Improve on algorithm P1 in Appendix 9B.

a. Develop an algorithm that requires \(2 n\) multiplications and \(n+1\) additions. Hint: \(x^{i+1}=x^{i} \times x\).

b. Develop an algorithm that requires only \(n+1\) multiplications and \(n+1\) additions. Hint: \(\mathrm{P}(x)=a_{0}+x \times q(x)\), where \(q(x)\) is a polynomial of degree \((n-1)\).

The remaining problems concern the knapsack public-key algorithm described in Appendix J.