CSCI 7000 - Cryptography - Spring 2008

Problem Set #4

Due: May 9th, 2008

• Using Fermat's Test, find the smallest 3-pseudoprime larger than 1278450012840012915. Call this number that you found, p.
• Now find the smallest 3-pseudoprime larger than 92830582375291128351. Call this number that you found, q.
• What is n? (Note: make sure you do the above two steps properly or you will get everything else wrong from here on out!)
• What is phi(n)?
• Let e=5. What is d?
• Let M=19571299103581789. What is the ciphertext C?
• Decipher C using d. Do you get M back?

2. Assume RSA. The public key is (411816231521, 5). The encoding method is to take 8 characters of plaintext and convert to an integer M1. Then encrypt under the public key as usual, to obtain C1. Then we take the next 8 characters of plaintext and convert to M2, which we encrypt to C2. And so forth.

The conversion works as follows: we treat a string of 8 alphabetical characters as a vector of 8 coordinates, each of which is between 0 and 25. (A is 0, B is 1, ..., Z is 25). For example, HEYTHERE converts to (7,4,24,19,7,4,17,4). We then convert this to an integer by taking these coordinates as the digits of a base-26 number. So HEYTHERE converts to the integer 7*26⁷ + 4*26⁶ + 24*26⁵ + 19*26⁴ + 7*26³ + 4*26² + 17*26 + 4 = 57752296086.

• Show that this conversion process always produces an integer between 0 and n-1 for the n given above in the public key.
• Assuming that the ciphertexts listed below are encrypted using the public key above and the given conversion process, recover the plaintext. Be sure and document your method.

  225803654487   27035345731   318496681005    222309193242   128671002039
  

3. We showed in class that RSA works. That is, that if you have some M relatively prime to n, then M^ed = M (mod n). Please show that RSA still works if M is any integer between 1 and n-1.

4. (Extra Credit) Fix some RSA modulus n=pq (where p and q are distinct primes) and some encryption exponent e which is relatively prime to phi(n). Show that the number of messages M, 0 <= M <= n-1, such that M^e = M (mod n) is equal to (1 + gcd(e-1, p-1))(1 + gcd(e-1, q-1)).