# A Model and Quantitative Framework for Evaluating Iterative Steganography
## 3 The Proposed Method  
### IIF (Incremental Information Function)  
The definition of the Incremental Information Function (IIF) is inspired by the concept of the Bit Error Rate (BER), which measures the ratio of incorrectly encoded bits to the total number of encoded bits in a message [19]. A lower BER indicates less information loss in the encoded message and higher robustness. In contrast, for the IIF, a higher function value signifies a greater amount of information decoded from the steganogram. The details of the relationship between the IIF and the BER are described in Section 3.5.  
First, we define the auxiliary function \(IIF_{(orig,dec)}\) (refer to Equation (15)), which assigns to a given steganogram, \(s^{i}\), and message, \(m\), the relative number of occurrences of decoded bits with the value \(dec\) that had the value \(orig\) in the original message, \(m\).  
\[IIF_{(orig,dec)}:\mathcal{S}\times\mathcal{M}\rightarrow\mathfrak{R}^{+} \tag{15}\]  
\[\begin{array}{c}\bigwedge_{\begin{subarray}{c}orig\in\{0,1\}\\ dec\in\{0,1\}\end{subarray}}\bigwedge_{i\in\{1...Z\}}IIF_{(orig,dec)}\big{(}s^ {i},\ m\big{)}=\sum\limits_{k=1}^{L}1_{\{m_{k}=orig\wedge f^{-1}(s^{i})_{k}= dec\}}/\,m(orig)\\ \sum\limits_{i\in\{1...Z\}}\bigwedge_{\begin{subarray}{c}orig\in\{0,1\} \\ dec\in\{0,1\}\end{subarray}}\sum\limits_{dec\in\{0,1\}}IIF_{(orig,dec)}\big{(}s ^{i},\ m\big{)}=1\end{array}\]  
whereorig_--the value of a bit in the original message \(m\); _dec_--the value of a bit in the decoded message \(m^{-1}\); _m_(_orig_)--the number of _orig_-bites in the message \(m\) (either \(m(0)\) or \(m(1)\)); _f_-1--the decoding function; _m_--the message encoded by function \(f\) into the cover \(c^{i}\). The \(IIF\) function (refer to Equation (16)) is defined as a weighted sum of the functions \(IIF_{(0,0)}\) and \(IIF_{(1,1)}\), where the weights are proportional to the number of bits with values \(0\) and \(1\) in the original message, \(m\). This represents the proportion of correctly decoded bits in the message, \(m^{-1}\). \[IIF:\mathcal{S}\times\mathcal{M}\rightarrow\langle 0,1\rangle\] (16) \[\bigwedge_{i\in\{1..Z\}}IIF\Big{(}s^{i},\ m\Big{)}=IIF_{(0,0)}\Big{(}s^{i},\ m \Big{)}\cdot\frac{m(0)}{L}+IIF_{(1,1)}\Big{(}s^{i},\ m\Big{)}\cdot\frac{m(1)}{ L}=\sum_{k=1}^{L}1_{\{m_{k}=f^{-1}(s^{i})_{k}\}}/L\] where _m_--the message decoded by the function \(f^{-1}\) from the steganogram, \(s^{i}\); _m_(0)--the number of \(0\)'s in the message, \(m\); _m_(1)--the number of \(1\)'s in the message, \(m\). In Equation (16), which defines the IIF, it is also possible to apply different weights, particularly in scenarios where the steganographic method processes \(0\)'s and \(1\)'s differently. For instance, in a hypothetical case where \(0\)'s are significantly less important than \(1\)'s, the weight for \(IIF_{(0,0)}\) might not be proportional to \(m(0)\) but could instead be, for example, one-tenth of the weight for the \(IIF_{(1,1)}\). Additionally, in some specific steganographic techniques, the values of the \(IIF_{(0,1)}\) and the \(IIF_{(1,0)}\) can be incorporated into the definition of the \(IIF\) function as \"penalties\" for errors in training the decoding algorithm. It should be noted that the \(IIF_{(0,1)}=1-IIF_{(0,0)}\) and \(IIF_{(1,0)}=1-IIF_{(1,1)}\).