# Dynamic Markov Model: Password Guessing Using Probability Adjustment Method
## 4 Experiments
### Effect of Parameter \(\alpha\)  
There exists a vital parameter \(\alpha\) that has a significant and decisive impact on the performance of the dynamic Markov model. This parameter represents the amplitude of each probability adjustment and directly controls the speed at which the probability distribution of passwords tends to be uniform. If \(\alpha\) is too large, most of the n-gram fragments cannot be adjusted, because the probability value is less than \(\alpha\), and the cover rate is greatly reduced (for example, when \(\alpha\) is \(10^{-5}\), the cover rate is less than 20%),making the model useless. On the contrary, if the value of the parameter is small, the effect difference between parameters is not apparent (for example, when \(\alpha\) is set as \(10^{-11}\) and \(10^{-12}\), the cover rate and repetition rate are almost the same), so we conducted experiments on parameter \(\alpha\) from \(10^{-5}\) to \(10^{-11}\), and tried a variety of different values, including \(10^{-5},10^{-6},10^{-7},10^{-8},10^{-9},10^{-10}\), and \(10^{-11}\). First, we used the training set to train the dynamic Markov models with different values of \(\alpha\), and we selected the model order \(n\) as 3. Next, we used these 7 password generative models to generate passwords. In order to evaluate the model comprehensively, we generated \(10^{9}\) passwords and calculate the \(CR\) and \(RR\) values in each scale. The experimental results are shown in Figures 4 and 5.  
The experimental results prove that the scale of parameter \(\alpha\) has a remarkable impact on the performance of the model. For the repetition rate, choosing a smaller \(\alpha\) would get a larger \(RR\) value. When \(\alpha\) is equal to \(10^{-5}\), \(RR\) is always below 0.5. As the parameter decreases, the gap between adjacent parameters becomes smaller. Both parameters \(10^{-11}\) and \(10^{-10}\) have essentially the same effect. As for the cover rate, a smaller \(\alpha\) means a higher cover rate. Analogously, the gap between neighboring parameters also becomes smaller as the parameter decreases. When parameter \(\alpha\) is selected as \(10^{-9}\), \(10^{-10}\), or \(10^{-11}\), they achieve the same and the best result. The best dynamic Markov model in the RockYou dataset is obtained when parameter \(\alpha\) is \(10^{-9}\).  
For the above results, we infer the core reason is that some high-probability passwords share the same n-gram fragment. After the dynamic Markov model randomly generates a high-probability password, it chooses an n-gram fragment to reduce its probability, which inevitably reduces the probability of those passwords that contain this n-gram fragment so they become low-probability passwords. This mechanism makes it difficult for these passwords to be randomly generated afterward, therefore the cover rate decreases. A larger parameter \(\alpha\) helps generate more unique passwords. Because the probability of some important n-gram fragments is reduced too much, some low-probability passwords have more chance to be generated, but these passwords usually have the limit potential for cracking. When parameter \(\alpha\) is selected appropriately, the password probability adjustment variation is relatively low, which can effectively prevent high-probability passwords from being submerged and reduce the repetition rate. This experimental phenomenon also shows us that the adjustment range of the password probability should be limited at a relatively flat level, so it is reasonable to randomly select one n-gram fragment instead of all n-gram fragments to adjust.  
Figure 4: Repetition rate for different \(\alpha\) values in 7 dynamic Markov models.