Create account

replied 1182d
| to avoid summing the infinite tail of the distribution...

![](

Converting to C code...

```
#include <math.h>
double AttackerSuccessProbability(doub
1 / 18
replied 1182d
|le q, int z)
{
double p = 1.0 - q;
double lambda = z * (q / p);
double sum = 1.0;
int i, k;
for (k = 0; k <= z; k++)
{
double poisson = exp(-lambda);
for (i = 1; i <= k; i++)
replied 1182d
| poisson *= lambda / i;
sum -= poisson * (1 - pow(q / p, z - k));
}
return sum;
}
```

Running some results, we can see the probability drop off exponentially with z.

```
q=0.1
z=
replied 1182d
|0 P=1.0000000
z=1 P=0.2045873
z=2 P=0.0509779
z=3 P=0.0131722
z=4 P=0.0034552
z=5 P=0.0009137
z=6 P=0.0002428
z=7 P=0.0000647
z=8 P=0.0000173
z=9 P=0.0000046
z=10 P=0.0000012
q=0.3
z=
1
1
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|0 P=1.0000000
z=5 P=0.1773523
z=10 P=0.0416605
z=15 P=0.0101008
z=20 P=0.0024804
z=25 P=0.0006132
z=30 P=0.0001522
z=35 P=0.0000379
z=40 P=0.0000095
z=45 P=0.0000024
z=50 P=0.0000006
1
1
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|```

Solving for P less than 0.1%...

```
P < 0.001
q=0.10 z=5
q=0.15 z=8
q=0.20 z=11
q=0.25 z=15
q=0.30 z=24
q=0.35 z=41
q=0.40 z=89
q=0.45 z=340
```

**12. Conclusion**
We have
2
2
replied 1182d
| proposed a system for electronic transactions without relying on trust. We started with
the usual framework of coins made from digital signatures, which provides strong control
3
2
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| of
ownership, but is incomplete without a way to prevent double-spending. To solve this, we
proposed a peer-to-peer network using proof-of-work to record a public history of
1
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| transactions
that quickly becomes computationally impractical for an attacker to change if honest nodes
control a majority of CPU power. The network is robust in its unstructured
1
1
replied 1182d
| simplicity. Nodes
work all at once with little coordination. They do not need to be identified, since messages are
not routed to any particular place and only need to be delivered on
1
replied 1182d
| a best effort basis. Nodes can
leave and rejoin the network at will, accepting the proof-of-work chain as proof of what
happened while they were gone. They vote with their CPU power,
1
1
replied 1182d
| expressing their acceptance of
valid blocks by working on extending them and rejecting invalid blocks by refusing to work on
them. Any needed rules and incentives can be enforced
1
1
replied 1182d
| with this consensus mechanism.

**References**
[1] W. Dai, "b-money," http://www.weidai.com/bmoney.txt, 1998.
[2] H. Massias, X.S. Avila, and J.-J. Quisquater, "Design of a secure
1
1
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| timestamping service with minimal
trust requirements," In 20th Symposium on Information Theory in the Benelux, May 1999.
[3] S. Haber, W.S. Stornetta, "How to time-stamp a digital
1
1
replied 1182d
| document," In Journal of Cryptology, vol 3, no
2, pages 99-111, 1991.
[4] D. Bayer, S. Haber, W.S. Stornetta, "Improving the efficiency and reliability of digital time-stamping,"
In
2
1
replied 1182d
| Sequences II: Methods in Communication, Security and Computer Science, pages 329-334, 1993.
[5] S. Haber, W.S. Stornetta, "Secure names for bit-strings," In Proceedings of the 4th
replied 1182d
| ACM Conference
on Computer and Communications Security, pages 28-35, April 1997.
[6] A. Back, "Hashcash - a denial of service counter-measure,"
http://www.hashcash.org/papers/hashcas
1
1
replied 1182d
|h.pdf, 2002.
[7] R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 Symposium on Security and
Privacy, IEEE Computer Society, pages 122-133, April 1980.
[8] W.
1
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| Feller, "An introduction to probability theory and its applications," 1957.
1
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