$1.$ A bank uses $($K$,$E$,$D$),$ a one$-$time pad symmetric encryption scheme with key space K$=\{0,1\}$ for their cryptographic needs. Observe that the encryption and decryption keys are identical as it is a symmetric OTP crypto system. The bank wishes to split the decryption key k in $\{0,1\}$ into two pieces p$1$ and p$2$ so that both are needed for decryption. The idea is that piece p$1$ can be given to one executive and p$2$ to another so that both must contribute their pieces for decryption to proceed. Following this idea, the bank generates a random k$1$ in $\{0,1\}$ and sets k$1$ k $$ k$1.$ Note that k$1$ k$1=$ k$.$ The bank can give k$1$ to one executive and k$1$ to another executive. Both must be present for decryption to proceed since, by itself, each piece contains no information about the secret key k $($note that each piece is a one$-$time pad encryption of k$,$ interesting!$).$

Now, suppose the bank wants to split k into three pieces p$1,$ p$2,$ and p$3$ so that any two of the pieces enable decryption using k$.$ This ensures that even if one executive is out sick, decryption can still succeed. To do so the bank generates two random pairs $($k$1,$k$1)$ and $($k$2,$k$2)$ as in the previous paragraph so that k$1$k$1=$k$2$k$2=$k$.$ How should the bank assign pieces so that any two pieces enable decryption using k$,$ but no single piece can decrypt?