ECE4122/6122 RSA Encryption


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Introduction. In this assignment we will use the GNU multi-precision arithmetic library to implement the wellknown public-key encryption algorithm known as Rivest-Shamir-Adelman (RSA). Our program will choose appropriate
public/private key pairs, choose random messages, encrypt the random message with the public key, decrypt the
random messagse with the private key, and verify that the decrypted message matches the original messages.
Choosing the RSA public/private key pair. To select the public key (n, d) and private key (n, e), do the following.
1. Create a variable of type gmp randclass to generate randomd numbers.
2. Select two random prime numbers p and q of size sz bits. (Note we will vary sz as described below). The
gmp randclass object has a method called get z bits which returns a random value of the specified size. The
GNU multi-precision library has a function mpz probab prime p to determine if a given number is “likely”
to be prime. For this assignment, use 100 iterations of the primality testing algorithm (the second parameter on
the mpz probab prime p function).
3. Compute n = p ∗ q.
4. Compute φ(n) = (p − 1) ∗ (q − 1)
5. Choose a random d of size sz * 2 and with d < φ(n). Note that d does not have to be prime. 6. Insure that the greatest common divisor of d and φ(n) is exactly 1. If not, choose another d. The GMP library has a function mpz gcd to compute the greatest common divisor. 7. Finally, compute e as the multiplicative inverse of d modulo φ(n). Use the GMP function mpz invert for this. At this point the pair (n, d) is the public key, and (n, e) is the private key (although this is arbitrary, we could use (n, d) as the private key and (n, e) as the public). Performing the Encryption and Decryption. Given a key (either public or private) (n, d), and a plaintext message m, the RSA encryption algorithm is simple. Just compute the ciphertext message c as c = md mod n. The GMP function mpz powm does this for us. Specific program details. 1. Start with sz as 32 bits, and double the size for each iteration up to and including 1024 bits. Recall that sz is the size of p and q, and n is p ∗ q, so n has twice as many bits. 2. For each value of sz, compute 100 different public/private key pairs as described above. 3. For each public/private key pair compute 100 random messages of size less than n bits. Again, you can use the get z bits function of the GMP random nubmer generator to do this. 4. For each random message, compute the ciphertext (using the RSA algorithm), and then decrypt the ciphertext message (again using the RSA algorithm) and verify the decrypted message is identical to the original plaintext message. 5. Important THe RSA has several Print functions you should use to print out the various keys,plaintext message and ciphertex message. 1 6. IMPORTANT. Feel free to add as much debug code (with screen prints) as you need to get the program debugged. However, the grading script will fail if extra prints are left in the submitted program. 7. Grad Students Only. Attempt to “break” the RSA algorithm for the 64 bit n values using a factoring algorithm of your choice. For this a skeleton will be provided. Copying the Project Skeletons 1. Log into using ssh and your prism log-in name. 2. Copy the files from the ECE6122 user account using the following command: /usr/bin/rsync -avu /nethome/ECE6122/RSA . Be sure to notice the period at the end of the above command. 3. Change your working directory to RSA cd RSA 4. Copy the provided to as follows: cp 5. Then edit to create your code for the project Turning in your Project. Use the same turnin script as prior assignments, specifically turnin-ece6211 or turninece4122 2