## 4 Theoretical Aspects of the Lossless Adaptive Binomial Data Compression Method
Therefore, in such a case, the compression of the equilibrium combination should be eliminated by transmitting it in its original form, which increases its average transmission compression speed. At the same time, \(q\) value information can be used \(k\) to improve the noise immunity of the transmitted equilibrium combination. In this case, comparing \(k\) with the number of units in the transmitted equilibrium combination determines its correctness. Table 4 shows examples of compression ratios \(K\) for various \(k\). The coefficient \(k\) is 1, 2, 16, 32, and 64. The presence of information about the number of units \(k\), transmitted separately from the compressed binary sequence, allows it to be used as a key, without which it is impossible to restore the original sequence. Therefore, it can be used to protect a compressed sequence. The key \(k\) can, in principle, be obtained by iterating its values, changing them from 0 to \(n\). However, this overkill still requires the expenditure of computing power and the corresponding program. Therefore, such a search can be used to protect confidential information that is not of particular importance or is rapidly losing its value. However, embedding this compression method into an information protection system from unauthorized access can significantly increase its resistance since the general enumeration of the protection system keys becomes more complicated. For each possible key from \(k\) of the compression system, the possible keys of the security system are sorted out, and only after that, in the absence of decryption, does the transition to a new possible key from \(k\) occur. In the worst case, the result is manifested in the need to enumerate the product of all possible keys \(k\) by the number of possible keys of the protected system. The use of binomial adaptive compression makes it possible to protect particular sequences from opening, and not all, as with conventional compression, which makes it possible to find the optimal \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\mathbf{n}\) & \(\mathbf{32}\) & \(\mathbf{64}\) & \(\mathbf{128}\) & \(\mathbf{256}\) & \(\mathbf{512}\) & \(\mathbf{1024}\) \\ \hline \hline \multirow{2}{*}{\(k=1\)} & \(k\) & \(\mathbf{1}\) & \(\mathbf{1}\) & \(\mathbf{1}\) & \(\mathbf{1}\) & \(\mathbf{1}\) & \(\mathbf{1}\) \\  & \(K\) & 3.20 & 5.33 & 9.14 & 16.00 & 28.44 & 51.20 \\ \hline \multirow{2}{*}{\(k=2\)} & \(k\) & \(\mathbf{2}\) & \(\mathbf{2}\) & \(\mathbf{2}\) & \(\mathbf{2}\) & \(\mathbf{2}\) & \(\mathbf{2}\) \\  & \(K\) & 2.29 & 3.76 & 6.40 & 11.13 & 19.69 & 35.31 \\ \hline \multirow{2}{*}{\(k=4\)} & \(k\) & \(\mathbf{4}\) & \(\mathbf{4}\) & \(\mathbf{4}\) & \(\mathbf{4}\) & \(\mathbf{4}\) & \(\mathbf{4}\) \\  & \(K\) & 1.58 & 2.53 & 4.21 & 7.23 & 12.67 & 22.55 \\ \hline \multirow{2}{*}{\(k=8\)} & \(k\) & \(\mathbf{8}\) & \(\mathbf{8}\) & \(\mathbf{8}\) & \(\mathbf{8}\) & \(\mathbf{8}\) & \(\mathbf{8}\) \\  & \(K\) & 1.12 & 1.68 & 2.70 & 4.52 & 7.80 & 13.71 \\ \hline \multirow{2}{*}{\(k=16\)} & \(k\) & \(\mathbf{16}\) & \(\mathbf{16}\) & \(\mathbf{16}\) & \(\mathbf{16}\) & \(\mathbf{16}\) & \(\mathbf{16}\) \\  & \(K\) & 0.93 & 1.16 & 1.74 & 2.81 & 4.72 & 8.15 \\ \hline \end{tabular}
\end{table} 
Table 4: The compression ratios \(K\) for various \(k\). ratio of protected and unprotected sequences from the point of view of the compression rate criterion.