***Modular arithmetic is fundamental to cryptography. In this system, you can only have integers. For example, in mod 14 system, the answer MUST be 0,1,2,3, . . . 9,11,12,13. Non-integer values have no place in this arithmetic. If you have an answer which is a floating point, such as 12.5, then you are doing something wrong.

***SHOW YOUR WORK!

1) 150 92 mod 14 (Hint: a b mod c = ((a mod c) (b mod c)) mod c)

2) 6 (4/11) mod 14 (Hint: In mod 14 system, a, a+14, a+28, a+42, a+56, etc. are all equivalent)

3) 24/17 mod 14 (Hint: First, simplify the numerator and denominator separately by applying the mod function independently, and then solve as in (#2) above).

4) 48 512 mod 14 (Hint: Try to compute the exponent in stages, each time simplifying it using the mod function. For example, to compute 48 mod 14, express 48 mod 14 (42 mod 14)4 mod 14, compute the one in the parenthesis, and repeat this process.

5) 510 68 mod 14 (same as #4 above)

6) Show the elements of groups Z13 and Z13 (Note that 13 is a prime number)

7) Show the elements of groups Z18 and Z18 (Note that 18 is NOT a prime number)

8) Find the order of 5 in Z13 (Hint: Order of an element in a finite group G is the smallest positive integer k such that ak = 1 where 1 is the identity element of G.)

9) Find (if it exists) the multiplicative inverse of 5 Z13 (integer ring) (Hint: For a Zn, its multiplicative inverse, if it exists, is defined as a1 such that a.a1 1 mod n.)

10) Is Z13 a cyclic group? If so, what is its order and the generator element? (Hint: group G which contains some element with maximum order ord() = |G| is said to be cyclic. Elements with maximum order are called generators.)