Algorithm 3 Image-encryption algorithm.

Input: Parameters of the hyperchaotic system $\{x_0,y_0,a,b\}$, plaintext image P, the initial values of the intermediate key $Pre$ and k. Output: Ciphertext image C and the secret key sequences $\{X,Y\}$. 1:$[M,N]=size\left(P\right)$;                                                 ▹ Get image size 2:$flag=zeros(M,N)$;                                          ▹ Initialize a zero matrix flag 3:$X=zeros(1,M\ast N)$; 4:$Y=zeros(1,M\ast N)$;                                   ▹ Initialize the secret key sequences $\{X,Y\}$ 5:for $i=1:M$ do 6:   for $j=1:N$ do 7:   Random values $\{x,y\}$ are obtained iteratively using system (1). 8:   $i_{next}=mod(floor(x\ast {10}^6),M)+1$; 9:   $j_{next}=mod(floor(y\ast {10}^6),N)+1$; 10:    while $flag(i_{next},j_{next})==1$ do 11:    Repeat steps 7 to 9; 12:    end while 13:    $C(i_{next},j_{next})=bitxor(mod(P(i,j)+k,256),Pre)$; 14:    $Pre=C(i_{next},j_{next})$; 15:    $flag(i_{next},j_{next})=1$; 16:    Record $\{x,y\}$ to sequences $\{X,Y\}$, respectively. 17:   end for 18:end for