Question

4. (20 pts) If a user of a comms system wants to utilize the RSA public key encryption with p = 11 and q = 7 find their public and private keys and encrypt the message, “cyber”, using the numeric value of 1 to represent an 'a', 2 = 'b', etc. Also, show the first step in the deciphering process of this message by the user (i.e. show the first calculation based off of the received cipher for each letter (i.e. show the first mathematical operation performed on the received cipher value sans the modulo calculation). You must show all of your work to receive credit. a 1 b 2 c 3 d 4 e 5 f g 6 7 h 8 i 9 j 10 k 11 l 12 m 13 n 14 o 15 p 16 q 17 r 18 s 19 t 20 u 21 v 22 w 23 x 24 y 25 z 26

Computer Engineering

Answer 1

First we have to generate the key

The given primes are p = 11 and q = 7

n = p x q = 11 x 7 = 77

$\phi (n)=(p-1)(q-1)=(11-1)(7-1)=60$

choose a number e less than n and relatively prime n

Let's choose e = 13

A number d is chosen such that

$ed\,\,mod\,\,\phi (n)=1$

So d = 37

So the public key is (e,n) = (13,77)

Private key = (d,n) = (37,77)

Now message m = cyber

After mapping to numerical values

m1 = 3

m2 = 25

m3 = 2

m4 = 5

m5 = 18

Now ciphertext is calculated as

c = m^e mod n

c1 = (3)¹³ mod 77 =  38

c2= (25)¹³ mod 77 = 60

c3 = (2)¹³ mod 77 = 30

c4 = (5)¹³ mod 77 = 26

c5 = (18)¹³ mod 77 = 46

Now decrypting the cipher text

m = c^d mod n

m1 = (38)³⁷ mod 77 =  3

m2= (60)³⁷ mod 77 = 25

m3 = (30)³⁷ mod 77 = 2

m4 = (26)³⁷ mod 77 = 5

m5 = (46)³⁷ mod 77 = 18