Question

Using RSA algorithm, what is the value of cipher text C, if the plain text M = 5 and p = 3, q = 11 & d = 7?
A) 33
B) 5
C) 25
D) 26
Answer: (D)
Please provide solution for this problem......

Answer 1

It is completely described in
https://en.wikipedia.org/wiki/RSA_(cryptosystem)

We are given:
M=5 (plain text)
p=3, q=11 (secret factors)
d=7 (private key exponent)
e=(1,\(\phi(n)\) ) such that e and \(\phi(n)\) are coprime
AND d.e\(\equiv\)1 (mod n)
public key is (n,e)
To encrypt, \(c=M^e (mod~~n)\)
where M is plain text, c=encrypted text
To decrypt, \(M=c^d (mod~~n)\)

We first calculate
n=pq=33
\(\phi_n(n)=(p-1)(q-1)=20\)
d=7, n=20, choose e=3 so that 7*3=21\(\equiv\)1 mod(20)
check: 3 is between 1 and 20, and is coprime with 20.
so
Public key = (n,e) = (33,3)
Private key = d = 7

Plain text = M = 5
encrypted text, c = M^e (mod n) = 5^3 mod (33) = 125 mod 33 = 26
decryption: M' = c^e mod n = 26^7 mod 33 = 5 = M (original plain text)

Answer 2

Thank you so much☺️

Answer 3

You're welcome! :)

Answer 4

But I have a problem I don't know how they calculate mod in last can u please explain

Answer 5

Mod is the calculation of the remainder when one integer is divided by another.

For example,
45 mod 11 equals 45-4*11=1
A more difficult example when we deal with big numbers:
To find 26^7 mod 33 is to calculate the remainder when 26^7 is divided by 33.
If you have a good calculator, you can find 26^7=8031810176.
If your calculator does not show that many digits, just type in 26^7 in google, it will give you the answer.
Then you divide 8031810176 by 33 to get 243388187.1515152
Take the integer portion 243388187 and multiply by 33 to get 8031810171.
Subtract from 8031810176 to get 26^7 mod 33 = 5.
You can also type in directly in Google 26^7 mod 33, it will return 5.

Answer 6

Thankyou so much ☺️☺️☺️

Answer 7

You're welcome! :)