serif;font-size:small;color:rgb(0,0,=
0)"><span style=3D"font-family:Arial,Helvetica,sans-serif;color:rgb(34,34,3=
4)">I&#39;ve previously done some analysis comparing Coin / Payment Pools w=
ith CTV to TapLeafUpdate which make CTV out favorably in terms of chain loa=
d and privacy.</span></div><div class=3D"gmail_default" style=3D"font-famil=
y:arial,helvetica,sans-serif;font-size:small;color:rgb(0,0,0)"><br></div><d=
iv class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;=
color:rgb(0,0,0)">On the &quot;anyone can withdraw themselves in O(1) trans=
actions&quot; front, is that if you contrast a CTV-style tree, the withdraw=
s are O(log(n)) but E[O(1)] for all participants, e.g. summing over the ent=
ire tree as it splits to evict a bad actor ends up being O(N) total work ov=
er N participants, so you do have to look at the exact transactions that co=
me out w.r.t. script size to determine which Payment Pool has overall less =
chain work to trustlessly withdraw. This is compounded by the fact that a T=
aproot for N participants uses a O(log N) witness.</div><div class=3D"gmail=
_default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"=
><br></div><div class=3D"gmail_default" style=3D"font-family:arial,helvetic=
a,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=
=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">Let&#39;s do o=
ut that basic math. First, let&#39;s assume we have 30 participants. The ba=
sic script for each node would be:</div><div class=3D"gmail_default" style=
=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div=
 class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;co=
lor:rgb(0,0,0)">TLUV: Taproot(Tweaked Key, {<span style=3D"font-family:Aria=
l,Helvetica,sans-serif;color:rgb(34,34,34)">&lt;KEY&gt; DUP &quot;&quot; 1 =
TLUV</span><br></div>=C2=A0 =C2=A0 CHECKSIGVERIFY<span class=3D"gmail-Apple=
-converted-space">=C2=A0</span><br>=C2=A0 =C2=A0 IN_OUT_AMOUNT SUB &lt;BAL&=
gt; GREATERTHANOREQUAL<span class=3D"gmail_default" style=3D"font-family:ar=
ial,helvetica,sans-serif;font-size:small;color:rgb(0,0,0)">, ...})</span><d=
iv><span style=3D"color:rgb(0,0,0);font-family:arial,helvetica,sans-serif">=
</span><br></div><div><div class=3D"gmail_default" style=3D"font-family:ari=
al,helvetica,sans-serif;color:rgb(0,0,0)">Under this, the first withdraw fo=
r TLUV would require in witnesses stack:</div><div class=3D"gmail_default" =
style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">Assume av=
erage amount is 0.005BTC, so we have 4.2 B users =3D 18.9 bits =3D3 bytes</=
div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-=
serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"fon=
t-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">1 signature (1+64 byt=
es) + (1 Script =3D (+ 1 <span style=3D"color:rgb(34,34,34);font-family:Ari=
al,Helvetica,sans-serif">1 32 1 1 1 1=C2=A0</span><span class=3D"gmail_defa=
ult">1</span><span class=3D"gmail-Apple-converted-space" style=3D"font-fami=
ly:Arial,Helvetica,sans-serif;color:rgb(34,34,34)">=C2=A0</span><span class=
=3D"gmail_default">1</span><span style=3D"font-family:Arial,Helvetica,sans-=
serif;color:rgb(34,34,34)">=C2=A01 3</span><span class=3D"gmail_default"> 1=
</span><span style=3D"font-family:Arial,Helvetica,sans-serif;color:rgb(34,3=
4,34)"> 1</span><span class=3D"gmail_default">) =3D 46 bytes)=C2=A0+ (1 tap=
root path =3D 2 + 33 + log2(N)*32)=C2=A0</span></div><div class=3D"gmail_de=
fault" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">=
=3D 146+log2(N)*32.</div><div class=3D"gmail_default" style=3D"font-family:=
arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_=
default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">=
now, because we delete the key, we need to sum this from N=3D0 to N=3D30:</=
div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-=
serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"fon=
t-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">&gt;&gt;&gt; sum([65+=
46+35+math.log(N,2)*32 for N in range(1, 31)])<br>7826.690154943152 bytes o=
f witness data<br></div><div class=3D"gmail_default" style=3D"font-family:a=
rial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_d=
efault" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">E=
ach transaction should have 1 input (40 bytes), 2 outputs (2* (34+8) =3D 84=
), 4 bytes locktime, 4 bytes version, 2 byte witness flag, 1 byte in counte=
r 1 byte out counter =C2=A0=3D 136 bytes (we already count witnesses above)=
</div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,san=
s-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"f=
ont-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div clas=
s=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;color:r=
gb(0,0,0)">136 * 30=C2=A0+ 7827 =3D 11907 bytes to withdraw all trustlessly=
</div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,san=
s-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"f=
ont-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">Now for CTV:</div><=
div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif=
;color:rgb(0,0,0)"><div class=3D"gmail_default">-CTV: Taproot(MuSigKey(subp=
arties), &lt;H splits pool with radix 4&gt; CTV)</div></div><div class=3D"g=
mail_default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0=
,0)"><br></div><div class=3D"gmail_default" style=3D"font-family:arial,helv=
etica,sans-serif;color:rgb(0,0,0)"><b>sidebar: </b><i>why radix 4? A while =
ago, I did the math out and a radix of 4 or 5 was optimal for bare script..=
. assuming this result holds with taproot.</i></div><div class=3D"gmail_def=
ault" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br=
></div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sa=
ns-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"=
font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">balance holders: 0=
..30</div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica=
,sans-serif;color:rgb(0,0,0)">you have a base set of transactions paying ou=
t: 0..4 4..8 8..12 12..16 16..20 20..24 24..27 27..30</div><div class=3D"gm=
ail_default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,=
0)">interior nodes covering: 0..16 16..30</div><div class=3D"gmail_default"=
 style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">root nod=
e covering: 0..30</div><div class=3D"gmail_default" style=3D"font-family:ar=
ial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_de=
fault" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">Th=
e witness for each of these looks like:</div><div class=3D"gmail_default" s=
tyle=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div>=
<div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-seri=
f;color:rgb(0,0,0)">(Taproot Script =3D 1+1+32+1)=C2=A0+ (Taproot Control =
=3D 33) =3D 68 bytes</div><div class=3D"gmail_default" style=3D"font-family=
:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail=
_default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"=
><div class=3D"gmail_default">A transaction with two outputs should have 1 =
input (40 bytes), 2 outputs (2* (34+8) =3D 84), 4 bytes locktime, 4 bytes v=
ersion, 2 byte witness flag, 1 byte in counter 1 byte out counter =C2=A0=3D=
 136 bytes=C2=A0+ 68 bytes witness =3D 204</div><div class=3D"gmail_default=
">A transaction with three outputs should have 1 input (40 bytes), 3 output=
s (3* (34+8) =3D 126), 4 bytes locktime, 4 bytes version, 2 byte witness fl=
ag, 1 byte in counter 1 byte out counter =C2=A0=3D 178 bytes=C2=A0+ 68 byte=
s witness =3D 246<br></div>A transaction with 4 outputs should have 1 input=
 (40 bytes), 4 outputs (4* (34+8) =3D 126), 4 bytes locktime, 4 bytes versi=
on, 2 byte witness flag, 1 byte in counter 1 byte out counter =C2=A0=3D 220=
 bytes +=C2=A068 bytes witness =3D 288<br class=3D"gmail-Apple-interchange-=
newline"></div><div class=3D"gmail_default" style=3D"font-family:arial,helv=
etica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" s=
tyle=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">204=C2=A0+=
 288*6=C2=A0+ 246*2 =3D=C2=A02424 bytes</div><div class=3D"gmail_default" s=
tyle=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div>=
<div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-seri=
f;color:rgb(0,0,0)">Therefore the CTV style pool is, in this example, about=
 5x more efficient in block space utilization as compared to TLUV at trustl=
essly withdrawing all participants. This extra space leaves lots of headroo=
m to e.g. including things like OP_TRUE anchor outputs (12*10) =3D 120 byte=
s total for CPFP; an optional script path with 2 inputs for a gas-paying in=
put (cost is around 32 bytes for taproot?). The design also scales beyond 3=
0 participants, where the advantage grows further (iirc, sum i =3D 0 to n l=
og i is relatively close to n log n).=C2=A0</div><div class=3D"gmail_defaul=
t" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></=
div><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-=
serif;color:rgb(0,0,0)">In the single withdrawal case, the cost to eject a =
single participant with CTV is 204+288 =3D 492 bytes, compared to 65+46+35+=
math.log(30,2)*32+136 =3D 439 bytes. The cost to eject a second participant=
 in CTV is much smaller as it amortizes -- worst case is 288, best case is =
0 (already expanded), whereas in TLUV there is limited amortization so it w=
ould be about 438 bytes.</div><div class=3D"gmail_default" style=3D"font-fa=
mily:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div><div class=3D"g=
mail_default" style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0=
,0)">The protocols are identical in the cooperative case.</div><div class=
=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;color:rg=
b(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"font-family:arial=
,helvetica,sans-serif;color:rgb(0,0,0)">In terms of privacy, the CTV versio=
n is a little bit worse. At every splitting, radix of the root nodes total =
value gets broadcast. So to eject a participant, you end up leaking a bit m=
ore information. However, it might be a reasonable assumption that if one o=
f your counterparties is uncooperative, they might dox you anyways. CTV tre=
es are also superior during updates for privacy in the cooperative case. Wi=
th the TLUV pool, you must know all tapleafs and the corresponding balances=
. Whereas in CTV trees, you only need to know the balances of the nodes abo=
ve you. E.g., we can update the balances</div><div class=3D"gmail_default" =
style=3D"font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div=
><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-ser=
if;color:rgb(0,0,0)">from: [[1 Alice, 2 Bob], [3 Carol, 4 Dave]]</div><div =
class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;col=
or:rgb(0,0,0)">to: [[2.5 Alice, 0.5 Bob], [3 Carol, 4 Dave]]</div><div clas=
s=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-serif;color:r=
gb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"font-family:aria=
l,helvetica,sans-serif;color:rgb(0,0,0)">without informing Carol or Dave ab=
out the updates in our subtree, just that our slice of participants signed =
off on it. So even in the 1 party uncooperative case, 3/4 the pool has upda=
te privacy against them, and the subtrees that share a branch with the unco=
operative also have this privacy recursively to an extent.</div><br class=
=3D"gmail-Apple-interchange-newline"><div class=3D"gmail_default" style=3D"=
font-family:arial,helvetica,sans-serif;color:rgb(0,0,0)">CTV&#39;s TXID sta=
bility also lets you embed payment channels a bit easier, but perhaps TLUV =
would be in an ANYPREVOUT universe w.r.t. channel protocols so that might m=
atter less across the designs. Nevertheless, I think it&#39;s exciting to t=
hink about these payment pools where every node is an updatable channel.</d=
iv><div class=3D"gmail_default" style=3D"font-family:arial,helvetica,sans-s=
erif;color:rgb(0,0,0)"><br></div><div class=3D"gmail_default" style=3D"font=
-family:arial,helvetica,sans-serif;color:rgb(0,0,0)"><br></div></div></div>

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