# Design of a Lightweight Cryptographic Scheme for Resource-Constrained Internet of Things Devices
## 3 Block Cipher SLA
### Permutation Layer
** In cryptanalysis, the Matsui's piling-up lemma is a principle used in linear cryptanalysis to construct linear relations to the action of block ciphers:

**Lemma (Pilling-up lemma)**

Let \(\epsilon_{i_{1}i_{2-\hat{\mathbf{x}}_{\text{k}}}}\) denote the bias of the random variable \(\text{X}_{i_{1}}\ \oplus\ldots\text{X}_{\text{k}}\). Then

\[\epsilon_{i_{1}i_{2-\hat{\mathbf{x}}_{\text{k}}}}=2^{\text{k}-1}\prod_{j=1}^{ \text{k}}\epsilon_{i_{j}}.\]

Let p be the probability of a linear characteristic. The correlation of the linear characteristic over S-box is given by \(\text{q}=(2\text{P}-1)^{2}\)[45]. From the input-output correlation of S-box, it is straightforward that any linear characteristic over S-box has a correlation of at most \(\left(2\times\frac{4}{16}-1\right)^{2}=2^{-2}\). The best way to resist against linear cryptanalysis is to increase the number of active S-boxes in the cipher scheme. The maximum probabilistic bias linear characteristic of SLA S-box equal to \(2^{-2}\) can be calculated similarly [18]. Therefore, the maximum probabilistic bias linear characteristic is estimated for three rounds as

\[2^{8}\times\left(2^{-2}\right)^{9}=2^{-10}\]

When applied to sixteen rounds, the maximum probabilistic bias linear characteristic is estimated as

\[\epsilon=2^{5}\times\left(2^{-10}\right)^{6}=2^{-55}\]

To determine the hardness of linear attack, compute the number of known plaintext as follows:

\[\text{N}_{\text{L}}=\frac{1}{\epsilon^{2}}\]

The number of known plaintext is specified for sixteen rounds of the SLA scheme as follows:

\[\text{N}_{\text{L}}=\frac{1}{\epsilon^{2}}=\frac{1}{\left(2^{-55}\right)^{2}}= 2^{110}\]

The available limit of known plaintext is \(2^{64}\). This number is lower than the desired number of known plaintext i.e., \(2^{110}\). So, the full number of rounds of the SLA scheme demonstrate solid resilience to linear attacks. ## References

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