 # Proof of Mining in Blockchain Systems

We propose a proof of mining system. Roughly speaking, in this system the mining stake mstak(A) with discrimination index a∈[0,1] of an account A is defined by the formula: mstak(A)=(1-a)·1/ NOM+a· NOBM(A)/L, where L is the length of the block-chain, NOM is the number of miners in the block-chain, and NOBM(A) is the number of blocks mined by A.

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## 1 Block-Chains

In this section we recall the notion of block-chain systems invented by Satoshi Nakamoto [Na].

###### Definition 1.1

A public key of a key pair in a public-key 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 public-key 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 public-key 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 public-key cryptography system is called the miner of the block.

###### Definition 1.6

Let be a block of a public-key cryptography system, and an account of that system. The balance of in is defined by the formula

 bal(A,B)=∑tx∈Tx(B)tx(A)+rwd(A,B),

where is the set of transactions of , and

 rwd(A,B)={1,A is the miner of B,0,otherewise;
###### Definition 1.7

Let be a sequence of blocks of a public-key cryptography system, and an account of that system. The balance of in is defined by the formula

 bal(A,C)=∑B∈Cbal(A,B).
###### Definition 1.8

A block-chain in a public-key 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 block-chain systems invented by Satoshi Nakamoto [Na].

###### Definition 2.1

Let be a block in a public-key cryptography system with a hash function, the maximum hash value, and be a positive number. If satisfies

 hash(B)≤MD,

then is called a PoW block of difficulty of that system.

###### Definition 2.2

Let be a block-chain in a public-key cryptography system with a hash function, a positive integer, and a sequence of positive numbers. If

 hash(Bi)≤MD[iL],

where is the maximum hash value, then is called a PoW block-chain 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 block-chain 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

 E(T)≈DP.

By the law of large numbers, we have the following theorem.

###### Theorem 2.6

Let be a large integer, , and a PoW block-chain 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 block-chain with period and difficulty vector is approximately almost surely.

###### Definition 2.7

Let be a positive integer. A PoW block-chain system of period is a public-key cryptography system with a hash function and a communication network between the accounts in which the accounts broadcast transactions, blocks and PoW block-chains of period .

###### Definition 2.8

Let be a PoW block-chain with period and difficulty vector . Then we call the difficulty of the segment .

Following Satoshi Nakamoto [Na], one can show that in a PoW block-chain system of period where the majority of the computing power favours the block-chain of largest difficulty, it is almost impossible for a block-chain with a difficulty less than the largest to grow to be a block-chain of largest difficulty.

## 3 Proof of Mining

In this section we propose the proof of mining block-chain, and prove its security.

###### Definition 3.1

Let be a block-chain in a public-key cryptography system with a hash function, and a positive integer. We set

 CL,n=(BnL,BnL+1,⋯,BnL+L−1).
###### Definition 3.2

Let be a block-chain in a public-key cryptography system with a hash function, and a positive integer. The number of blocks mined by an account in is

 NOBM(A,CL,n)=∑nL≤i
###### Definition 3.3

Let be a block-chain in a public-key cryptography system with a hash function, and a positive integer. The number of miners in is:

 NOM(CL,n)=|{A:NOBM(A,CL,n)>0}|.
###### Definition 3.4

Let be a block-chain in a public-key cryptography system with a hash function, a positive integer, and . We define the mining-stake of an account in with discrimination index by the formula:

 mstak(A,CL,n,a)=(1−a)⋅1NOM(CL,n)+a⋅NOBM(A,CL,n)L.
###### Definition 3.5

Let be a block-chain in a public-key cryptography system with a hash function, a positive integer, and a sequence of positive numbers. If

 hash(Bi)≤MD[iL]×mstak(A,CL,[iL]−1,a),

where is the maximum hash value and is the miner of , then is called a PoM block-chain 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 block-chain 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 block-chain with period , difficulty vector , and discrimination index . Then

 E(T)≈DNmstak(S,CL,N−1,a),

where

 mstak(S,CL,N−1,a)=∑A∈Smstak(A,CL,N−1,a)

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 block-chain 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 block-chain with period , difficulty vector , and discrimination index is approximately

 k⋅DNmstak(S,CL,N−1,a)

almost surely.

It is easy to prove the following.

###### Lemma 3.8

Let be a PoM block-chain with period , and discrimination index . Let be a set of accounts. Then

 mstak(S,CL,n,a)>12

if and only if

 NOBM(S,CL,n)L>1−aa(12(1−a)−|S|NOM(CL,n)),

where

 NOBM(S,CL,n)=∑A∈SNOBM(A,CL,na)

From the above lemma one can infer the following.

###### Corollary 3.9

Let be a PoM block-chain with period , and discrimination index . Let be a set of accounts such that

 |S|NOM(CL,n)≥12(1−a).

Then

 mstak(S,CL,n,a)>12.
###### Definition 3.10

Let be a positive integer, and . A PoM block-chain system of period and discrimination index is a public-key cryptography system with a hash function and a communication network between the accounts in which the accounts broadcast transactions, blocks and PoW block-chains of period and discrimination index .

###### Definition 3.11

Let be a PoM block-chain with period and difficulty vector . Then we call the difficulty of the segment .

Following Satoshi Nakamoto [Na], one can show that in a PoM block-chain system of period and index where the majority of the accounts favours the block-chain of largest difficulty, it is almost impossible for a block-chain with a difficulty less than the largest to grow to be a block-chain of largest difficulty.

## 4 Conclusion

We have proposed a proof of mining block-chain system. We have shown that the proof of mining block-chain system is secure. The proof of mining block-chain system is more efficient than the proof of work system, but a litter less efficient than the the proof of stake block-chain systems. The proof of stake systems have been studied by many authors [KN, BGM, NXT, Mi, BPS, DGKR, KRDO, Bu, Po].