**Homework for Mathematics of Information Security **

M360: Fall 2004

** HW 4: due Friday 9/17**

READ: Sections 2.3, B3 Ex.4 (on Friday we will learn how to decrypt the Vigenere cipher on the computer).

BOOK PROBLEMS: Section 2.13 #16, 3.11 #3a,#4, 3.12 #5.

OTHER PROBLEMS:

A. Find two numbers n and n+35 which have no primes between them.

B. Show that 207=x^2+y^2+z^2 has no solutions if x,y,z are integers.
Hint: look mod 8.

C. Multiple encryption:
Pick two numbers m and n.

What is the period if you encrypt a message first with a Vigenere cipher
of period m and then encrypt the ciphertext with a Vigenere cipher of period n?

Your answer should depend on m and n.

D. For an alphabet of N=27 letters, find the number of shift ciphers, affine ciphers, substitution ciphers,
and vigenere ciphers with period 20.

Do the same for N=30.

E. What is the dot product of (1,0,0,....0) with A0?

What is the dot product of (1,1,1,....1) with A0? What do these numbers represent?

Bonus:

1. Find formulae for problem D in the case of an arbitrary number N of letters.

2. Using data on page 21, find which vector Ai is closest to the vector A0 (other than A0 itself).
What is the angle between them?