Question

For solving the problems, you are required to use the following formalization of the RSA public-key cryptosystem.

In the ** RSA public-key cryptosystem**,
each participants creates his public key and secret key according
to the following steps:

· Select two very large prime number p and q. The number of bits needed to represent p and q might be 1024.

· Compute

n = pq

(n) = (p – 1) (q – 1).

The formula for (n) is owing to Theorem: The number of elements in is given by Euler’s totient function, which is

where the product is over all primes that divide n, including n if n is prime.

· Choose a small prime number as an encryption component g, that is relatively prime to (n). That means,

gcd(g, (n) ) = 1, i.e.,

gcd(g, (p-1)(q-1)) = 1.

· Compute the multiplicative inverse That is,

The inverse exists and is unique.

That is, the decryption component h = g^{-1} mod (n).

· Let pkey = (n, g) by the public key, and skey = (n, h) be the secret key.

· For any message M mod
n, the encryption of M is C = M^{g} mod n.

· The decryption of C
is M = C^{h} mod n.

End of the formalization of the RSA public-key cryptosystem.

**Let g = 59, p = 991 and q = 997.**

Problem B: [20 pts.]

Given a plaintext **M = 5065747269**, what is the
encryption of M, using C = M^{g} mod n. Show in details how
you derive C, which is the ciphertext of the plaintext M.

Answer #1

Ans:

here p=991 and q=997

And the Encryption key g=59

so,now

n=p*q=991*997 = 988027

(n)= (p-1)*(q-1) =990*996 = 986040

And now we selected public key(encryption key) g =59

Now, private key (decryption key) h was given by,

g=h^{-1} mod(n)

or this can be wrriten as g*h =1 mod (n)

59*h =1 mod 986040

59 * 584939 = 1 mod 986040 (true)

so, decryption key (h) =584939

Now,

given that plaintext M = 5065747269,

Now Ciphertext was given by,

C=M^{g} mod n = 5065747269 ^{59} mod 988027 =
433940

**so, the Ciphertext C= 433940**

Below is an example of key generation, encryption, and
decryption using RSA. For the examples below, fill in the
blanks to indicate what each part is or answer the
question.
Public key is (23, 11) What is 23 called?
_______________, What is 11 called?_______________
Private key is (23, 13) What is 23
called?_______________, What is 13
called?_______________
23 can be part of the public key because it is very hard
to _______________ large prime numbers.
ENCRYPT (m) = m^e mod...

Let p=3 and q=17 and let an RSA public-key cryptosystem be
given.
1. Why is the number 8 not a valid encryption-key?
2. We encrypt the number M=8 with the help of the encryption-key
e=3.
Why is the encrypted message C=2?
3. Why is the decryption key d for the encryption-key, (e=3),
equal to 11?
https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Encryption

Suppose your RSA Public-key factors are p =6323 and q = 2833,
and the public exponent e is 31. Suppose you were sent the
Ciphertext 6627708. Write a program that takes the above parameters
as input and implements the RSA decryption function to recover the
plaintext.
IN PYTHON

Suppose the totient for a given public key (52,494,503;7) is
inadvertently released and is equal to 52,479,768. What are p and
q? What is the decryption exponent d? For extra credit what is the
plaintext message of the ciphertext C=15,114,578 encrypted with the
given public key?

Let p=11, q=17, n = pq = 187. Your (awful) public RSA encryption
key is (e=107, n=187). (a) What is your private decryption key? (b)
You receive the encrypted message: 100 Decrypt the message. (In
other words, what was the original message, before it was
encrypted? Just give me a number, don’t convert it to letters).

1. Dexter wants to set up his own public keys. He chooses p = 23
and q = 37 with e = 5. Answer the following using RSA cryptographic
method: a) Encrypt the message ‘100’ and find the CIPHER number (C)
to be sent. ‘100’ should be taken together as M while computing
encryption [5 Marks] C = Me mod pq (Use This Formula) b) Find the
decryption key ‘d’, using extended Euclidean GCD algorithm. [10
Marks] c) Now decrypt...

Discrete Math
In this problem, we will implement the RSA algorithm to encrypt
and decrypt the message ”148”.For this exercise, you may want to
use some kind of calculator that can compute the mod function.
1. Set the primes p and q as follows:p=31 and q=47. What are the
values for N and φ?
2.The value for e is chosen to be 11. Use Euclid’s algorithm to
verify that e and φ are relatively prime and to find d, the...

4. RSA For some of the questions below you may want to use a
high precision calculator e.g. a. Given n = 55 we choose e = 7 to
complete the public key for RSA. Explain why this value works but e
= 8 doesn’t.
______________________________________________________________________________________________________
______________________________________________________________________________________________________
b. What is d given e = 7? ______________________ Remember: you
need to find and inverse of e (mod φ(n)). Show work.
c. What is the number 2 encrypted as? _______________. Show
work.

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