$11\mathrm{.}7$ In Figure $11\mathrm{.}12,$ it is assumed that an array of $8064-$bit words is available to store the

values of $W_t,$ so that they can be precomputed at the beginning of the processing of

a block. Now assume that space is at a premium. As an alternative, consider the use

of a $16-$word circular buffer that is initially loaded with $W_0$ through $W_{15}.$ Design an

algorithm that, for each step $t,$ computes the required input value $W_t.$

$11\mathrm{.}8$ For SHA$-512,$ show the equations for the values of $W_{16},W_{18},W_{23},$ and $W_{31}.$