Asymmetric ciphers

Question 1

The RSA cryptosystem is still the most often used public key cryptosystem today.

a) Explain the RSA cryptosystem.
b) Alice wants to encipher the message m= 1234 by using the RSA cryptosystem and send it to Bob. Let Bob’s public key be (e,n) = (29,67·71). Compute the private key for Bob and encipher Alice’s message to Bob.
c) Decipher the ciphertext that Bob received from Alice.

Answer 1

IMT4552+Cryptology2+09august2016.pdf

Question 2

Letp=73, q=79 be prime numbers. Alice wants to encipher the message m—4321 by using the RSA cryptosystem and send it to Bob. If Bob’s public key is (e,n) =(41,73•79), do the following:

a) Compute the private key for Bob.
b) Encipher the message m.
c) Decipher the ciphertext that Bob received from Alice.

Question 3

Let (e,n) be the Alice’s public key in the RSA system. How does the enemy cryptanalyst get the Alice’s private key if she knows the factorization of the publicly known integer n=pq, where p and q are large primes?

Question 4

Letp=97, q 101 be prime numbers chosen by Bob and kept secret. Alice wants to encipher the message m 2548 by using the RSA cryptosystem and send it to Bob. Bob has chosen the integer e=89.

a) Why is Bob’s choice of the integer e appropriate? What is Bob’s public key?
b) Compute the private key for Bob.
c) Encipher the message m.

Bob wants to decipher the Alice’s message. Perform decipherment!

Question 5

Letp=79, q=89 be prime numbers. Alice wants to encipher the message m 1221 by using the RSA cryptosystem and send it to Bob. If Bob’s public key is (e,n) =(61,79•89), do the following:

a) Compute the private key for Bob.
b) Encipher the message m.

Question 6

Letp=71, q—73 be prime numbers. Alice wants to encipher the message m 2222 by using the RSA cryptosystem and send it to Bob. If Bob’s public key is (e,n) —(47,71 •73), do the following:

a) Compute the private key for Bob.
b) Encipher the message m.
c) Bob wants to decipher the Alice’s message. Perform decipherment!

Question 7

Alice wants to encipher the message m 1234 by using the RSA cryptosystem and send it to Bob. If Bob’s public key is (e,n) =(17,41 •43), do the following:

a) Compute the private key for Bob.
b) Encipher the message m.
c) Bob wants to decipher the Alice’s message. Perform decipherment!

Question 8

Letp=59, q 61 be prime numbers. Alice wants to encipher the message m=1234 by using the RSA cryptosystem and send it to Bob. If Bob’s public key is (e,n) =(31,59•61), do the following:

a) Compute the private key for Bob.
b) Encipher the message m.
c) Bob wants to decipher the Alice’s message. Perform decipherment!

Answer 8

http://C:\Users\tedah\Desktop\eksoppgiflash\IMT4532_Cryptology_1_Solution_08_10_2012_Master.pdf

Question 9

Explain how we break the RSA cryptosystem if we know the factorization of n, where n=pq is the second part of the public key (e,n) of the recipient. (7 points) b) Factorize the integer n=3599 using Fermat factorization.

Question 10

Compute the Jacobi symbol using the properties of the Jacobi symbol presented 2191 below:

Let n be an odd positive integer and let a,b20. Then the following identities hold

Answer 10

IMT4552+Cryptology2+07januar2016.pdf

Question 11

a) Factorize the integer n=2021 using Fermat factorization.
b) Let q=ll, and cc=2 be a generator of Z{l. Alice and Bob want to establish a common secret key by means of the Diffie-Hellman secret key establishment algorithm. If Alice generates a random number x=5 and Bob generates a random number y=6, compute the common secret that they are going to obtain after completing the algorithm.

Answer 11

IMT4552+Cryptology2+inkl+l_prcent_F8sningsforslag+14mars2016.pdf

Question 12

Compute the Jacobi symbol 2191985 ) using the properties of the Jacobi symbol presented below:

Let n be an odd positive integer and let a,b20. Then the following identities hold: