# Design of a Lightweight Cryptographic Scheme for Resource-Constrained Internet of Things Devices
## 3 Block Cipher SLA
### Permutation Layer
The bias (aka, correlation or imbalance) of a Boolean function \(f\) with \(n-\) variables is defined as

\[\varepsilon(f)=\sum_{x\in\mathbb{F}_{2}^{n}}(-1)^{f(x)}=2^{n}-2\square\sqcup(f) \tag{12}\]In other words,

\[\Pr_{\mathrm{X}}[\mathrm{f}(\mathrm{X})=1]=\frac{\exists\sqcup(\mathrm{f})}{2^{n}} =\frac{1}{2}(1-\frac{\varepsilon(\mathrm{f})}{2^{n}}) \tag{13}\]
2. Differential Cryptanalysis

Differential cryptanalysis [26; 27] is the main form of attack on symmetric block ciphers. Differential paths are formed by considering the differences between inputs and outputs with a high probability for each round. The S-box where the differences between the inputs or the outputs are nonzero is called the active S-box. SLA presents a minimal bound on the number of so-called \"active\" S-boxes defined in a differential characteristic to assess the hardness of the SLA scheme differential cryptanalysis. Appendix E shows the linear/differential relations SLA scheme S-box. Table 5 shows the differential characteristics SLA scheme, and Table 6 shows the minimal number of active S-boxes in the differential characteristic. **Theorem 2**. : _For sixteen rounds of SLA, it features 48 active S-boxes, and a maximum probabilistic differential characteristic for the sixteen rounds is \(2^{-96}\)._

Theorem 2 is formally proved in Appendix D.

Autocorrelation. The autocorrelation transform taken concerning to a \(\in\mathbb{F}_{2}^{n}\), of Boolean function \(\mathrm{f}\) with \(n-\) variable is denoted by \(\hat{\mathrm{r}}_{\mathrm{f}}(\mathrm{a})\) and defined as (14):

\[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]0

Differential uniformity. Given differential uniformity for any vectorial Boolean function \(\mathbb{F}\in\mathcal{F}\mathrm{n},\mathrm{m}\) for any \(\mathrm{a}\in\mathbb{F}_{2}^{n}\) into \(\mathrm{b}\in\mathbb{F}_{2}^{m}\), we defined in (15):

\[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]1

Then, the differential spectrum of \(\mathbb{F}\) is the multi-set \(\{\delta\{\mathrm{a},\mathrm{b}\};\mathrm{a}\in\mathbb{F}_{2}^{n}\backslash \{0\};\mathrm{b}\in\mathbb{F}_{2}^{m}\}\), and its maximum (16):

\[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]2
3. Algebraic degree. An algebraic degree of a vectorial Boolean function \(\mathbb{F}\in\mathcal{F}\mathrm{n},\mathrm{m}\) is the number of variables in the longest item of its ANF, denoted by \(\mathcal{D}\ \varepsilon\mathcal{G}(\mathbb{F})\)[28]. Nonlinearity. The nonlinearity of Boolean function \(\mathrm{f}\in\mathbb{F}_{\mathrm{m}}\) is defined as the Hamming distance between \(\mathrm{f}\) and the set \(\mathcal{A}_{\mathrm{n}}\) of all affine functions (or linear) [29] in (17)

\[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]3
4. Nonlinearity. The nonlinearity of the vectorial Boolean function \(\mathbb{F}\in\mathcal{F}_{\mathrm{n},\mathrm{m}}\) is the minimal of all component functions of \(\mathbb{F}\)[28], and the Walsh spectrum is used to calculate it in the manner outlined below in (18): \[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]4 (18)
5. Correlation Immunity. The correlation immunity of a Boolean function \(\mathrm{f}\in\mathbb{F}_{\mathrm{n}}\) is defined as a measurement of how uncorrelated outputs are with a certain subset of its inputs. If \(\mathrm{f}\) is balanced, and \(\mathrm{t}-\mathcal{CI}\), then so-called \(\mathrm{t}-\) resilient [30]. This criterion is from the Walsh spectrum in the manner outlined in (19): \[[k_{63}k_{62}k_{61}k_{60}k_{59}]=[k_{63}k_{62}k_{61}k_{60}k_{59}]\bigoplus\ \ RC^{i} \tag{5}\]5 (19)Balancedness. The vectorial Boolean function \(\mathbb{F}\in\mathcal{F}_{\text{n,m}}\) is balanced if its outputs are distributed uniformly over \(\mathbb{F}_{2}^{\text{m}}\).