Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 1 Solutions - Ex1.7

- 3 mins

Exercise 1.7

We consider the ring Z4. Construct a table which describes the addition of all elements in the ring with each other:

  1. Construct the multiplication table for \(\mathbb{Z}_4\).
  2. Construct the addition and multiplication tables for \(\mathbb{Z}_5\).
  3. Construct the addition and multiplication tables for \(\mathbb{Z}_6\).
  4. There are elements in Z4 and Z6 without a multiplicative inverse. Which elements are these? Why does a multiplicative inverse exist for all non-zero elements in \(\mathbb{Z}_5\)?

Solution

This solution is verified as correct by the official Solutions for Odd-Numbered Questions manual.

1. Multiplication table for \(\mathbb{Z}_4\):

2. Addition and Multiplication Tables for \(\mathbb{Z}_5\):

3. Addition and Multiplication Tables for \(\mathbb{Z}_6\):

4. To determine whether an element has a multiplicative inverse, we must check if it can be multiplied with another element to produce 1:

Elements without a multiplicative inverse in \(\mathbb{Z}_4\) are 2 and 0

Elements without a multiplicative inverse in \(\mathbb{Z}_6\) are 2, 3, 4 and 0

For all non-zero elements of \(\mathbb{Z}_5\), there exists a multiplicative inverse because 5 is a prime. Hence, all non-zero elements smaller than 5 are relatively prime to 5.


Thomas Busby

Thomas Busby

I write about computing stuff

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