### CPSC 430 Spring 2003Homework 1

Here is a clarification of the homework. Refer to the copy of p. 37 (from 2nd edition of Stallings) that was distributed in class.

As we discussed in class, using the Hill cipher C = KP, where C (the ciphertext ) and P (the plaintext) are column vectors of length m and K (the key) is an m x m matrix. This process is used repeatedly with the same K to produces blocks of length m of C from blocks of length m of P. The goal of the work on p. 37 is to find K where m=2, P="friday", and C="PQCFKY".

For ease of typing I will write matrices in row order, so if M is a 2x2 matrix I would write it as (m11 m12)(m21 m22).

1. First verify the calculations at the bottom of the page, which are all mod 26. Show all work.
1. Find the inverse of X where X = (5 17)(8 3) Do not just verify the inverse that is given.
2. Find the product of (9 1)(2 15) and (15 16)(2 5).

2. Page 37 concludes with the statement that K = (7 19)(8 3). Verify or disprove this.

3. How can the true K be derived from the K that is given? What is this relationship called?

4. A single change in notation, applied in the appropriate places, will make the presentation on p. 37 correct (although it will still be somewhat different from that in the 3rd edition). What is this change, and where does it need to be applied? What property of matrix multiplication are you applying?