Introduction
Our contributions
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We first look into the results by investigating the cryptographic properties of some standard S-boxes.
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Secondly, the TOPSIS method based on the IVPF set is applied to analyze the results to reach the final decision.
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We ranked best nonlinear confusion component of block ciphers which can be utilized in any modern information confidentiality mechanism.
Some basic preliminaries
Limitations of existing score and accuracy function
Improved score function
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If Q (\({I}_{P1}\)) < Q (\({I}_{P2}\)), then \({I}_{P1}\) <\({I}_{P2}\).
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If Q (\({I}_{P1}\)) > Q (\({I}_{P2}\)), then \({I}_{P1}\) >\({I}_{P2}\).
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If Q (\({I}_{P1}\)) = Q (\({I}_{P2}\)), then \({I}_{P1}\) \(\sim \) \({I}_{P2}\).
Multi-criteria decision-making
Cryptographic properties of S-boxes
Nonlinearity
Strict avalanche criterion (SAC)
Bit-independent criterion
Sum of square and absolute indicator
Algebraic degree
Algebraic immunity
Transparency order
Robustness to differential cryptanalysis
Signal to noise ratio
Confusion coefficient variance
Selection of optimum nonlinear confusion component based on interval-valued Pythagorean fuzzy set
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Step 1
\(\left[\begin{array}{cccccc}(\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.8}\right],\left[\mathrm{0.3,0.5}\right])& (\left[\mathrm{0.6,0.8}\right],\left[\mathrm{0.3,0.5}\right])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[0.2,0.4])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.2,0.4}\right],[\mathrm{0.7,0.8}])& (\left[0.4,0.6\right],[\mathrm{0.5,0.6}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.2,0.5}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.5,0.6}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[0.3,0.5])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[0.\mathrm{6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& \left(\left[\mathrm{0.5,0.6}\right],\left[\mathrm{0.4,0.5}\right]\right)& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0}.5])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.8}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.6}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])& \left(\left[\mathrm{0.7,0.8}\right],\left[\mathrm{0.2,0.4}\right]\right)& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\end{array}\right]\) |
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Step 2
\(\left[\begin{array}{cccccc}(\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.8}\right],\left[\mathrm{0.3,0.5}\right])& (\left[\mathrm{0.6,0.8}\right],\left[\mathrm{0.3,0.5}\right])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[0.\mathrm{7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])\\ (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])\\ (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.7,0.8}\right],[0.\mathrm{2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])\\ (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.2,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.4}])& (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.6,0.7}\right],[\mathrm{0.3,0.5}])& \left(\left[\mathrm{0.5,0.6}\right],\left[\mathrm{0.4,0.5}\right]\right)& (\left[0.6,0.7\right],[\mathrm{0.3,0.5}])& (\left[\mathrm{0.7,0.8}\right],[\mathrm{0.2,0.4}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.5}])& (\left[\mathrm{0.6,0.8}\right],[\mathrm{0.3,0.5}])\\ (\left[\mathrm{0.4,0.6}\right],[\mathrm{0.5,0.6}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.6}])& (\left[\mathrm{0.5,0.6}\right],[\mathrm{0.4,0.6}])& \left(\left[\mathrm{0.2,0.4}\right],\left[\mathrm{0.7,0.8}\right]\right)& (\left[\mathrm{0.3,0.5}\right],[\mathrm{0.6,0.7}])\end{array}\right]\) |
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Step 3
\( R = \left[\begin{array}{cccccc} 0.7266& 0.7266& 0.7266& 0.1689& 0.1689& -0.0796\\ 0.7266& 0.7266& 0.7266& 0.1689& 0.4948& 0.4948\\ 0.7266& 0.4163& 0.7266& 0.5893& 0.4163& 0.7266\\ 0.7266& 0.4163& 0.7266& 0.7266& 0.4163& 0.7266\\ -0.4163& -0.4163& -0.4163& 0.7266& 0.0796& 0.0796\\ -0.4163& -0.4163& -0.4651& 0.4163& 0& 0.2343\\ 0.4163& 0.4163& 0.4163& 0.4163& 0.4163& 0.4163\\ 0.4163& 0.4163& 0.4163& 0.4163& 0.4163& 0.4163\\ 0.4163& 0.4163& 0.4163& 0.0796& 0.1689& 0.4163\\ 0.4163& 0.4163& 0.4163& 0.4163& 0.4163& 0.4163\\ 0.1689& 0.1689& 0.0796& -0.0796& 0.4977& 0.4163\\ 0.4163& 0.1689& 0.4163& 0.7266& 0.1689& 0.4948\\ -0.0796& -0.4163& -0.3245& 0.0796& -0.7266& -0.4163\end{array}\right] \) |
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Step 4
\({S}_{1}\) | \({S}_{2}\) | \({S}_{3}\) | \({S}_{4}\) | \({S}_{5}\) | \({S}_{6}\) | |
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\(D({S}_{n}, {s}^{+})\) | 0.1610 | 0.1932 | 0.1823 | 0.1604 | 0.2074 | 0.1615 |
\(D({S}_{n}, {s}^{-})\) | 0.0965 | 0.0714 | 0.0967 | 0.0794 | 0.0533 | 0.081 |
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Step 5
\({S}_{1}\) | \({S}_{2}\) | \({S}_{3}\) | \({S}_{4}\) | \({S}_{5}\) | \({S}_{6}\) | |
---|---|---|---|---|---|---|
\({\mathrm{RCC}}_{\mathrm{i}}\) | 0.3747 | 0.2699 | 0.3467 | 0.3312 | 0.2045 | 0.3358 |
Rank | 1 | 5 | 2 | 4 | 6 | 3 |
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Step 6