# Cryptography: Basic Concepts in Number Theory and Finite Fields

All answers must be in complete sentences or, if it is a problem, work must be shown.
1.Briefly describe the following: 1.group
2.ring
3.field

2.For the group of all permutations of n distinct symbols,
1. what is the number of elements in ?

2. show that is not abeian for n > 2.

3. For each of the following equations, find an integer x that satisfies the equation.
1. 5x = 4 (mod 3)

2. 7x = 6 (mod 5)

3. 9x = 8 (mod 7)

4. What is the smallest positive integer that has exactly k divisors, for ?

5.
Find the multiplicative inverse of each nonzero element in .

6.
a.) Determine gcd (24140, 16762). b.) Determine gcd (4655, 12075)

7.
Develop a set of tables similar to Table 4.5 for GF(5).

8.
Using the extended Euclidean algorithm, find the multiplicative inverse of
1.
1234 mod 4321

2.
24140 mod 40902

3.
550 mod 1769

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