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From: Erik Aronesty <erik@q32.com>
Date: Sat, 3 Jul 2021 07:31:04 -0400
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To: Jeremy <jlrubin@mit.edu>, 
 Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
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Subject: Re: [bitcoin-dev] CheckSigFromStack for Arithmetic Values
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i may be ignorant here but i have a question:

Given that schnorr signatures now allow signers to perform complex
arithmetic signing operations out-of-band using their own communications
techniques, couldn't you just perform the publishing and accumulation of
these signature components without using a bitcoin script?

In other words, push the effort of combination and computation off of the
bitcoin network and nodes.


On Sat, Jul 3, 2021 at 12:01 AM Jeremy via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> wrote:

> Yep -- sorry for the confusing notation but seems like you got it. C++
> templates have this issue too btw :)
>
> One cool thing is that if you have op_add for arbitrary width integers or
> op_cat you can also make a quantum proof signature by signing the signatu=
re
> made with checksig with the lamport.
>
> There are a couple gotchas wrt crypto assumptions on that but I'll write
> it up soon =F0=9F=99=82 it also works better in segwit V0 because there's=
 no keypath
> spend -- that breaks the quantum proofness of this scheme.
>
> On Fri, Jul 2, 2021, 4:58 PM ZmnSCPxj <ZmnSCPxj@protonmail.com> wrote:
>
>> Good morning Jeremy,
>>
>> > Dear Bitcoin Devs,
>> >
>> > It recently occurred to me that it's possible to do a lamport signatur=
e
>> in script for arithmetic values by using a binary expanded representatio=
n.
>> There are some applications that might benefit from this and I don't rec=
all
>> seeing it discussed elsewhere, but would be happy for a citation/referen=
ce
>> to the technique.
>> >
>> > blog post here, https://rubin.io/blog/2021/07/02/signing-5-bytes/,
>> text reproduced below
>> >
>> > There are two insights in this post:
>> > 1. to use a bitwise expansion of the number
>> > 2. to use a lamport signature
>> > Let's look at the code in python and then translate to bitcoin script:
>> > ```python
>> > def add_bit(idx, preimage, image_0, image_1):
>> >     s =3D sha256(preimage)
>> >     if s =3D=3D image_1:
>> >         return (1 << idx)
>> >     if s =3D=3D image_0:
>> >         return 0
>> >     else:
>> >         assert False
>> > def get_signed_number(witnesses : List[Hash], keys : List[Tuple[Hash,
>> Hash]]):
>> >     acc =3D 0
>> >     for (idx, preimage) in enumerate(witnesses):
>> >         acc +=3D add_bit(idx, preimage, keys[idx][0], keys[idx][1])
>> >     return x
>> > ```
>> > So what's going on here? The signer generates a key which is a list of
>> pairs of
>> > hash images to create the script.
>> > To sign, the signer provides a witness of a list of preimages that
>> match one or the other.
>> > During validation, the network adds up a weighted value per preimage
>> and checks
>> > that there are no left out values.
>> > Let's imagine a concrete use case: I want a third party to post-hoc
>> sign a sequence lock. This is 16 bits.
>> > I can form the following script:
>> > ```
>> > <pk> checksigverify
>> > 0
>> > SWAP sha256 DUP <H(K_0_1)> EQUAL IF DROP <1> ADD ELSE <H(K_0_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_1_1)> EQUAL IF DROP <1<<1> ADD ELSE <H(K_1_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_2_1)> EQUAL IF DROP <1<<2> ADD ELSE <H(K_2_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_3_1)> EQUAL IF DROP <1<<3> ADD ELSE <H(K_3_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_4_1)> EQUAL IF DROP <1<<4> ADD ELSE <H(K_4_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_5_1)> EQUAL IF DROP <1<<5> ADD ELSE <H(K_5_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_6_1)> EQUAL IF DROP <1<<6> ADD ELSE <H(K_6_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_7_1)> EQUAL IF DROP <1<<7> ADD ELSE <H(K_7_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_8_1)> EQUAL IF DROP <1<<8> ADD ELSE <H(K_8_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_9_1)> EQUAL IF DROP <1<<9> ADD ELSE <H(K_9_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_10_1)> EQUAL IF DROP <1<<10> ADD ELSE <H(K_10_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_11_1)> EQUAL IF DROP <1<<11> ADD ELSE <H(K_11_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_12_1)> EQUAL IF DROP <1<<12> ADD ELSE <H(K_12_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_13_1)> EQUAL IF DROP <1<<13> ADD ELSE <H(K_13_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_14_1)> EQUAL IF DROP <1<<14> ADD ELSE <H(K_14_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_15_1)> EQUAL IF DROP <1<<15> ADD ELSE <H(K_15_0)>
>> EQUALVERIFY ENDIF
>> > CHECKSEQUENCEVERIFY
>> > ```
>>
>> This took a bit of thinking to understand, mostly because you use the
>> `<<` operator in a syntax that uses `< >` as delimiters, which was mildl=
y
>> confusing --- at first I thought you were pushing some kind of nested
>> SCRIPT representation, but in any case, replacing it with the actual
>> numbers is a little less confusing on the syntax front, and I think (hop=
e?)
>> most people who can understand `1<<1` have also memorized the first few
>> powers of 2....
>>
>> > ```
>> > <pk> checksigverify
>> > 0
>> > SWAP sha256 DUP <H(K_0_1)> EQUAL IF DROP <1> ADD ELSE <H(K_0_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_1_1)> EQUAL IF DROP <2> ADD ELSE <H(K_1_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_2_1)> EQUAL IF DROP <4> ADD ELSE <H(K_2_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_3_1)> EQUAL IF DROP <8> ADD ELSE <H(K_3_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_4_1)> EQUAL IF DROP <16> ADD ELSE <H(K_4_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_5_1)> EQUAL IF DROP <32> ADD ELSE <H(K_5_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_6_1)> EQUAL IF DROP <64> ADD ELSE <H(K_6_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_7_1)> EQUAL IF DROP <128> ADD ELSE <H(K_7_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_8_1)> EQUAL IF DROP <256> ADD ELSE <H(K_8_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_9_1)> EQUAL IF DROP <512> ADD ELSE <H(K_9_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_10_1)> EQUAL IF DROP <1024> ADD ELSE <H(K_10_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_11_1)> EQUAL IF DROP <2048> ADD ELSE <H(K_11_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_12_1)> EQUAL IF DROP <4096> ADD ELSE <H(K_12_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_13_1)> EQUAL IF DROP <8192> ADD ELSE <H(K_13_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_14_1)> EQUAL IF DROP <16384> ADD ELSE <H(K_14_0)>
>> EQUALVERIFY ENDIF
>> > SWAP sha256 DUP <H(K_15_1)> EQUAL IF DROP <32768> ADD ELSE <H(K_15_0)>
>> EQUALVERIFY ENDIF
>> > CHECKSEQUENCEVERIFY
>> > ```
>>
>> On the other hand LOL WTF, this is cool.
>>
>> Basically you are showing that if we enable something as innocuous as
>> `OP_ADD`, we can implement Lamport signatures for **arbitrary** values
>> representable in small binary numbers (16 bits in the above example).
>>
>> I was thinking "why not Merkle signatures" since the pubkey would be muc=
h
>> smaller but the signature would be much larger, but (a) the SCRIPT would=
 be
>> much more complicated and (b) in modern Bitcoin, the above SCRIPT would =
be
>> in the witness stack anyway so there is no advantage to pushing the size
>> towards the signature rather than the pubkey, they all have the same
>> weight, and since both Lamport and Merkle are single-use-only and we do =
not
>> want to encourage pubkey reuse even if they were not, the Merkle has muc=
h
>> larger signature size, so Merkle sigs end up more expensive.
>>
>> Regards,
>> ZmnSCPxj
>>
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists.linuxfoundation.org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>

--000000000000fc4fc005c63668a6
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<div dir=3D"ltr">i may be ignorant here but i have a question:<div><br></di=
v><div>Given that schnorr signatures now allow signers to perform complex a=
rithmetic signing operations out-of-band using their own communications tec=
hniques, couldn&#39;t you just perform the publishing and accumulation of t=
hese signature components without using a bitcoin script?</div><div><br></d=
iv><div>In other=C2=A0words, push the effort of combination and computation=
 off of the bitcoin network and nodes.</div><div><br></div></div><br><div c=
lass=3D"gmail_quote"><div dir=3D"ltr" class=3D"gmail_attr">On Sat, Jul 3, 2=
021 at 12:01 AM Jeremy via bitcoin-dev &lt;<a href=3D"mailto:bitcoin-dev@li=
sts.linuxfoundation.org">bitcoin-dev@lists.linuxfoundation.org</a>&gt; wrot=
e:<br></div><blockquote class=3D"gmail_quote" style=3D"margin:0px 0px 0px 0=
.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir=3D"a=
uto">Yep -- sorry for the confusing notation but seems like you got it. C++=
 templates have this issue too btw :)<div dir=3D"auto"><br></div><div dir=
=3D"auto">One cool thing is that if you have op_add for arbitrary width int=
egers or op_cat you can also make a quantum proof signature by signing the =
signature made with checksig with the lamport.</div><div dir=3D"auto"><br><=
/div><div dir=3D"auto">There are a couple gotchas wrt crypto assumptions on=
 that but I&#39;ll write it up soon =F0=9F=99=82 it also works better in se=
gwit V0 because there&#39;s no keypath spend -- that breaks the quantum pro=
ofness of this scheme.</div></div><br><div class=3D"gmail_quote"><div dir=
=3D"ltr" class=3D"gmail_attr">On Fri, Jul 2, 2021, 4:58 PM ZmnSCPxj &lt;<a =
href=3D"mailto:ZmnSCPxj@protonmail.com" target=3D"_blank">ZmnSCPxj@protonma=
il.com</a>&gt; wrote:<br></div><blockquote class=3D"gmail_quote" style=3D"m=
argin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left=
:1ex">Good morning Jeremy,<br>
<br>
&gt; Dear Bitcoin Devs,<br>
&gt;<br>
&gt; It recently occurred to me that it&#39;s possible to do a lamport sign=
ature in script for arithmetic values by using a binary expanded representa=
tion. There are some applications that might benefit from this and I don&#3=
9;t recall seeing it discussed elsewhere, but would be happy for a citation=
/reference to the technique.<br>
&gt;<br>
&gt; blog post here, <a href=3D"https://rubin.io/blog/2021/07/02/signing-5-=
bytes/" rel=3D"noreferrer noreferrer" target=3D"_blank">https://rubin.io/bl=
og/2021/07/02/signing-5-bytes/</a>, text reproduced below<br>
&gt;<br>
&gt; There are two insights in this post:<br>
&gt; 1. to use a bitwise expansion of the number<br>
&gt; 2. to use a lamport signature<br>
&gt; Let&#39;s look at the code in python and then translate to bitcoin scr=
ipt:<br>
&gt; ```python<br>
&gt; def add_bit(idx, preimage, image_0, image_1):<br>
&gt; =C2=A0 =C2=A0 s =3D sha256(preimage)<br>
&gt; =C2=A0 =C2=A0 if s =3D=3D image_1:<br>
&gt; =C2=A0 =C2=A0 =C2=A0 =C2=A0 return (1 &lt;&lt; idx)<br>
&gt; =C2=A0 =C2=A0 if s =3D=3D image_0:<br>
&gt; =C2=A0 =C2=A0 =C2=A0 =C2=A0 return 0<br>
&gt; =C2=A0 =C2=A0 else:<br>
&gt; =C2=A0 =C2=A0 =C2=A0 =C2=A0 assert False<br>
&gt; def get_signed_number(witnesses : List[Hash], keys : List[Tuple[Hash, =
Hash]]):<br>
&gt; =C2=A0 =C2=A0 acc =3D 0<br>
&gt; =C2=A0 =C2=A0 for (idx, preimage) in enumerate(witnesses):<br>
&gt; =C2=A0 =C2=A0 =C2=A0 =C2=A0 acc +=3D add_bit(idx, preimage, keys[idx][=
0], keys[idx][1])<br>
&gt; =C2=A0 =C2=A0 return x<br>
&gt; ```<br>
&gt; So what&#39;s going on here? The signer generates a key which is a lis=
t of pairs of<br>
&gt; hash images to create the script.<br>
&gt; To sign, the signer provides a witness of a list of preimages that mat=
ch one or the other.<br>
&gt; During validation, the network adds up a weighted value per preimage a=
nd checks<br>
&gt; that there are no left out values.<br>
&gt; Let&#39;s imagine a concrete use case: I want a third party to post-ho=
c sign a sequence lock. This is 16 bits.<br>
&gt; I can form the following script:<br>
&gt; ```<br>
&gt; &lt;pk&gt; checksigverify<br>
&gt; 0<br>
&gt; SWAP sha256 DUP &lt;H(K_0_1)&gt; EQUAL IF DROP &lt;1&gt; ADD ELSE &lt;=
H(K_0_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_1_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;1&gt; ADD =
ELSE &lt;H(K_1_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_2_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;2&gt; ADD =
ELSE &lt;H(K_2_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_3_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;3&gt; ADD =
ELSE &lt;H(K_3_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_4_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;4&gt; ADD =
ELSE &lt;H(K_4_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_5_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;5&gt; ADD =
ELSE &lt;H(K_5_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_6_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;6&gt; ADD =
ELSE &lt;H(K_6_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_7_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;7&gt; ADD =
ELSE &lt;H(K_7_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_8_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;8&gt; ADD =
ELSE &lt;H(K_8_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_9_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;9&gt; ADD =
ELSE &lt;H(K_9_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_10_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;10&gt; AD=
D ELSE &lt;H(K_10_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_11_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;11&gt; AD=
D ELSE &lt;H(K_11_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_12_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;12&gt; AD=
D ELSE &lt;H(K_12_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_13_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;13&gt; AD=
D ELSE &lt;H(K_13_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_14_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;14&gt; AD=
D ELSE &lt;H(K_14_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_15_1)&gt; EQUAL IF DROP &lt;1&lt;&lt;15&gt; AD=
D ELSE &lt;H(K_15_0)&gt; EQUALVERIFY ENDIF<br>
&gt; CHECKSEQUENCEVERIFY<br>
&gt; ```<br>
<br>
This took a bit of thinking to understand, mostly because you use the `&lt;=
&lt;` operator in a syntax that uses `&lt; &gt;` as delimiters, which was m=
ildly confusing --- at first I thought you were pushing some kind of nested=
 SCRIPT representation, but in any case, replacing it with the actual numbe=
rs is a little less confusing on the syntax front, and I think (hope?) most=
 people who can understand `1&lt;&lt;1` have also memorized the first few p=
owers of 2....<br>
<br>
&gt; ```<br>
&gt; &lt;pk&gt; checksigverify<br>
&gt; 0<br>
&gt; SWAP sha256 DUP &lt;H(K_0_1)&gt; EQUAL IF DROP &lt;1&gt; ADD ELSE &lt;=
H(K_0_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_1_1)&gt; EQUAL IF DROP &lt;2&gt; ADD ELSE &lt;=
H(K_1_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_2_1)&gt; EQUAL IF DROP &lt;4&gt; ADD ELSE &lt;=
H(K_2_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_3_1)&gt; EQUAL IF DROP &lt;8&gt; ADD ELSE &lt;=
H(K_3_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_4_1)&gt; EQUAL IF DROP &lt;16&gt; ADD ELSE &lt=
;H(K_4_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_5_1)&gt; EQUAL IF DROP &lt;32&gt; ADD ELSE &lt=
;H(K_5_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_6_1)&gt; EQUAL IF DROP &lt;64&gt; ADD ELSE &lt=
;H(K_6_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_7_1)&gt; EQUAL IF DROP &lt;128&gt; ADD ELSE &l=
t;H(K_7_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_8_1)&gt; EQUAL IF DROP &lt;256&gt; ADD ELSE &l=
t;H(K_8_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_9_1)&gt; EQUAL IF DROP &lt;512&gt; ADD ELSE &l=
t;H(K_9_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_10_1)&gt; EQUAL IF DROP &lt;1024&gt; ADD ELSE =
&lt;H(K_10_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_11_1)&gt; EQUAL IF DROP &lt;2048&gt; ADD ELSE =
&lt;H(K_11_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_12_1)&gt; EQUAL IF DROP &lt;4096&gt; ADD ELSE =
&lt;H(K_12_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_13_1)&gt; EQUAL IF DROP &lt;8192&gt; ADD ELSE =
&lt;H(K_13_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_14_1)&gt; EQUAL IF DROP &lt;16384&gt; ADD ELSE=
 &lt;H(K_14_0)&gt; EQUALVERIFY ENDIF<br>
&gt; SWAP sha256 DUP &lt;H(K_15_1)&gt; EQUAL IF DROP &lt;32768&gt; ADD ELSE=
 &lt;H(K_15_0)&gt; EQUALVERIFY ENDIF<br>
&gt; CHECKSEQUENCEVERIFY<br>
&gt; ```<br>
<br>
On the other hand LOL WTF, this is cool.<br>
<br>
Basically you are showing that if we enable something as innocuous as `OP_A=
DD`, we can implement Lamport signatures for **arbitrary** values represent=
able in small binary numbers (16 bits in the above example).<br>
<br>
I was thinking &quot;why not Merkle signatures&quot; since the pubkey would=
 be much smaller but the signature would be much larger, but (a) the SCRIPT=
 would be much more complicated and (b) in modern Bitcoin, the above SCRIPT=
 would be in the witness stack anyway so there is no advantage to pushing t=
he size towards the signature rather than the pubkey, they all have the sam=
e weight, and since both Lamport and Merkle are single-use-only and we do n=
ot want to encourage pubkey reuse even if they were not, the Merkle has muc=
h larger signature size, so Merkle sigs end up more expensive.<br>
<br>
Regards,<br>
ZmnSCPxj<br>
</blockquote></div>
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</blockquote></div>

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