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From: nopara73 <adam.ficsor73@gmail.com>
Date: Fri, 27 Dec 2019 14:03:49 -0400
Message-ID: <CAEPKjgdtgDbyLoj6FV+cjY1Djca_FBtd9Kt_eB4zWU+at=wfYQ@mail.gmail.com>
To: Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
Content-Type: multipart/alternative; boundary="00000000000086d4cc059ab35144"
Subject: [bitcoin-dev] Non-equal value CoinJoins. Opinions.
Precedence: list

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The CashFusion research came out of the Bitcoin Cash camp, thus this
probably went under the radar of many of you. I would like to ask your
opinions on the research's claim that, if non-equal value coinjoins can be
really relied on for privacy or not.

(Btw, there were also similar ideas in the Knapsack paper in 2017:
https://www.comsys.rwth-aachen.de/fileadmin/papers/2017/2017-maurer-trustco=
m-coinjoin.pdf
 )

https://github.com/cashshuffle/spec/blob/master/CASHFUSION.md#avoiding-amou=
nt-linkages-through-combinatorics


I copy the most relevant paragraphs here:

  ---------BEGIN QUOTE ---------


Consider a transaction where 10 people have each brought 10 inputs of
arbitary amounts in the neighborhood of ~0.1 BCH. One input might be
0.03771049 BCH; the next might be 0.24881232 BCH, etc. All parties have
chosen to consolidate their coins, so the transaction has 10 outputs of
around 1 BCH. So the transaction has 100 inputs, and 10 outputs. The first
output might be 0.91128495, the next could be 1.79783710, etc.

Now, there are 100!/(10!)^10 ~=3D 10^92 ways to partition the inputs into a
list of 10 sets of 10 inputs, but only a tiny fraction of these partitions
will produce the precise output list. So, how many ways produce this exact
output list? We can estimate with some napkin math. First, recognize that
for each partitioning, each output will typically land in a range of ~10^8
discrete possibilities (around 1 BCH wide, with a 0.00000001 BCH
resolution). The first 9 outputs all have this range of possibilities, and
the last will be constrained by the others. So, the 10^92 possibilies will
land somewhere within a 9-dimensional grid that cointains (10^8)^9=3D10^72
possible distinct sites, one site which is our actual output list. Since we
are stuffing 10^92 possibilties into a grid that contains only 10^72 sites,
then this means on average, each site will have 10^20 possibilities.

Based on the example above, we can see that not only are there a huge
number of partitions, but that even with a fast algorithm that could find
matching partitions, it would produce around 10^20 possible valid
configurations. With 10^20 possibilities, there is essentially no linkage.
The Cash Fusion scheme actually extends this obfuscation even further. Not
only can players bring many inputs, they can also have multiple outputs.
---------END QUOTE ---------
--=20
Best,
=C3=81d=C3=A1m

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<div dir=3D"ltr">The CashFusion research came out of the Bitcoin Cash camp,=
 thus this probably went under the radar of many of you. I would like to as=
k your opinions on the research&#39;s claim that, if non-equal value coinjo=
ins can be really relied on for privacy or not.<br><br>(Btw, there were als=
o similar ideas in the Knapsack paper in 2017:=C2=A0<a href=3D"https://www.=
comsys.rwth-aachen.de/fileadmin/papers/2017/2017-maurer-trustcom-coinjoin.p=
df">https://www.comsys.rwth-aachen.de/fileadmin/papers/2017/2017-maurer-tru=
stcom-coinjoin.pdf</a>=C2=A0)=C2=A0<br><br><a href=3D"https://github.com/ca=
shshuffle/spec/blob/master/CASHFUSION.md#avoiding-amount-linkages-through-c=
ombinatorics">https://github.com/cashshuffle/spec/blob/master/CASHFUSION.md=
#avoiding-amount-linkages-through-combinatorics</a>=C2=A0=C2=A0<br><br>I co=
py the most relevant paragraphs here:<br clear=3D"all"><div><br>=C2=A0 ----=
-----BEGIN QUOTE

---------=C2=A0<br>=C2=A0<br><p style=3D"box-sizing:border-box;margin-top:0=
px;margin-bottom:16px;color:rgb(36,41,46);font-family:-apple-system,BlinkMa=
cSystemFont,&quot;Segoe UI&quot;,Helvetica,Arial,sans-serif,&quot;Apple Col=
or Emoji&quot;,&quot;Segoe UI Emoji&quot;;font-size:16px">Consider a transa=
ction where 10 people have each brought 10 inputs of arbitary amounts in th=
e neighborhood of ~0.1 BCH. One input might be 0.03771049 BCH; the next mig=
ht be 0.24881232 BCH, etc. All parties have chosen to consolidate their coi=
ns, so the transaction has 10 outputs of around 1 BCH. So the transaction h=
as 100 inputs, and 10 outputs. The first output might be 0.91128495, the ne=
xt could be 1.79783710, etc.</p><p style=3D"box-sizing:border-box;margin-to=
p:0px;margin-bottom:16px;color:rgb(36,41,46);font-family:-apple-system,Blin=
kMacSystemFont,&quot;Segoe UI&quot;,Helvetica,Arial,sans-serif,&quot;Apple =
Color Emoji&quot;,&quot;Segoe UI Emoji&quot;;font-size:16px">Now, there are=
 100!/(10!)^10 ~=3D 10^92 ways to partition the inputs into a list of 10 se=
ts of 10 inputs, but only a tiny fraction of these partitions will produce =
the precise output list. So, how many ways produce this exact output list? =
We can estimate with some napkin math. First, recognize that for each parti=
tioning, each output will typically land in a range of ~10^8 discrete possi=
bilities (around 1 BCH wide, with a 0.00000001 BCH resolution). The first 9=
 outputs all have this range of possibilities, and the last will be constra=
ined by the others. So, the 10^92 possibilies will land somewhere within a =
9-dimensional grid that cointains (10^8)^9=3D10^72 possible distinct sites,=
 one site which is our actual output list. Since we are stuffing 10^92 poss=
ibilties into a grid that contains only 10^72 sites, then this means on ave=
rage, each site will have 10^20 possibilities.</p><p style=3D"box-sizing:bo=
rder-box;margin-top:0px;margin-bottom:16px;color:rgb(36,41,46);font-family:=
-apple-system,BlinkMacSystemFont,&quot;Segoe UI&quot;,Helvetica,Arial,sans-=
serif,&quot;Apple Color Emoji&quot;,&quot;Segoe UI Emoji&quot;;font-size:16=
px">Based on the example above, we can see that not only are there a huge n=
umber of partitions, but that even with a fast algorithm that could find ma=
tching partitions, it would produce around 10^20 possible valid configurati=
ons. With 10^20 possibilities, there is essentially no linkage. The Cash Fu=
sion scheme actually extends this obfuscation even further. Not only can pl=
ayers bring many inputs, they can also have multiple outputs.</p>---------E=
ND QUOTE

---------

</div>-- <br><div dir=3D"ltr" class=3D"gmail_signature" data-smartmail=3D"g=
mail_signature"><div dir=3D"ltr"><div><div dir=3D"ltr"><div><div dir=3D"ltr=
"><div><div><span style=3D"font-size:13.3333339691162px">Best,<br>=C3=81d=
=C3=A1m</span></div></div></div></div></div></div></div></div></div>

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