To determine:The decoded message for the following encoded messages. E=456

Question

To determine:The decoded message for the following encoded messages.  E=456

Nathaniel Kramer · Accepted Answer

Given information:
p=17, q=59, and s=3
Explanation:
Since, gcd (3,16)=1, there exist integers a and x such that
3a+16x=1.
We know that 3(11)-16(2)=1.
Thus, a=11.
Also, 3(39)-58(2)=1
Thus, b=39.
It is given that E=456.
So,

E^{a} = {(456)}^{11} = 10 (b mod 17)

.
Also,

E^{b} = {(456)}^{39} = 42 (b mod 59)

.
Now the problem is reduced to solvinga pair of congruences.

x \equiv 10 (b mod 17)

x \equiv 42 (b mod 59)

We have, 7(17)-2(59)=1. Thus, x=42.7(17)-10.2(59) is the solution of the above pair of congruences modulo 1003.
x=42.7(17)-10.2(59)
x=4998-1180
x=3818

x = 809 (b mod 1003)

Thus, M=809.
We conclude that the decoded message for E =456 is HI.

To determine:The decoded message for the following encoded messages. E=456

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