Proof of Mining in Blockchain Systems

TL;DR

This paper introduces a new proof of mining system that combines the number of blocks mined by an account and the total number of miners, with a discrimination index to determine mining stake.

Contribution

It proposes a novel proof of mining mechanism incorporating a weighted formula based on mining activity and network participation.

Findings

Defines a new mining stake formula with discrimination index

Provides a theoretical framework for proof of mining systems

Suggests potential fairness improvements in blockchain mining

Abstract

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

Equations

$$\mathrm{mstak}(A)=(1-a)\cdot \frac{1}{\mathrm{NOM}}+a\cdot \frac{\mathrm{NOBM}(A)}{L},$$

$$\mathrm{bal}(A,B)=\sum _{\mathrm{tx}\in \mathrm{Tx}(B)}\mathrm{tx}(A)+\mathrm{rwd}(A,B),$$

$${\rm rwd}(A,B)=\left\{\begin{array}[]{ll}1,&A\hbox{ is the miner of }B,\\ 0,&\hbox{otherewise;}\end{array}\right.$$

$$\mathrm{bal}(A,C)=\sum _{B\in C}\mathrm{bal}(A,B).$$

$$\mathrm{hash}(B)\le \frac{M}{D},$$

$$\mathrm{hash}(B_i)\le \frac{M}{D_{[\frac{i}{L}]}},$$

$$\mathrm{E}(T)\approx \frac{D}{P}.$$

$$C_{L,n}=(B_{nL},B_{nL+1},\cdots ,B_{nL+L-1}).$$

$$\mathrm{NOBM}(A,C_{L,n})=\sum _{nL\le i<nL+L}\mathrm{rwd}(A,B_i).$$

$$\mathrm{NOM}(C_{L,n})=|\{A:\mathrm{NOBM}(A,C_{L,n})>0\}|.$$

$$\mathrm{mstak}(A,C_{L,n},a)=(1-a)\cdot \frac{1}{\mathrm{NOM}(C_{L,n})}+a\cdot \frac{\mathrm{NOBM}(A,C_{L,n})}{L}.$$

$$\mathrm{hash}(B_i)\le \frac{M}{D_{[\frac{i}{L}]}}\times \mathrm{mstak}(A,C_{L,[\frac{i}{L}]-1},a),$$

$$\mathrm{E}(T)\approx \frac{D_N}{\mathrm{mstak}(S,C_{L,N-1},a)},$$

$$\mathrm{mstak}(S,C_{L,N-1},a)=\sum _{A\in S}\mathrm{mstak}(A,C_{L,N-1},a)$$

$$\frac{k\cdot D_N}{\mathrm{mstak}(S,C_{L,N-1},a)}$$

$$\mathrm{mstak}(S,C_{L,n},a)>\frac{1}{2}$$

$$\frac{\mathrm{NOBM}(S,C_{L,n})}{L}>\frac{1-a}{a}(\frac{1}{2(1-a)}-\frac{|S|}{\mathrm{NOM}({\mathrm{C}}_{\mathrm{L},\mathrm{n}})}),$$

$$\mathrm{NOBM}(S,C_{L,n})=\sum _{A\in S}\mathrm{NOBM}(A,C_{L,n}a)$$

$$\frac{|S|}{\mathrm{NOM}({\mathrm{C}}_{\mathrm{L},\mathrm{n}})}\ge \frac{1}{2(1-a)}.$$

$$\mathrm{mstak}(S,C_{L,n},a)>\frac{1}{2}.$$

Full text

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where $L$ is the length of the block-chain, ${\rm NOM}$ is the number of miners in the block-chain, and ${\rm NOBM}(A)$ is the number of blocks mined by $A$

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

[TABLE]

where ${\rm Tx}(B)$ is the set of transactions of $B$, and

[TABLE]

Definition 1.7

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

[TABLE]

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

[TABLE]

then $B$ is called a PoW block of difficulty $D$ of that system.

Definition 2.2

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m})$ be a block-chain in a public-key cryptography system with a hash function, $L$ a positive integer, and $D=(D_{0},D_{1},D_{2},\cdots,D_{[\frac{m}{L}]})$ a sequence of positive numbers. If

[TABLE]

where $M$ is the maximum hash value, then $C$ is called a PoW block-chain with period $L$ and difficulty vector $D$ 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 $T$ be the time for a set of accounts with total computing power $P$ to find a PoW block of difficulty $D$. Then

[TABLE]

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

Theorem 2.6

Let $L$ be a large integer, $NL\leq m<NL+L$, and $C=(B_{0},B_{1},\cdots,B_{m-1})$ a PoW block-chain with period $L$ and difficulty vector $D=(D_{0},D_{1},D_{2},\cdots,D_{N})$. Let $k$ be a large integer such that $m+k-1<NL+L$. Then the time for a set of accounts with total computing power $P$ to find blocks $B_{m},B_{m+1},\cdots,B_{m+k-1}$ such that $(B_{0},B_{1},\cdots,B_{m+k-1})$ is a PoW block-chain with period $L$ and difficulty vector $D$ is approximately $\frac{k\cdot D_{N}}{P}$ almost surely.

Definition 2.7

Let $L$ be a positive integer. A PoW block-chain system of period $L$ 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 $L$.

Definition 2.8

Let $C=(B_{0},B_{1},\cdots,B_{m})$ be a PoW block-chain with period $L$ and difficulty vector $(D_{0},D_{1},D_{2},\cdots,D_{[\frac{m}{L}]})$. Then we call $\sum_{i=k}^{m}D_{[\frac{i}{L}]}$ the difficulty of the segment $(B_{k},B_{k+1},\cdots,B_{m})$.

Following Satoshi Nakamoto [Na], one can show that in a PoW block-chain system of period $L$ 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 $C=(B_{0},B_{1},B_{2},\cdots,B_{m})$ be a block-chain in a public-key cryptography system with a hash function, and $L$ a positive integer. We set

[TABLE]

Definition 3.2

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m})$ be a block-chain in a public-key cryptography system with a hash function, and $L$ a positive integer. The number of blocks mined by an account $A$ in $C_{L,n}$ is

[TABLE]

Definition 3.3

Let $C$ be a block-chain in a public-key cryptography system with a hash function, and $L$ a positive integer. The number of miners in $C_{L,n}$ is:

[TABLE]

Definition 3.4

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m})$ be a block-chain in a public-key cryptography system with a hash function, $L$ a positive integer, and $a\in[0,1]$. We define the mining-stake of an account $A$ in $C_{L,n}$ with discrimination index $a$ by the formula:

[TABLE]

Definition 3.5

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m})$ be a block-chain in a public-key cryptography system with a hash function, $L$ a positive integer, and $D=(D_{0},D_{1},D_{2},\cdots,D_{[\frac{m}{L}]})$ a sequence of positive numbers. If

[TABLE]

where $M$ is the maximum hash value and $A$ is the miner of $B_{i}$, then $C$ is called a PoM block-chain with period $L$, difficulty vector $D$, and discrimination index $a$.

It is easy to prove the following.

Lemma 3.6

$L$* be a positive integer, $D=(D_{0},D_{1},D_{2},\cdots,D_{N})$ a sequence of positive numbers, and $a\in[0,1]$. Let $NL\leq m<NL+L$, and $C=(B_{0},B_{1},B_{2},\cdots,B_{m-1})$ a PoM block-chain with period $L$, difficulty vector $D$, and discrimination index $a$. Let $T$ be the time for a set $S$ of accounts to find a block $B_{m}$ such that $(B_{0},B_{1},\cdots,B_{m})$ is a PoM block-chain with period $L$, difficulty vector $D$, and discrimination index $a$. Then*

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where

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

Theorem 3.7

Let $L$ be a positive integer, $D=(D_{0},D_{1},D_{2},\cdots,D_{N})$ a sequence of positive numbers, and $a\in[0,1]$. Let $NL\leq m<NL+L$, and $C=(B_{0},B_{1},B_{2},\cdots,B_{m-1})$ a PoM block-chain with period $L$, difficulty vector $D$, and discrimination index $a$. Let $k$ be a large integer such that $m+k-1<NL+L$. Then the time for a set $S$ of accounts to find block $B_{m},B_{m+1},\cdots,B_{m+k-1}$ such that $(B_{0},B_{1},\cdots,B_{m+k-1})$ is a PoM block-chain with period $L$, difficulty vector $D$, and discrimination index $a$ is approximately

[TABLE]

almost surely.

It is easy to prove the following.

Lemma 3.8

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m-1})$ be a PoM block-chain with period $L$, and discrimination index $a\neq 0$. Let $S$ be a set of accounts. Then

[TABLE]

if and only if

[TABLE]

where

[TABLE]

From the above lemma one can infer the following.

Corollary 3.9

Let $C=(B_{0},B_{1},B_{2},\cdots,B_{m-1})$ be a PoM block-chain with period $L$, and discrimination index $a\neq 0$. Let $S$ be a set of accounts such that

[TABLE]

Then

[TABLE]

Definition 3.10

Let $L$ be a positive integer, and $a\in[0,1]$. A PoM block-chain system of period $L$ and discrimination index $a$ 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 $L$ and discrimination index $a$.

Definition 3.11

Let $C=(B_{0},B_{1},\cdots,B_{m})$ be a PoM block-chain with period $L$ and difficulty vector $(D_{0},D_{1},D_{2},\cdots,D_{[\frac{m}{L}]})$. Then we call $\sum_{i=k}^{m}D_{[\frac{i}{L}]}$ the difficulty of the segment $(B_{k},B_{k+1},\cdots,B_{m})$.

Following Satoshi Nakamoto [Na], one can show that in a PoM block-chain system of period $L$ and index $a$ 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].