Compute additive and multiplicative inverses of 7 and 9 in Z11
(mod 11).
Find out whether or not 4 and 7 have multiplicative inverse in
Z14 (mod 14).
Let S be the set of even integers under the operations of
addition and multiplication. Is S a ring? Is it commutative? Is it
a field? Justify your answer.
Compute the multiplicative inverse of 9 under modulo 31 using
the extended Euclid’s algorithm.
0 mod 35 = 〈0 mod 5, 0 mod 7〉 12 mod 35 = 〈2 mod 5, 5 mod 7〉 24
mod 35 = 〈4 mod 5, 3 mod 7〉
1 mod 35 = 〈1 mod 5, 1 mod 7〉 13 mod 35 = 〈3 mod 5, 6 mod 7〉 25 mod
35 = 〈0 mod 5, 4 mod 7〉
2 mod 35 = 〈2 mod 5, 2 mod 7〉 14 mod 35 = 〈4 mod 5, 0 mod 7〉...
a) Use Fermat’s little theorem to compute 52003 mod 7,52003 mod 11, and 52003 mod 13.
b) Use your results from part (a) and the Chinese remaindertheorem to find 52003 mod 1001. (Note that1001 = 7 ⋅ 11 ⋅ 13.)
Problem Solving Set #1 (10 pts each) a. Find the multiplicative
inverse of 1234 in GF(4321) using the extended Euclidean algorithm
b. Does the multiplicative inverse of 24140 in GF(40902) exist?
Prove your answer. c. Is x4 + 1 irreducible over GF(2)? Prove your
answer. d. Find (x3 + x + 1)-1 in GF(24 ) mod x4 + x + 1 using the
extended Euclidean algorithm e. Find (x3 + x + 1)-1 in GF(28 ) mod
x8 + x4...
Write a program( preferably in C++) using the
extended Euclidean algorithm to find the multiplicative inverse of
a mod n. Your program should allow user to enter a and n.
Note: For this question please make sure the code compiles and
runs, it is not copied and pasted from elsewhere( I will be
checking!). Thanks
1. Find the multiplicative inverse of 14 in GF(31) domain using
Fermat’s little theorem. Show your work.
2 Using Euler’s theorem to find the following exponential: 4200
mod 27. Show how you have employed Euler’s theorem here.