Question

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 n What is m? _______________, what
is n? _______________ (generic name, not a number)**

**DECRYPT (c) = c^d mod n What is c?
_______________**

**What does "mod"
mean?__________________________________________________________________________________________________________________**

**Everything in Bold is What I am Lost on? Teacher gave
confusing example to solve.**

EXAMPLE

An RSA public-key / private-key pair can be generated by the following steps:

1. Generate a pair of large, random primes p and q. NOTE for examples we use small prime numbers due to the size of the number generated.

**p=3, q=5**

2. Compute the modulus n as n = pq.

**n = p*q, 15=3*5**

3. Select an odd public exponent e between 3 and n-1 that is relatively prime to p-1 and q-1.

e is between 3 and 14, e is relatively prime to 2 (3-1) and 4 (5-1)

φ(n)=2*4=8

**e=7, could also be 9 or 11 or 13**

4. Compute the private exponent d from e, p and q. (See below.)

**e^-1 = d (mod φ(n))**

**7^-1 = d (mod 8)**

**7 * d = 1 (mod 8)** What number multiplied by 7
and then divided by 8 equals 1?

Multiples of 8 |
8 |
16 |
24 |
32 |
48 |
56 |
64 |
72 |
80 |
88 |
96 |
104 |
|||

Multiples of 7 |
7 |
14 |
21 |
28 |
35 |
42 |
49 |
56 |
63 |
70 |
77 |
84 |
91 |
98 |
105 |

**7 * 15 = 1 (mod 8) = 105/8= 13 remainder 1**

**d = 15**

5. Output (n, e) as the public key and (n, d) as the private key.

**PUBLIC KEY IS (15, 7) (Modulus, e), PRIVATE KEY IS (15,
15) (Modulus,d)**

The encryption operation in the RSA cryptosystem is exponentiation to the e th power modulo n:

**Plaintext = 8**

ciphertext = ENCRYPT (m) = m^e mod n

**c = 8^7 mod 15 = 2,097,152 mod 15 = 2
(ciphertext)**

The decryption operation is exponentiation to the dth power modulo n: m = DECRYPT (c) = c^d mod n.

**m= 2^15 mod 15 = 32,768 mod 15 = 8
(plaintext)**

Answer #1

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...

RSA Cryptography
Perform encryption and decryption to find private decryption key
d.
a. p = 3, q = 13, e = 7
b. p = 787, q = 631, e = 234559

Alice is sending message “HIDE” to Bob. Perform encryption and
decryption using RSA algorithm, with the following information:
Parameters p q e 11 5 7 Present all the information that you will
need to encrypt and decrypt only the first letter from text.

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...

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

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).

You want to encrypt a message using RSA using your private key
of d=3,n=35. Your message is 15. What is your ciphertext?

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

Bob has an RSA public key of
(n, e) = (1363, 87)
(a) What is Bob’s private key?
(b) Bob receives the ciphertext which has been encrypted with
his public key
893, 1265, 406, 171, 980, 1040, 12, 1152, 573
Decrypt the message.
(You can use an appropriate package such as Matlab or Wolfram Alpha
to do the calculations)

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...

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