Image Encryption Using the Chaotic Josephus Matrix

Algorithm 1

The generation of a Josephus permutation sequence.

Input: 𝑡, the initial total number of persons in a circle

     𝑠, the starting position in the circle

     𝑛, the counting period

Output:  𝑞_𝜋, the Josephus permutation sequence according to parameter set (𝑡,𝑠,𝑛)

count = 0; done = 0; pos = 𝑠; label = zeros(1, 𝑡); 𝑞_𝜋=[]; % initial settings

while (∼done) % main loop

   todo = label(pos);

   if (todo == 0) % if this person has not been taken out

      count = count + 1;

      if (count == 𝑛) % if this is the 𝑛th person

         𝑞_𝜋(end + 1) = pos; count = 0; label(pos) = 1;

         if (length(𝑞_𝜋) == 𝑡)

            done = 1;

         end

      end

   end

   pos = pos + 1;

   if (pos > 𝑡)

      pos = 1;

   end

end