Math 455.1 Homework Set 5 Spring, 2009
Due: Wednesday, April 8 – postponed (start of class)
•Work individually, that is, not in a team. See the About >Homework sets page on
the Math 455 web site regarding collaboration and plagiarism.
•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 affine ciphers, be sure to follow the convention of first adding and then multiplying.
And use the convention that Ais coded numerically as 0, Bis coded as 1, etc. Also,
work with the numerical equivalents of the text letters and not the letters themselves.
See notebook AffineEncrypt.nb.
1. (a) Use the definition of congruence (in terms of divisibility) to prove the following
for all integers a, b, c and all integers m > 1: If c+a≡c+b(mod m), then a≡b
(mod m).
(b) Is the analog of (a) for multiplication, instead of addition, also true? If so, prove
it (in general); if not, give a counterexample.
2. Use an affine cipher with additive key 23 and multiplicative key 5 to encrypt the
plaintext:
Government is, or ought to be, instituted for the common benefit,
protection, and security, of the people, nation, or community.
For your convenience, this text is available in the notebook Set5#2.nb. You must first
strip out all spaces and punctuation, convert all letters to upper-case, and encode into
integers; then encrypt into integers and decode the result into ciphertext split into
groups of 5 letters. You may use Mathematica for some or all of the work.
3. Use notebook Set5#3.nb to obtain your own, individual ciphertext. That ciphertext
was obtained by using either a multiplicative cipher or an additive cipher. Break the
cipher and decrypt the ciphertext.
The ciphertext is short, so you may or may not be able to exploit letter frequencies.
Try up to the three most likely possibilities for the most frequent letter in the ci-
phertext (and include the results from those trials). If you cannot succeed after three
trials—and only in that case—apply “brute force”, that is, try all possibilities. See
notebook BreakAffineCipher.nb for examples of breaking such ciphers.
4. Alice uses the affine encryption system with key (4,3)—that is, multiplicative key 4
and additive key 3—to send you the ciphertext message SCCEGOMME. She tells you she
is using this system with this key.
(a) Try to decrypt (into plaintext letters, eventually) the numerical form of the
ciphertext. You should find that you cannot do so unambiguously. So determine
at least four of the possibilities for the plaintext message that Alice sent. (Do not
rely upon the meaning of the message—Alice could have sent nonsense words!)
