Handin problems, due Wednesday, 4/17/96

1. Give the 12 bit codeunits produced by the Limpel-Zev compaction of the string "aaaaaaaaaa".
2. Give the encryption of those 12 bit codeunits using the RSA method and keys generated by rsa.map (with maple) when p := 103 and q := 53.
3. Then verify correctness by decrypting and uncompressing. [show your work]
4. What does the encrypted text C look like if it happens in building the keys that d is a prime (so that sh = 1)?
5. In building RSA keys, what is bad about it turning out that d is the square of a prime number (for example when p := 101 and q := 103)?

Claim problems, rsa 1 and rsa 2

1. • Explain how you could break a RSA key set if you could factor n. That is, you know P = [ph, n], p, and q. How do you compute sh?
   • It may be, though, that there is a much easier way to find sh, a way which does not involve finding out p and q. It turns out that this idea is essentially false. The problem "factor a large number n which is known to be the product of two primes" is of approximately the same difficulty as the problem "find sh, knowing ph and n". Give evidence for this claimed "equivalent difficulty" of these two problems by showing that if ph, sh and n are known it is not too much trouble to find out p and q. Hint: Assume d-1 is easy to factor and usually has a fair number of small factors. and a few larger ones. For example for ph = 5, sh = 1061, and n = 5459, we see that d-1 factors as 2*2*2*3*13*17. How can you use this to find p and q?
2. Break the code (i.e. find sh) when the public key is P = [5, 13529].