Math 455.1 Homework Set 6 Spring, 2009
Corrected: 5 April 2009
Due: Monday, April 13 (start of class)
•Work either in a team or individually.
•For Mathematica work here, turn in printed pages. Try to place associated written
work directly onto such printed pages (or include text cells there).
•For RSA systems, follow all our conventions: letters should be encoded into integers
in {0,1,2. . . , 25}as usual; and consecutive pairs of the resulting integers should be
joined into 2-digit to 4-digit integers before encryption is applied.
1. Use the method shown in class, applying Fermat’s Little Theorem, to find each of
the following modular powers of 8 as efficiently as possible—without first actually
computing 8 to the given powers.
(a) 82003 mod 7
(b) 82003 mod 17
2. Already known (and proved in class):
Proposition 1. If cis relatively prime to m, then [c]has a multiplicative inverse in
Zm.
Corollary 1. If mis prime, then every nonzero element of Zmhas a multiplicative
inverse.
Prove the following two converses of those results:
Proposition 2. If [c]has a multiplicative inverse in Zm, then cis relatively prime to
m.
(Suggestion to get started: Assume that [c] has a multiplicative inverse in Zm. Express
this in terms of a congruence modulo m.)
Corollary 2. If every nonzero element of Zmhas a multiplicative inverse, then mis
prime.
3. A bungling cipher bureau issues to Bob the public RSA key (n, e) = (3239,17) (which
is rather insecure). Assist Alice by encrypting the following message that she wants
to send Bob to Bob.
What’s another word for Thesaurus?
4. (a) Starting with the primes p= 41 and q= 67, generate for Bob a suitable RSA
public key (n, e) with eas small as possible and yet satisfying the usual require-
ments.
(b) Help Alice send the number 2418 securely to Bob: use that public key of Bob’s
to encrypt the number.
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