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**Additional resources for Defending Against Statistical Steganalysis**

**Example text**

The transition matrix for Example 9 is GG, GG GG, GG 1 GG, Gg 1/4 GG, gg 0 P= Gg, Gg 1/16 Gg, gg 0 gg, gg 0 GG, Gg 0 1/2 0 1/4 0 0 GG, gg 0 0 0 1/8 0 0 Gg, Gg 0 1/4 1 1/4 1/4 0 Gg, gg 0 0 0 1/4 1/2 0 gg, gg 0 0 0 . 1/16 1/4 1 Imagine a game in which your fortune is the number of G’s in the state that you are in. This is a fair game. For example, when you are in state Gg,gg your fortune is 1. On the next step it becomes 2 with probability 1/4, 1 with probability 1/2, and 0 with probability 1/4.

B) w = (1/n, · · · , 1/n). (c) From the program Ergodic we obtain 0 0 0 1 4 M = 2 6 36 4 4 1 4 0 4 6 6 2 6 4 0 4 6 3 6 6 4 0 4 4 4 6 6 . 4 0 This is consistent with the conjecture that mij = d(n − d), where d is the clockwise distance from i to j. 11. Yes, the reverse transition matrix is the same matrix. 13. Assume that w is a fixed vector for P. Then wi p∗ij = i i wi wj pji = wi wj pji = wj , i so w is a fixed vector for P*. Thus if w* is the unique fixed vector for P* we must have w = w*.

D) m1,5 = 7 9. (a) if n is odd, P is regular. If n is even, P is ergodic but not regular. (b) w = (1/n, · · · , 1/n). (c) From the program Ergodic we obtain 0 0 0 1 4 M = 2 6 36 4 4 1 4 0 4 6 6 2 6 4 0 4 6 3 6 6 4 0 4 4 4 6 6 . 4 0 This is consistent with the conjecture that mij = d(n − d), where d is the clockwise distance from i to j. 11. Yes, the reverse transition matrix is the same matrix. 13. Assume that w is a fixed vector for P. Then wi p∗ij = i i wi wj pji = wi wj pji = wj , i so w is a fixed vector for P*.