Quantified Analysis of Security Issues and Its Mitigation in Blockchain Using Game Theory

Abstract

Storing data in the Blockchain is indeed one of the good security measures for data. However, blockchain itself could be under different types of security threats. The mining of the block into the longest chain is a constrained task. Typically, the nodes having high investments are selected as potential miners in the blockchain. A miner or a pool of miners is assigned for this mining job. The challenge lies in working with the honest miners against the continuous negative influence of dishonest miners. There have been considerable efforts in the existing literature that tries to overcome such security threats. Game theory is used and incorporated towards this by many researchers. This manuscript aims to analyze different security threats of blockchain mining and the possible approaches that have claimed to overcome these. We also analyzed and correlated some of the selected well-cited solution approaches that uses game theory and presented a comparative performance analysis among those.

Annexure 1

1.1 Formalism in the Model Proposed by B. Johnson, A. Laszka et al. [4]

The payoff is defined for B and S of DDoS attack and computation is

Table 3. The pay-off matrix of mining of B and S

Table 4. Payoff Matrix for B and S with Imperfect DDoS and Linear Costs

1.2 Formalism in the Model Proposed by B. Johnson, A. Laszka et al. [6]

The Short term policy The calculated utility function of B and S are $U_{B}^{\left( k \right)}$ and $U_{S}^{\left( k \right)}$ respectively in kth iteration are given by the equn

$$ U_{B}^{\left( k \right)} { } = \frac{{S_{B}^{\left( k \right)} \cdot \left( {1 - a_{S}^{\left( k \right)} } \right) }}{{1 - S_{B}^{\left( k \right)} \cdot a_{S}^{\left( k \right)} - S_{S}^{\left( k \right)} \cdot a_{B}^{\left( k \right)} }}{ } - {\text{ C}} \cdot a_{B}^{\left( k \right)} $$

(1a)

$$ U_{S}^{\left( k \right)} { } = \frac{{S_{S}^{\left( k \right)} \cdot \left( {1 - a_{B}^{\left( k \right)} } \right) }}{{1 - S_{S}^{\left( k \right)} \cdot a_{B}^{\left( k \right)} - S_{B}^{\left( k \right)} \cdot a_{S}^{\left( k \right)} }}{ } - {\text{ C}} \cdot a_{S}^{\left( k \right)} $$

(1b)

The Long term policy The calculated size of B and S are $S_{B}^{{\left( {k + 1} \right)}}$ and $S_{S}^{{\left( {k + 1} \right)}}$ respectively in k_th iteration are given by the equⁿ

• a) Migration of miner into B pool

  $$ S_{B}^{{\left( {k + 1} \right)}} { } = { }S_{B}^{\left( k \right)} + A_{B} \cdot [\left( {1 - { }S_{B}^{\left( k \right)} } \right) \cdot {\text{ M }} + S_{S}^{\left( k \right)} \cdot a_{B}^{\left( k \right)} \left( {1 - {\text{M}}} \right) $$

  (2a)

• b) Migration of miner out of B pool

  $$ S_{B}^{{\left( {k + 1} \right)}} { } = { }S_{B}^{\left( k \right)} - { }S_{B}^{\left( k \right)} \cdot { }\left( {1 - { }A_{B} } \right) \cdot \left[ {{\text{ M }} + \cdot a_{S}^{\left( k \right)} \left( {1 - {\text{M}}} \right)} \right] $$

  (2b)

• c) Migration of miner into S pool

  $$ S_{S}^{{\left( {k + 1} \right)}} { } = { }S_{S}^{\left( k \right)} + A_{S} \cdot [\left( {1 - { }S_{S}^{\left( k \right)} } \right) \cdot {\text{ M }} + S_{B}^{\left( k \right)} \cdot a_{S}^{\left( k \right)} \left( {1 - {\text{M}}} \right) $$

  (2c)

• d) Migration of miner out of S pool

  $$ S_{S}^{{\left( {k + 1} \right)}} { } = { }S_{S}^{\left( k \right)} - { }S_{S}^{\left( k \right)} \cdot { }\left( {1 - { }A_{S} } \right) \cdot \left[ {{\text{ M }} + \cdot a_{B}^{\left( k \right)} \left( {1 - {\text{M}}} \right)} \right] $$

  (2d)

In case of peaceful equilibrium where ($a_{S, }$ $a_{B}$) = (0,0)

$$ {\text{C }} \ge { }\frac{{A_{B} A_{S} }}{{{\text{Min}}\left( {{\text{M}},1 - A_{S},1 - A_{B} } \right)}}. $$

(3a)

In case of one side attack equilibrium where ($a_{S, }$ $a_{B}$) = (0,1)

$$ {\text{C}} \le \frac{{A_{B} A_{S} }}{{\left( {1 - A_{S} } \right)^{2} }}{ } \cdot {\text{ Min}}\left( {{\text{M}},1 - A_{S} } \right) $$

(3b)

$$ {\text{C}} \le \frac{{A_{B} A_{S} }}{{\left( {1 - A_{B} } \right)^{2} }}{ } \cdot {\text{ Min}}\left( {{\text{M}},1 - A_{B} } \right) $$

(3c)