# Homework #4

This corrects typos in problem 4, where “conditional probabilities” should be “conditional entropies”, and problem 5 item (b), in which there should be set braces around the formula to the right of the equal sign.

Due: March 5, 2021
Points: 100

## Questions

1. (25 points) Suppose composite machine catdog (see Section 9.4.1) receives no HIGH inputs. Does it emit the same value from the left and the right? If so, prove it; if not, give a counterexample.

2. (25 points) Consider again the algorithm in Figure 9–7. The power used is another side channel for most instantiations of this algorithm. Explain how this side channel works. How might you add sufficient noise to it to render it unusable?

3. (15 points) Prove that for n = 2, H(X) is maximal when p1 = p2 = 1/2.

4. (15 points) Consider the statement
```
if (x = 1) and (y = 1) then z := 1
```
where x and y can each be 0 or 1, with both equally likely and z is initially 0. Compute the conditional entropies H(x | z′) and H(y | z′).

5. (20 points) Let L = (SL, ≤L) be a lattice. Define:
1. SIL = { [a, b] | a, bSaL b }
2. IL = { ([a1, b1], [a2, b2]) | a1L a2b1L b2 }
3. lubIL([a1, b1], [a2, b2]) = (lubL(a1, a2), lubL(b1, b2))
4. glbIL([a1, b1], [a2, b2]) = (glbL(a1, a2), glbL(b1, b2))
Prove that the structure IL = (SIL, ≤IL) is a lattice.

 Matt Bishop Office: 2209 Watershed Sciences Phone: +1 (530) 752-8060 Email: mabishop@ucdavis.edu
ECS 235B, Foundations of Computer and Information Security
Version of February 23, 2021 at 12:21AM