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Date: Wed, 19 Mar 2025 02:07:56 -0700 (PDT)
From: weichu deng <weichudeng@stu2024.jnu.edu.cn>
To: Bitcoin Development Mailing List <bitcoindev@googlegroups.com>
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Subject: =?UTF-8?Q?Re=3A_=5Bbitcoindev=5D_New_Proposal=EF=BC=9AString_Substring_Sea?=
	=?UTF-8?Q?rch_in_Bitcoin_Script_=2D_OP=5FISSUBSTR?=
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Hi Rijndael,

=20

Thanks for your example=20

[witness: foobar  foo] bar CAT EQ

=20

Yes, the unfixed string can be checked against a target substring in your=
=20
example. However, if the target substring is located in the middle of the=
=20
unfixed string, how to check it? In other words, how to have the same=20
function as =E2=80=9Cfoobar ob ISSUBSTR=E2=80=9D with CAT if =E2=80=9Cfooba=
r=E2=80=9D is unfixed?

=20

For example, suppose that a lucky draw game has the rule: if anyone has a=
=20
publicKey which includes a special substring "goodluck", he/she will be=20
awarded.=20

This game can be easily implemented with OP_ISSUBSTR as follow.

- LockScript: OP_DUP goodluck OP_ISSUBSTR...

- UnlockScript: signature publicKey

How to implement it with OP_CAT?

=20

*Regards*

Weichu deng

weichudeng@stu2024.jnu.edu.cn

=20

=E5=9C=A82025=E5=B9=B43=E6=9C=8819=E6=97=A5=E6=98=9F=E6=9C=9F=E4=B8=89 UTC+=
8 10:28:25<Rijndael> =E5=86=99=E9=81=93=EF=BC=9A

> Stack elements in Taproot are limited to 520 bytes. The current proposal=
=20
> for re-activating OP_CAT includes this restriction: creating a string=20
> longer than 520 bytes with CAT will cause the script to fail.
>
> With either CAT or ISSUBSTR, you can either commit to the substrings or=
=20
> provide them at spend-time as witness data (and allow them to be unfixed =
in=20
> the script).
>
> Fixed: FOOBAR BAR ISSUBSTR =3D=3D FOOBAR FOO BAR CAT EQ
> Variable: [witness: FOOBAR] BAR ISSUBSTR =3D=3D [witness: FOOBAR FOO] BAR=
 CAT=20
> EQ
>
>
> rijndael=20
>
>
> On Mar 18, 2025, at 11:32=E2=80=AFAM, weichu deng <weich...@stu2024.jnu.e=
du.cn>=20
> wrote:
>
> Hi, Peter Todd
> Thanks for your feedback. I agree that "Bitcoin scripts are about=20
> validation. Not computation."
> String search and concatenation are equivalent in some cases, such as in=
=20
> the example you provided.
> However, it is still necessary to introduce the OP_ISSUBSTR operation=20
> separately.
> One example is converting a non-deterministic signature to a deterministi=
c=20
> one.
> Another case is when the substring in question is located in the middle o=
f=20
> the checked string.
> CAT cannot replace ISSUBSTR for the following reasons:
>
>    1. The security of CAT is still controversial. It can easily generate=
=20
>    overly long strings, potentially causing a stack overflow. Additionall=
y,=20
>    whether OP_CAT will be restored is still under discussion.
>    2. The other substring (bar) must be known in advance.
>   =20
>
> With respect,
>
> Weichu Deng
>
> weich...@stu2024.jnu.edu.cn
> =E5=9C=A82025=E5=B9=B43=E6=9C=8818=E6=97=A5=E6=98=9F=E6=9C=9F=E4=BA=8C UT=
C+8 01:01:16<Peter Todd> =E5=86=99=E9=81=93=EF=BC=9A
>
> On Mon, Mar 17, 2025 at 09:14:05AM -0700, weichu deng wrote:
> >=20
> >=20
> > Dear fellow Bitcoin developers,
> >=20
> >=20
> >=20
> > I am pleased to present a new BIP proposal. This proposal introduces a=
=20
> new=20
> > opcode for Bitcoin scripts: OP_ISSUBSTR.
> >=20
> >=20
> > *Abstract*
> >=20
> > This BIP introduces two string opcodes, OP_ISSUBSTR and OP_ISSUBSTRVERI=
FY
> =20
> > (similar to the relationship between OP_EQUAL and OP_EQUALVERIFY), to=
=20
> > determine whether one string is a substring of another. As these opcode=
s=20
> do=20
> > not alter any blockchain state, they are secure.
>
> Bitcoin scripts are about validation. Not computation.
>
> This means that substring search and concatenation are equivalent. For
> every script that validates a substring search, you can instead
> concatenate the substring with the rest of the string, and validate
> equality instead.
>
> Basically speaking:
>
> foobar foo IsSubStr
>
> is equivalent to:
>
> foobar foo bar Cat Equal
>
> A real-world example would be more complex. But I hope that illustrates
> my point sufficiently.
>
> --=20
> https://petertodd.org 'peter'[:-1]@petertodd.org
>
>
> --=20
> You received this message because you are subscribed to the Google Groups=
=20
> "Bitcoin Development Mailing List" group.
> To unsubscribe from this group and stop receiving emails from it, send an=
=20
> email to bitcoindev+...@googlegroups.com.
> To view this discussion visit=20
> https://groups.google.com/d/msgid/bitcoindev/678d40e3-3e22-4d55-82c0-b25c=
cafb87ecn%40googlegroups.com=20
> <https://groups.google.com/d/msgid/bitcoindev/678d40e3-3e22-4d55-82c0-b25=
ccafb87ecn%40googlegroups.com?utm_medium=3Demail&utm_source=3Dfooter>
> .
>
>
>

--=20
You received this message because you are subscribed to the Google Groups "=
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To unsubscribe from this group and stop receiving emails from it, send an e=
mail to bitcoindev+unsubscribe@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/bitcoindev/=
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<p><span lang=3D"EN-US">Hi Rijndael,</span></p>

<p><span lang=3D"EN-US">=C2=A0</span></p>

<p><span lang=3D"EN-US">Thanks for your example </span></p>

<p style=3D"text-indent: 12pt;"><span lang=3D"EN-US">[witness: foobar =C2=
=A0foo] bar CAT
EQ</span></p>

<p><span lang=3D"EN-US">=C2=A0</span></p>

<p><span lang=3D"EN-US">Yes, the unfixed string can be checked
against a target substring in your example. However, if the target substrin=
g is
located in the middle of the unfixed string, how to check it? In other word=
s,
how to have the same function as =E2=80=9Cfoobar ob ISSUBSTR=E2=80=9D with =
CAT if =E2=80=9Cfoobar=E2=80=9D is
unfixed?</span></p>

<p><span lang=3D"EN-US">=C2=A0</span></p>

<p><span lang=3D"EN-US">For example, suppose that a lucky draw game
has the rule: if anyone has a publicKey which includes a special substring
"goodluck", he/she will be awarded. </span></p>

<p><span lang=3D"EN-US">This game can be easily implemented with
OP_ISSUBSTR as follow.</span></p>

<p style=3D"text-indent: 48pt;"><span lang=3D"EN-US">- LockScript: OP_DUP g=
oodluck OP_ISSUBSTR...</span></p>

<p style=3D"text-indent: 48pt;"><span lang=3D"EN-US">- UnlockScript: signat=
ure publicKey</span></p>

<p><span lang=3D"EN-US">How to implement it with OP_CAT?<b></b></span></p>

<p><b><span lang=3D"EN-US">=C2=A0</span></b></p>

<p><b><span lang=3D"EN-US">Regards</span></b><span lang=3D"EN-US"></span></=
p>

<p><span lang=3D"EN-US">Weichu deng</span></p>

<p><span lang=3D"EN-US"><a href=3D"mailto:weichudeng@stu2024.jnu.edu.cn" ta=
rget=3D"_blank">weichudeng@stu2024.jnu.edu.cn</a></span></p>

<p><span lang=3D"EN-US">=C2=A0</span></p><br /><div class=3D"gmail_quote"><=
div dir=3D"auto" class=3D"gmail_attr">=E5=9C=A82025=E5=B9=B43=E6=9C=8819=E6=
=97=A5=E6=98=9F=E6=9C=9F=E4=B8=89 UTC+8 10:28:25&lt;Rijndael> =E5=86=99=E9=
=81=93=EF=BC=9A<br/></div><blockquote class=3D"gmail_quote" style=3D"margin=
: 0 0 0 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex=
;"><div dir=3D"ltr"><div dir=3D"auto" style=3D"line-break:after-white-space=
">Stack elements in Taproot are limited to 520 bytes. The current proposal =
for re-activating OP_CAT includes this restriction: creating a string longe=
r than 520 bytes with CAT will cause the script to fail.<div><br></div><div=
>With either CAT or ISSUBSTR, you can either commit to the substrings or pr=
ovide them at spend-time as witness data (and allow them to be unfixed in t=
he script).</div><div><br></div><div>Fixed: FOOBAR BAR ISSUBSTR =3D=3D FOOB=
AR FOO BAR CAT EQ</div><div>Variable: [witness: FOOBAR] BAR ISSUBSTR =3D=3D=
 [witness: FOOBAR FOO] BAR CAT EQ</div><div><br></div><div><br></div><div>r=
ijndael=C2=A0</div><div><br><div style=3D"direction:ltr"><br><blockquote ty=
pe=3D"cite"></blockquote></div></div></div></div><div dir=3D"ltr"><div dir=
=3D"auto" style=3D"line-break:after-white-space"><div><div style=3D"directi=
on:ltr"><blockquote type=3D"cite"><div>On Mar 18, 2025, at 11:32=E2=80=AFAM=
, weichu deng &lt;<a href data-email-masked rel=3D"nofollow">weich...@stu20=
24.jnu.edu.cn</a>&gt; wrote:</div><br></blockquote></div></div></div></div>=
<div dir=3D"ltr"><div dir=3D"auto" style=3D"line-break:after-white-space"><=
div><div style=3D"direction:ltr"><blockquote type=3D"cite"><div><div style=
=3D"font-size:12px;font-style:normal;font-variant-caps:normal;font-weight:4=
00;text-align:start;text-indent:0px;text-transform:none;white-space:pre-wra=
p;word-spacing:0px;text-decoration:none;padding:0px;border:0px;font-variant=
-numeric:inherit;font-variant-east-asian:inherit;font-variant-alternates:in=
herit;font-stretch:inherit;line-height:24px;font-family:-apple-system,Blink=
MacSystemFont,&quot;Helvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Ari=
al,Roboto,&quot;PingFang SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;M=
icrosoft YaHei&quot;,sans-serif;font-size-adjust:inherit;font-kerning:inher=
it;font-feature-settings:inherit;vertical-align:baseline;margin:0px 0px 0.8=
59em;letter-spacing:0.25px;max-width:100%;overflow:auto hidden;color:rgb(6,=
6,7);word-break:break-word">Hi, <span style=3D"color:rgb(32,33,36);font-fam=
ily:Roboto,Arial,sans-serif">Peter Todd</span></div><div style=3D"font-size=
:12px;font-style:normal;font-variant-caps:normal;font-weight:400;text-align=
:start;text-indent:0px;text-transform:none;white-space:pre-wrap;word-spacin=
g:0px;text-decoration:none;padding:0px;border:0px;font-variant-numeric:inhe=
rit;font-variant-east-asian:inherit;font-variant-alternates:inherit;font-st=
retch:inherit;line-height:24px;font-family:-apple-system,BlinkMacSystemFont=
,&quot;Helvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Arial,Roboto,&qu=
ot;PingFang SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;Microsoft YaHe=
i&quot;,sans-serif;font-size-adjust:inherit;font-kerning:inherit;font-featu=
re-settings:inherit;vertical-align:baseline;margin:0px 0px 0.859em;letter-s=
pacing:0.25px;max-width:100%;overflow:auto hidden;color:rgb(6,6,7);word-bre=
ak:break-word">Thanks for your feedback. I agree that &quot;Bitcoin scripts=
 are about validation. Not computation.&quot;</div><div style=3D"font-size:=
12px;font-style:normal;font-variant-caps:normal;font-weight:400;text-align:=
start;text-indent:0px;text-transform:none;white-space:pre-wrap;word-spacing=
:0px;text-decoration:none;padding:0px;border:0px;font-variant-numeric:inher=
it;font-variant-east-asian:inherit;font-variant-alternates:inherit;font-str=
etch:inherit;line-height:24px;font-family:-apple-system,BlinkMacSystemFont,=
&quot;Helvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Arial,Roboto,&quo=
t;PingFang SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;Microsoft YaHei=
&quot;,sans-serif;font-size-adjust:inherit;font-kerning:inherit;font-featur=
e-settings:inherit;vertical-align:baseline;margin:0px 0px 0.859em;letter-sp=
acing:0.25px;max-width:100%;overflow:auto hidden;color:rgb(6,6,7);word-brea=
k:break-word">String search and concatenation are equivalent in some cases,=
 such as in the example you provided.</div><div style=3D"font-size:12px;fon=
t-style:normal;font-variant-caps:normal;font-weight:400;text-align:start;te=
xt-indent:0px;text-transform:none;white-space:pre-wrap;word-spacing:0px;tex=
t-decoration:none;padding:0px;border:0px;font-variant-numeric:inherit;font-=
variant-east-asian:inherit;font-variant-alternates:inherit;font-stretch:inh=
erit;line-height:24px;font-family:-apple-system,BlinkMacSystemFont,&quot;He=
lvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Arial,Roboto,&quot;PingFa=
ng SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;Microsoft YaHei&quot;,s=
ans-serif;font-size-adjust:inherit;font-kerning:inherit;font-feature-settin=
gs:inherit;vertical-align:baseline;margin:0px 0px 0.859em;letter-spacing:0.=
25px;max-width:100%;overflow:auto hidden;color:rgb(6,6,7);word-break:break-=
word">However, it is still necessary to introduce the OP_ISSUBSTR operation=
 separately.</div><div style=3D"font-size:12px;font-style:normal;font-varia=
nt-caps:normal;font-weight:400;text-align:start;text-indent:0px;text-transf=
orm:none;white-space:pre-wrap;word-spacing:0px;text-decoration:none;padding=
:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inheri=
t;font-variant-alternates:inherit;font-stretch:inherit;line-height:24px;fon=
t-family:-apple-system,BlinkMacSystemFont,&quot;Helvetica Neue&quot;,Helvet=
ica,&quot;Segoe UI&quot;,Arial,Roboto,&quot;PingFang SC&quot;,MIUI,&quot;Hi=
ragino Sans GB&quot;,&quot;Microsoft YaHei&quot;,sans-serif;font-size-adjus=
t:inherit;font-kerning:inherit;font-feature-settings:inherit;vertical-align=
:baseline;margin:0px 0px 0.859em;letter-spacing:0.25px;max-width:100%;overf=
low:auto hidden;color:rgb(6,6,7);word-break:break-word">One example is conv=
erting a non-deterministic signature to a deterministic one.</div><div styl=
e=3D"font-size:12px;font-style:normal;font-variant-caps:normal;font-weight:=
400;text-align:start;text-indent:0px;text-transform:none;white-space:pre-wr=
ap;word-spacing:0px;text-decoration:none;padding:0px;border:0px;font-varian=
t-numeric:inherit;font-variant-east-asian:inherit;font-variant-alternates:i=
nherit;font-stretch:inherit;line-height:24px;font-family:-apple-system,Blin=
kMacSystemFont,&quot;Helvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Ar=
ial,Roboto,&quot;PingFang SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;=
Microsoft YaHei&quot;,sans-serif;font-size-adjust:inherit;font-kerning:inhe=
rit;font-feature-settings:inherit;vertical-align:baseline;margin:0px 0px 0.=
859em;letter-spacing:0.25px;max-width:100%;overflow:auto hidden;color:rgb(6=
,6,7);word-break:break-word">Another case is when the substring in question=
 is located in the middle of the checked string.</div><div style=3D"font-si=
ze:12px;font-style:normal;font-variant-caps:normal;font-weight:400;text-ali=
gn:start;text-indent:0px;text-transform:none;white-space:pre-wrap;word-spac=
ing:0px;text-decoration:none;padding:0px;border:0px;font-variant-numeric:in=
herit;font-variant-east-asian:inherit;font-variant-alternates:inherit;font-=
stretch:inherit;line-height:24px;font-family:-apple-system,BlinkMacSystemFo=
nt,&quot;Helvetica Neue&quot;,Helvetica,&quot;Segoe UI&quot;,Arial,Roboto,&=
quot;PingFang SC&quot;,MIUI,&quot;Hiragino Sans GB&quot;,&quot;Microsoft Ya=
Hei&quot;,sans-serif;font-size-adjust:inherit;font-kerning:inherit;font-fea=
ture-settings:inherit;vertical-align:baseline;margin:0px 0px 0.859em;letter=
-spacing:0.25px;max-width:100%;overflow:auto hidden;color:rgb(6,6,7);word-b=
reak:break-word">CAT cannot replace ISSUBSTR for the following reasons:</di=
v><ol start=3D"1" style=3D"font-size:12px;font-style:normal;font-variant-ca=
ps:normal;font-weight:400;text-align:start;text-indent:0px;text-transform:n=
one;white-space:normal;word-spacing:0px;text-decoration:none;padding:0px 0p=
x 0px 2em;border:0px;font-variant-numeric:inherit;font-variant-east-asian:i=
nherit;font-variant-alternates:inherit;font-stretch:inherit;line-height:inh=
erit;font-family:-apple-system,BlinkMacSystemFont,&quot;Helvetica Neue&quot=
;,Helvetica,&quot;Segoe UI&quot;,Arial,Roboto,&quot;PingFang SC&quot;,MIUI,=
&quot;Hiragino Sans GB&quot;,&quot;Microsoft YaHei&quot;,sans-serif;font-si=
ze-adjust:inherit;font-kerning:inherit;font-feature-settings:inherit;vertic=
al-align:baseline;list-style-position:initial;margin:0.859em 0px 0px;color:=
rgb(6,6,7);letter-spacing:0.5px"><li style=3D"margin:0px;padding:0px;border=
:0px;font-style:inherit;font-variant-ligatures:inherit;font-variant-caps:in=
herit;font-variant-alternates:inherit;font-variant-numeric:inherit;font-var=
iant-east-asian:inherit;font-weight:inherit;font-stretch:inherit;font-size:=
inherit;line-height:24px;font-family:inherit;font-size-adjust:inherit;font-=
kerning:inherit;font-feature-settings:inherit;vertical-align:baseline;lette=
r-spacing:0.25px"><div style=3D"padding:0px;border:0px;font-style:inherit;f=
ont-variant-ligatures:inherit;font-variant-caps:inherit;font-variant-altern=
ates:inherit;font-variant-numeric:inherit;font-variant-east-asian:inherit;f=
ont-weight:inherit;font-stretch:inherit;line-height:24px;font-family:inheri=
t;font-size-adjust:inherit;font-kerning:inherit;font-feature-settings:inher=
it;vertical-align:baseline;margin:0px;max-width:100%;overflow:visible;word-=
break:break-word">The security of CAT is still controversial. It can easily=
 generate overly long strings, potentially causing a stack overflow. Additi=
onally, whether OP_CAT will be restored is still under discussion.</div></l=
i><li style=3D"margin:0px;padding:0px;border:0px;font-style:inherit;font-va=
riant-ligatures:inherit;font-variant-caps:inherit;font-variant-alternates:i=
nherit;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-we=
ight:inherit;font-stretch:inherit;font-size:inherit;line-height:24px;font-f=
amily:inherit;font-size-adjust:inherit;font-kerning:inherit;font-feature-se=
ttings:inherit;vertical-align:baseline;letter-spacing:0.25px"><div style=3D=
"padding:0px;border:0px;font-style:inherit;font-variant-ligatures:inherit;f=
ont-variant-caps:inherit;font-variant-alternates:inherit;font-variant-numer=
ic:inherit;font-variant-east-asian:inherit;font-weight:inherit;font-stretch=
:inherit;line-height:24px;font-family:inherit;font-size-adjust:inherit;font=
-kerning:inherit;font-feature-settings:inherit;vertical-align:baseline;marg=
in:0px;max-width:100%;overflow:visible;word-break:break-word">The other sub=
string (bar) must be known in advance.</div></li></ol><div style=3D"font-fa=
mily:Helvetica;font-size:12px;font-style:normal;font-variant-caps:normal;fo=
nt-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-t=
ransform:none;white-space:normal;word-spacing:0px;text-decoration:none"><fo=
nt color=3D"#060607" face=3D"-apple-system, BlinkMacSystemFont, Helvetica N=
eue, Helvetica, Segoe UI, Arial, Roboto, PingFang SC, MIUI, Hiragino Sans G=
B, Microsoft YaHei, sans-serif"><span style=3D"letter-spacing:0.25px"><br><=
/span></font></div><div style=3D"font-family:Helvetica;font-size:12px;font-=
style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal=
;text-align:start;text-indent:0px;text-transform:none;white-space:normal;wo=
rd-spacing:0px;text-decoration:none"><p><span lang=3D"EN-US">With respect,<=
/span></p><p><span lang=3D"EN-US">Weichu Deng</span></p><p><span lang=3D"EN=
-US"><a href data-email-masked rel=3D"nofollow">weich...@stu2024.jnu.edu.cn=
</a></span></p></div><div style=3D"font-family:Helvetica;font-size:12px;fon=
t-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:norm=
al;text-align:start;text-indent:0px;text-transform:none;white-space:normal;=
word-spacing:0px;text-decoration:none"><div dir=3D"auto">=E5=9C=A82025=E5=
=B9=B43=E6=9C=8818=E6=97=A5=E6=98=9F=E6=9C=9F=E4=BA=8C UTC+8 01:01:16&lt;Pe=
ter Todd&gt; =E5=86=99=E9=81=93=EF=BC=9A<br></div><blockquote style=3D"marg=
in:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-l=
eft-color:rgb(204,204,204);padding-left:1ex">On Mon, Mar 17, 2025 at 09:14:=
05AM -0700, weichu deng wrote:<br>&gt;<span>=C2=A0</span><br>&gt;<span>=C2=
=A0</span><br>&gt; Dear fellow Bitcoin developers,<br>&gt;<span>=C2=A0</spa=
n><br>&gt;<span>=C2=A0</span><br>&gt;<span>=C2=A0</span><br>&gt; I am pleas=
ed to present a new BIP proposal. This proposal introduces a new<span>=C2=
=A0</span><br>&gt; opcode for Bitcoin scripts: OP_ISSUBSTR.<br>&gt;<span>=
=C2=A0</span><br>&gt;<span>=C2=A0</span><br>&gt; *Abstract*<br>&gt;<span>=
=C2=A0</span><br>&gt; This BIP introduces two string opcodes, OP_ISSUBSTR a=
nd OP_ISSUBSTRVERIFY<span>=C2=A0</span><br>&gt; (similar to the relationshi=
p between OP_EQUAL and OP_EQUALVERIFY), to<span>=C2=A0</span><br>&gt; deter=
mine whether one string is a substring of another. As these opcodes do<span=
>=C2=A0</span><br>&gt; not alter any blockchain state, they are secure.<br>=
<br>Bitcoin scripts are about validation. Not computation.<br><br>This mean=
s that substring search and concatenation are equivalent. For<br>every scri=
pt that validates a substring search, you can instead<br>concatenate the su=
bstring with the rest of the string, and validate<br>equality instead.<br><=
br>Basically speaking:<br><br>foobar foo IsSubStr<br><br>is equivalent to:<=
br><br>foobar foo bar Cat Equal<br><br>A real-world example would be more c=
omplex. But I hope that illustrates<br>my point sufficiently.<br><br>--<spa=
n>=C2=A0</span><br><a href=3D"https://petertodd.org/" rel=3D"nofollow" targ=
et=3D"_blank" data-saferedirecturl=3D"https://www.google.com/url?hl=3Dzh-CN=
&amp;q=3Dhttps://petertodd.org/&amp;source=3Dgmail&amp;ust=3D17424615095790=
00&amp;usg=3DAOvVaw3nmXfOkH6THUCD0y0ysiv6">https://petertodd.org</a><span>=
=C2=A0</span>&#39;peter&#39;[:-1]@<a href=3D"http://petertodd.org/" rel=3D"=
nofollow" target=3D"_blank" data-saferedirecturl=3D"https://www.google.com/=
url?hl=3Dzh-CN&amp;q=3Dhttp://petertodd.org/&amp;source=3Dgmail&amp;ust=3D1=
742461509579000&amp;usg=3DAOvVaw1KKF8hKgJW2bEOfwEuZimQ">petertodd.org</a><b=
r></blockquote></div><div style=3D"font-family:Helvetica;font-size:12px;fon=
t-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:norm=
al;text-align:start;text-indent:0px;text-transform:none;white-space:normal;=
word-spacing:0px;text-decoration:none"><br></div></div></blockquote></div><=
/div></div></div><div dir=3D"ltr"><div dir=3D"auto" style=3D"line-break:aft=
er-white-space"><div><div style=3D"direction:ltr"><blockquote type=3D"cite"=
><div><span style=3D"font-family:Helvetica;font-size:12px;font-style:normal=
;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:=
start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0=
px;text-decoration:none;float:none;display:inline!important">--<span>=C2=A0=
</span></span><br style=3D"font-family:Helvetica;font-size:12px;font-style:=
normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-=
align:start;text-indent:0px;text-transform:none;white-space:normal;word-spa=
cing:0px;text-decoration:none"><span style=3D"font-family:Helvetica;font-si=
ze:12px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-s=
pacing:normal;text-align:start;text-indent:0px;text-transform:none;white-sp=
ace:normal;word-spacing:0px;text-decoration:none;float:none;display:inline!=
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</blockquote></div>

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