$3.$ This problem deals with the A$5/1$ cipher. Justify your answers.

a$)$ On average, how often does the X register step?

b$)$ On average, how often does the Y register step?

c$)$ On average, how often does the Z register step?

d$)$ On average, how often do all three registers step?

e$)$ On average, how often do exactly two registers step?

f$)$ On average, how often does exactly one register step?

g$)$ On average, how often does no register step?

$7.$ Implement the RC$4$ algorithm. Suppose the key consists of the following seven bytes: $(0$x$1$A$,0$x$2$B$,0$x$3$C$,0$x$4$D$,0$x$5$E$,0$x$6$F$,0$x$77).$ For each of the following, give S in the form of a $16\backslash$times $16$ array where each entry is in hex.

a$)$ List the permutation S and indices i and j after the initialization phase has completed.

b$)$ List the permutation S and indices i and j after the first $100$ bytes of keystream have been generated.

c$)$ List the permutation S and indices i and j after the first $1000$ bytes of keystream $($that is$,900$ additional keystream bytes after your solution to part b$)$ have been generated.