Codes Ciphers and Cryptography Chapter 1 Homework Questions

 

This homework assignment is due the week of September 20, 2004 at the beginning of class. You may discuss problems with other people in our class, but must turn in your own work. Be as neat and complete as possible.

 

  1. Read Edgar Allan Poe’s “A Few Words on Secret Writing” which can be found on line at http://www.eapoe.org/works/essays/fwsw0741.htm.  This article appeared in Graham’s Magazine in July 1841, pp. 33-38.  What examples of cryptography and steganography does he consider?  What methods does he suggest for cryptanalysis of a cipher?

 

  1. On pages 389-391 of the article “Encryption Using a Variant of the Turning-Grille Method” by Stephen Fratini which appeared in Mathematics Magazine, Volume 75, Number 5, December 2002, Fratini discusses the number of different grids that can be constructed.  He says that two grids are equivalent if one can be obtained from the other by a rotation.  Thus, one grid is equivalent to three others, so when counting the number of “different” possible grids, one has to divide by 4.  Show that the number of “different” grids in an n x n   grid with n odd is 4k-1 where k = (n+1)(n-1)/4.  (Hint:  First sketch some samples of n x n grids for specific choices of n.  Then sketch the general case.)

 

  1. A soldier of ancient Sparta has been captured with a strip of leather bearing the following message.  Decipher the message.

 

LGOABIAIRFRENPKEHSEGHEAIENIIMTSESNCAGTMACSNEOSDRUYN

RTLYBOEYOYPTFRCHLTLTAAAAOEHLEVCGTESSEKRY.

 

  1. Here are two more substitution ciphers that involve modular arithmetic.

a.       Multiplicative cipher:  This cipher works like the additive cipher, except instead of adding m, mod 26, you multiply by m, mod 26.  Make a cipher with m = 3.

                                                               i.            Does this method of creating a cipher work for all integers m = 0, 1, 2, …, 25? 

                                                              ii.            For which integers does it work? 

                                                            iii.            Can you show why it fails for certain integers?

                                                             iv.            How do you decipher an encrypted message?

b.       Affine cipher:  This cipher works in two steps.  Step 1:  add an integer m mod 26.  Step 2:  multiply by an integer n mod 26.

                                                               i.            Make an affine cipher with m = 14 and n = 3.

                                                              ii.            How many different affine ciphers are there? (Hint: Fundamental Principle of Counting.)

 

  1. Here is an example of a rail fence cipher:

 

Cleartext:

 

I

 

 

 

I

 

 

 

O

 

 

 

I

 

 

 

R

 

 

S

 

H

 

S

 

G

 

O

 

C

 

P

 

E

 

?

 

 

T

 

 

 

A

 

 

 

D

 

 

 

H

 

 

 

 

 

 

 

Ciphertext:  IIOIRSHSGOCPE?TADH

 

Is this a transposition cipher or a substitution cipher?  Suppose you intercept this message and suspect it is a rail fence cipher:  TCHHEKTELEHIIMICSA.  Decrypt the message!

 

  1. Decipher each of the following messages:

a.       SBKF SFAF SFZF

b.       EIM QCJ CII SAQ KMM IBK GJI FSQ HAT GMG JJI VAP WVI GJK MHA VJG TUI

c.        HEXFX CG ICHHIX WAJQH HEPH HEX BFAQIXT AZ BFAHXVHCYS PYW GXVJFCYS VATT

JYCVPHCAYG LCII VAYHCYJX HA SFAL WJFCYS HEX VATCYS NXPFG YAH AYIN CY CHG

HFPWCHCAYPI TCICHPFN PYW BAICHCVPI FAIXG QJH PIGA CY HEX BJQICV PYW

VATTXFVCPI WATPCYG (Hint:  A keyword was used for this cipher!)