1. Solving equations and systems of equations:
1a. Use Newton's method to design an algorithm for computing acrsine, i.e., for solving the equation sin(x) = a for an unknown x.
1b. Use the algorithm for computing 1/b that we had in class (and that is implemented in the computers) to perform the first two steps of computing the ratio 1 / 1.01.
1c. Use Newton's method to solve the following system of non-linear equations:
2a. Use numerical differentiation to compute the derivatives of the function V(x1, x2) at the midpoint x1 = 10, x2 = 20; compare with the actual derivative.
2b. Use linearization technique and your estimate for the derivative to estimate the range of the variance.
2c. Use naive interval computations to estimate this range.
2d. Use mean value form to estimate this range.
3a. Use the Lagrange multiplier technique for find the general formula for the values Δ1 and Δ2 for which the inaccuracy a1 * Δ1 + a2 * Δ2 of estimating a statistical characteristic C is the smallest under the condition that the cell contains k records, i.e., that ρ(x) * 2n * Δ1 * Δ2 = k.
3b. Apply your formula to the example when k = 10 and a1 = a2 = 1. What will then the resulting inaccuracy be?
4a. Explain what is k-anonymity, and why it is important. If k increases, will we get more or less privacy protection? Explain your answer.
4b. Explain what is l-diversity, and why it is important. If l increases, will we get more or less privacy protection? Explain your answer.
4c. Explain what is differential privacy.
5a. Use this algorithm to eliminate outliers from the following database:
5b. Explain how outlier elimination is used in computer security.
6a. In class, we derived a general formula for the optimal sizes Δi of a privacy-enhancing box for which the inaccuracy in the resulting estimation of a statistical characteristic C is the smallest possible under the condition of k-anonymity: Δi = c / ai, where ai is the absolute value of the partial derivative of C with respect to the i-th variable xi, and c = (1/2) * √(k * a1 * a2 * ...) / ρ(x) , where ρ(x) is the data density, i.e., number of record per unit volume.
Use the general formula to find the sizes of the cell that provides the smallest possible inaccuracy in computing the covariance between age and blood pressure for adults. Assume that:
6b. What if, in addition to k-anonymity, we also require l-diversity, with l = 10, and the thresholds ε1 = 0.001 and ε2 = 0.0001?
7a. What percentage of privacy do we lose if someone detects the second digit x in the blood pressure 1xy? the third digit y? Assume that the blood pressure is between 100 and 200.
7b. Based on the computations from Part a, it may look like the last decimal digit -- the remainder of dividing a number by 10 -- is irrelevant, and should not be protected much. Explain that it is not so: what if someone knows the remainder by 10 and the remainder in the hex system (when divided by 16), and the blood pressure is between 110 and 180 -- can we then uniquely determine the blood pressure? Explain your answer.
8a. Use the efficient raising-to-the-power algorithm to compute 521 mod 13. Where is this algorithm used in RSA coding?
8b. Use Euclid's algorithm to compute the greatest common divisor gcd(13, 21). Then find a number d for which d * 13 mod 21 = 1. Where is this used in RSA coding?
9a. Briefly describe your project for this class.
9b. Briefly describe someone else's project for this class.