Elon Musk talks Crypto：What is Bitcoin? v4

Given our assumption that p > q, the probability drops exponentially as the number of blocks the

attacker has to catch up with increases. With the odds against him, if he doesn't make a lucky

lunge forward early on, his chances become vanishingly small as he falls further behind.

We now consider how long the recipient of a new transaction needs to wait before being

sufficiently certain the sender can't change the transaction. We assume the sender is an attacker

who wants to make the recipient believe he paid him for a while, 

then switch it to pay back to

himself after some time has passed. The receiver will be alerted when that happens, but the

sender hopes it will be too late.

The receiver generates a new key pair and gives the public key to the sender shortly before

signing. 

This prevents the sender from preparing a chain of blocks ahead of time by working on

it continuously until he is lucky enough to get far enough ahead, then executing the transaction at

that moment. Once the transaction is sent,

 the dishonest sender starts working in secret on a

parallel chain containing an alternate version of his transaction.

The recipient waits until the transaction has been added to a block and z blocks have been

linked after it. He doesn't know the exact amount of progress the attacker has made,

 but

assuming the honest blocks took the average expected time per block, the attacker's potential

progress will be a Poisson distribution with expected value:

To get the probability the attacker could still catch up now, we multiply the Poisson density for

each amount of progress he could have made by the probability he could catch up from that point:

Rearranging to avoid summing the infinite tail of the distribution...

Converting to C code...

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

Solving for P less than 0.1%...

12. Conclusion

We have 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 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 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 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 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, 

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 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 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 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 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 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/hashcash.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. Feller, "An introduction to probability theory and its applications," 1957.