A linear cipher is defined by the congruence C a P b mod 2

A linear cipher is defined by the congruence C = a P + b (mod 26), where a and b are integers with gcd(a, 26) = 1. Show that the corresponding decrypting congruence is P s a\'(C - b) (mod 26), where the integer a\' satisfies aa\' 1 (mod 26). Using the linear cipher C = 5P + 11 (mod 26), encrypt the message NUMBER THEORY IS EASY. Decrypt the message RZQTGU HOZTKGH DJ KTKMMTG, which was produced using the linear cipher C = 3P + 7 (mod 26).

Solution

we need to show

a\' (C-b) = P mod(26)

C - b = aP (mod 26)

So a\' (C-b) = a\' a P (mod 26)

Now if a\'a = 1 mod 26

this shows a\' (C-b) = P (mod 26)

N = 14 , U = 21, M = 13, B=2 , E= 5, R = 18, T = 20, H = 8, E = 5, O = 15, R=18, Y = 25, I = 9, S= 19, E=5,A=1,S=19,Y=25

On encrypting by 5P + 11, multiplying by 5 and add 11 take mod

N = 3 = C, U =12 = L , M = 24= X, B=21= U , E= 10= J, R = 23= W, T = 7=G, H = 25=Y, E = 10=J, O = 8= H, R=23=W, Y = 6=F, I = 4= D, S= 2= B, E=10= J,A=16=P,S=2=B,Y=6=F

CLXUJW GYJHWF DB JPBF

3) aa\' = 1 mod 26

solve 3x = 1 mod 26

x = 9

decrypt by subtract 7 and multiply by 9 and take mod 26