RSA encryption/decryption algebra questions?

Oct 2018
1
0
UK
Hello, I have been trying to get my head around these 2 questions for 2 days now and I really need to complete these last 2 set of questions however I have looked everywhere for tutorials and cannot find any solution to work it out. Hope you all can give me a helping hand!(Hi)
I have attached the questions to this thread as an PNG image.
Many thanks!

 

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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The first problem asks you to find the smallest value of x such that 5x= 1 (mod 21). That means that 5x= 1+ 21k for some integer k. We can write that as 5x- 21k= 1. That's a "Diophantine" equation (an equation to be satisfied by integers). 5 divides into 21 4 times with remainder 1: 21- 5(4)= -(-21)+5(-4)= 5(-4)- 21(-1)= 1. So the equation is satisfied by x= -4 and k= -1. But then x= -4+ 21n, k= -1+ 5n also satisfy the equation for any integer, n: 5(-4+ 21n)- 21(-1+ 5n)= -20+ 105n+ 21- 105n= 1. The smallest positive value of x is -4+ 21(1)= 17.

(5*17= 85= 4(21)+ 1)


For Q4: what is "RSA decryption"?