1.7. We consider the ring Z4. Construct a table which describes the addition of
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Question
1.7. We consider the ring Z4. Construct a table which describes the addition of all elements in the ring with each other:
26 1 IntroductiontoCryptographyandDataSecurity
1: Construct the multiplication table for Z4.
2: Construct the addition and multiplication tables for Z5.
3: Construct the addition and multiplication tables for Z6.
4: There are elements in Z4 and Z6 without a multiplicative inverse. Which elements are these? Why does a multiplicative inverse exist for all nonzero elements in Z5?
3 1-12 001 : +0123Explanation / Answer
1) Multiplication table for Z4.
Construction of Cayley tables are simple. In the main row and columns. Based on the operation the remainder is stored inside the table
Multiplication (mod) 4
2)Addition and multiplication tables for Z5.
Addition table of Z5
Multiplication table of Z5
3) Addition and multiplication tables for Z6.
Addition tables for Z6.
Multiplication table of Z6
4)
Multiplicative Inverse of integer is the integer which upon apllication of the operation gives the Identity of the operation.Identity of Addition is 0 and Identity of Multiplication is 1.
In Multiplication table of Z4 there is no idenity element for elements 0,2 after multiplication. Therefore it doesn't have multiplicative inverse.
In Multiplication table of Z6 there is no idenity element for element 0,2,3,4 after multiplication. Therefore it doesn't have multiplicative inverse.
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