1 BlockChains
In this section we recall the notion of blockchain systems invented by Satoshi Nakamoto [Na].
Definition 1.1
A public key of a key pair in a publickey cryptography system is called an account of that system.
Definition 1.2
A function of mass 0 on a finite set of accounts of a publickey cryptography system is called a transaction of that system.
Definition 1.3
The signed version of transaction is the digital signature of the transaction signed by accounts on which the transaction is negative.
Definition 1.4
A block of a publickey cryptography system is a data containing a specified account, a finite set of transactions, and the signed versions of the transactions.
Definition 1.5
The specified account in a block of a publickey cryptography system is called the miner of the block.
Definition 1.6
Let be a block of a publickey cryptography system, and an account of that system. The balance of in is defined by the formula
where is the set of transactions of , and
Definition 1.7
Let be a sequence of blocks of a publickey cryptography system, and an account of that system. The balance of in is defined by the formula
Definition 1.8
A blockchain in a publickey cryptography system with a hash function is a sequence of blocks in which the hash of each block is contained in the next block and in which the balance of each account is nonnegative.
2 Proof of Work
In this section we recall the notion of proof of work blockchain systems invented by Satoshi Nakamoto [Na].
Definition 2.1
Let be a block in a publickey cryptography system with a hash function, the maximum hash value, and be a positive number. If satisfies
then is called a PoW block of difficulty of that system.
Definition 2.2
Let be a blockchain in a publickey cryptography system with a hash function, a positive integer, and a sequence of positive numbers. If
where is the maximum hash value, then is called a PoW blockchain with period
and difficulty vector
of that system.Definition 2.3
The computing power of a CPU with respect to a hash function is the inverse of the time it completes a single hash operation.
Definition 2.4
The computing power of an account of a blockchain system with a hash function is the sum of computing powers of all its CPU’s.
It is easy to prove the following.
Lemma 2.5
Let be the time for a set of accounts with total computing power to find a PoW block of difficulty . Then
By the law of large numbers, we have the following theorem.
Theorem 2.6
Let be a large integer, , and a PoW blockchain with period and difficulty vector . Let be a large integer such that . Then the time for a set of accounts with total computing power to find blocks such that is a PoW blockchain with period and difficulty vector is approximately almost surely.
Definition 2.7
Let be a positive integer. A PoW blockchain system of period is a publickey cryptography system with a hash function and a communication network between the accounts in which the accounts broadcast transactions, blocks and PoW blockchains of period .
Definition 2.8
Let be a PoW blockchain with period and difficulty vector . Then we call the difficulty of the segment .
Following Satoshi Nakamoto [Na], one can show that in a PoW blockchain system of period where the majority of the computing power favours the blockchain of largest difficulty, it is almost impossible for a blockchain with a difficulty less than the largest to grow to be a blockchain of largest difficulty.
3 Proof of Mining
In this section we propose the proof of mining blockchain, and prove its security.
Definition 3.1
Let be a blockchain in a publickey cryptography system with a hash function, and a positive integer. We set
Definition 3.2
Let be a blockchain in a publickey cryptography system with a hash function, and a positive integer. The number of blocks mined by an account in is
Definition 3.3
Let be a blockchain in a publickey cryptography system with a hash function, and a positive integer. The number of miners in is:
Definition 3.4
Let be a blockchain in a publickey cryptography system with a hash function, a positive integer, and . We define the miningstake of an account in with discrimination index by the formula:
Definition 3.5
Let be a blockchain in a publickey cryptography system with a hash function, a positive integer, and a sequence of positive numbers. If
where is the maximum hash value and is the miner of , then is called a PoM blockchain with period , difficulty vector , and discrimination index .
It is easy to prove the following.
Lemma 3.6
be a positive integer, a sequence of positive numbers, and . Let , and a PoM blockchain with period , difficulty vector , and discrimination index . Let be the time for a set of accounts to find a block such that is a PoM blockchain with period , difficulty vector , and discrimination index . Then
where
By the law of large numbers, we have the following theorem.
Theorem 3.7
Let be a positive integer, a sequence of positive numbers, and . Let , and a PoM blockchain with period , difficulty vector , and discrimination index . Let be a large integer such that . Then the time for a set of accounts to find block such that is a PoM blockchain with period , difficulty vector , and discrimination index is approximately
almost surely.
It is easy to prove the following.
Lemma 3.8
Let be a PoM blockchain with period , and discrimination index . Let be a set of accounts. Then
if and only if
where
From the above lemma one can infer the following.
Corollary 3.9
Let be a PoM blockchain with period , and discrimination index . Let be a set of accounts such that
Then
Definition 3.10
Let be a positive integer, and . A PoM blockchain system of period and discrimination index is a publickey cryptography system with a hash function and a communication network between the accounts in which the accounts broadcast transactions, blocks and PoW blockchains of period and discrimination index .
Definition 3.11
Let be a PoM blockchain with period and difficulty vector . Then we call the difficulty of the segment .
Following Satoshi Nakamoto [Na], one can show that in a PoM blockchain system of period and index where the majority of the accounts favours the blockchain of largest difficulty, it is almost impossible for a blockchain with a difficulty less than the largest to grow to be a blockchain of largest difficulty.
4 Conclusion
We have proposed a proof of mining blockchain system. We have shown that the proof of mining blockchain system is secure. The proof of mining blockchain system is more efficient than the proof of work system, but a litter less efficient than the the proof of stake blockchain systems. The proof of stake systems have been studied by many authors [KN, BGM, NXT, Mi, BPS, DGKR, KRDO, Bu, Po].
References
 [BGM] I. Bentov, A. Gabizon, and A. Mizrahi, Cryptocurrencies without of proof of work , CoRR, abs/1406.5694, 2014.
 [BPS] I. Bentov, R. Pass, and E. Shi, Snow white: Provably secure proof of stake , http://eprint.iacr.org/2016919, 2016.

[Bu]
V. Buterin, Longrange attacks: The serious problem with adaptive proof of work ,
https://download.wpsoftware.net/bitcion/old.pos.pdf, 2014. 
[NXT]
The NXT Community, NXT whitepaper ,
https://bravenewcoin.com/assets/Whitepapers/NxtWhitepaperv122rev4.pdf, 2014.  [DGKR] B. David, P. Gaz̆i, A. Kiayias, and A. Russell, Ouroboros praos: An adaptivelysecure semisynchronous proof of stake protocol , http://eprint.iacr.org/2017573, 2017.
 [KN] S. King, and S. Nadal, Ppcoin: Peertopeer cryptocurrency with proof of stake , https://ppcoin.net/assets/paper/ppcoinpaper.pdf, 2012.
 [KRDO] A. Kiayias, A. Russell, B. David, and R. Oliynykov, Ouroboros: A provably secure proof of stake blockchain protocol , In J. Kakz and S. Shacham, editors, CRYPTO 2017, Part I, vol. 10401 of LNCS,357388, Springer, Heidelberg, 2017.
 [Mi] S. Micali, ALGORAND: The efficient and demacradic leger , CoRR, abs/1607.0134, 2016.
 [Po] A. Poelstra, Distributed consensus from proof of stake is impssible , https://download.wpsoftware.net/bitcion/old.pos.pdf, 2014.

[Na]
S. Nakamoto, A peertopeer cash system ,
http://bitcoin.org/bitcoin.pdf, 2008.
Comments
There are no comments yet.