Cryptology - I: Exercise 2 - Statistics & Distributions

Instructors: R.E. Newman-Wolfe and M.S. Schmalz

In this class, we will use statistics as a key tool in cryptanalysis. A summary of statistical concepts covered over the past week can be found in Review of Basic Statistics.

In Class #5 (Wed 4 Sep 96), we concluded our brief review of basic statistics. Here follow a few exercises that we strongly recommend you work through. Answers will be posted to this Web page in 5-7 days, and problems like this could appear on a homework assignment or exam.

Problem 1. Moments and histograms.

Let image a = (1,2,4,5,3,4,5,2,3,3,4,3,2,4,4,5). Compute the following:
1. The histogram h(a);
2. The cumulative histogram c(h);
3. (a) mean(a), (b) standard-dev(a),
   (c) mode(a), (d) median(a);
4. Central moments m₁ through m₄ of h; and
5. Skewness and kurtosis of h.

Problem 2. Do Problem 1 for b = (2,3,3,3,4,3,2,4,4,4,5,3,5,2,1,5). (Note: This is not busy work, as we will use the result in Problem 3).

Problem 3. Recall the results of Problems 1 and 2, above.

Compute the following tests:
1. t-test of significant (or insignificant) difference between mean(a) and mean(b) at confidence k=0.05;
2. Pearson product-moment correlation of a and b.

Problem 4. Write programs in C or C++ to compute the Pearson product-moment correlation and the t-test. You will need these later on when we do cryptanalysis of substitutions and transpositions.