Checkerboard Tutorial

Background

The Checkerboard cipher is a simple substitution cipher based on a 5 x 5 Polybius Square. It was invented by the Greek historian Polybius in the 2nd century BC.

The Encipherment System:

The substitution alphabet(s) is(are) based on a 5×5 keyed Polybius square that is formed in any of the usual ways (I/J are considered equivalent). The rows and columns are labeled by either a single 5 letter word/sequence (Checkerboard) or by two five letter words/sequences (Double Checkerboard). In the case of the simple checkerboard, each plain text letter has a single cipher digraph replacement. In the case of the double checkerboard, each plain text letter has 4 possible cipher digraph replacements. This tutorial will consider only the simple checkerboard.

The plain text is enciphered one letter at a time by replacing it with the two letter digraph that corresponds to that letters Row and Column designators. Deciphering is the reverse of enciphering.

Example:

  • Keyword: SONGBIRD
  • Row Designator: ROBIN
  • Column Designator: VIREO
  • Plain Text: The early bird gets the worm.

Form the Keysquare:

   V I R E O  
  ----------
R :s o n g b   
O :i r d a c    
B :e f h k l     
I :m p q t u      
N :v w x y z      
    
P: t  h  e  e  a  r  l  y  b  i  r  d  g  e  t  s  t  h  e  w  o  r  m
C: IE BR BV BV OE OI BO NE RO OV OI OR RE BV IE RV IE BR BV NI RI OI IV

Message:  IE BR BV BV OE OI BO NE RO OV OI OR RE BV IE RV IE BR BV NI RI OI IV

The student should study the above example to become familiar with the encipherment and decipherment methods since this knowledge is critical for the cryptanalysis of the cipher.

Cryptanalysis

The cryptanalysis of this cipher primarily involves reconstructing the key square. This is done by matching pieces of the cipher text with plain text and reconstructing the square from the identified plain/cipher text equivalences.

Example Analysis

The example we will be working on is from the log of a puzzle type Geocache in New England. The cipher is:

Checkerboard. Hard Knocks Cache (GCRTRX) MSCREP

EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
ET TU II RQ  (Location)

Note the following:

  1. The plain text ‘location’ has been given as a crib. Checkerboard ciphers published in the ‘Cryptogram’ are normally provided with a healthy crib.
  2. The latitude and longitude coordinates of an object (the ‘cache’) are enciphered in this CON. Consequently, we should expect the plain text to primarily be long hand numbers.
  3. The cache log states that the cache is within a few miles of 43 degrees 17.382 minutes North and 070 degrees 51.264 minutes West. This point is within the town of Berwick, Maine.
  4. Most geocaches are hidden in stumps, under logs, in stonewalls, or under trees. Caches can contain a log book, ‘stash’, geocoins, and/or travelbugs. Some geocachers who lives near this cache are named ‘TigQuilt’ (short for TiggerQuilt) and “Pax Et Bonum’.
  5. The notation GCRTRX is the waypoint designation for the cache. It has nothing to do with the CON.

Row/Column Indicators

The first order of business is to determine the letters that are being used to designate the rows and columns. From this, we may be able to determine the actual designator words by anagrammimg. Examination of the cipher text digraphs reveals the following:

   Rows:  E, I, G, T, R
Columns:  I, U, Q, T, L

Anagramming these letters provides only, TIGER and QUILT as possible words. We will start the decrypt assuming these are the row and column designators.

Crib Location

The crib ‘location’ has the 8 letters with ‘o’ appearing twice. We need to search through the cipher text to find a sequence of 8 digraphs which has the 2nd and 7th digraph the same. It is quickly evident that the start of the cipher text meets this condition. Specifically, if we place the crib at position 1, the first ten digraphs provide:

 CT:  EI IU GQ TU TT EQ IU II IU GI 
 PT:   l  o  c  a  t  i  o  n  o  *

Note that the first letter following the crib would also be an ‘o’. This suggests the phrase “location of’. This extended crib would provide:

 EI = l, IU = o, GQ = c, TU = a, TT = t, EQ = i, II = n, and  GI = f.

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :* a * * t   
I :* o n * *   
G :c * f * *   
E :i * l * *   
R :* * * * *   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  *  *  i  *  *  *  c  i  *  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  *  i  n  *  t  *  *  o  n  *  *  i  *  *  o  i  n  t  *  * 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 *  *  n  t  *  o  f  i  *  *  n  o  *  t  *  a  n  *  f  i  *  *  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  *  o  *  o  t  f  i  *  *  f  i  *  *  f  o  *  *  *  *  *  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  *  *  i  *  a  t  t  *  *  *  a  *  *  o  f  a  l  a  *  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 *  *  t  *  *  *  *  *  i  l  *  i  n  t  *  *  a  *  *  a  *  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 *  *  o  *  *  *  *  *  *  *  c  t  *  t  o  t  *  *  *  *  t  *  

ET TU II RQ  
 *  a  n  *  

The first line of the plain text reads, “Location of the cac**…”. This suggest the fourth word is cache. This provides:

  GT = h and TL =e

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :* a * e t   
I :* o n * *   
G :c * f * h   
E :i * l * *   
R :* * * * *   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  h  e  i  *  *  e  c  i  *  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  *  i  n  *  t  e  *  o  n  e  *  i  *  *  o  i  n  t  *  e 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 *  e  n  t  *  o  f  i  *  e  n  o  *  t  h  a  n  *  f  i  *  e  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  *  o  *  o  t  f  i  *  e  f  i  *  e  f  o  *  *  *  e  *  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  h  e  i  *  a  t  t  h  e  *  a  *  e  o  f  a  l  a  *  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 *  e  t  *  e  e  *  h  i  l  e  i  n  t  h  e  a  *  e  a  *  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 *  *  o  *  *  *  e  *  *  e  c  t  *  t  o  t  h  e  *  e  t  e  

ET TU II RQ  
 *  a  n  *  

Now, the end of line 3 and the beginning of line four reads “fi*et*o”. This suggests the phrase “five two” and provides RU = v and RI = w.

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :* a * e t   
I :* o n * *   
G :c * f * h   
E :i * l * *   
R :* v w * *   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  h  e  i  *  *  e  c  i  *  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  *  i  n  *  t  e  *  o  n  e  *  i  *  *  o  i  n  t  *  e 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 v  e  n  t  w  o  f  i  v  e  n  o  *  t  h  a  n  *  f  i  v  e  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  w  o  *  o  t  f  i  v  e  f  i  v  e  f  o  *  *  w  e  *  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  h  e  i  *  a  t  t  h  e  *  a  *  e  o  f  a  l  a  *  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 *  e  t  *  e  e  w  h  i  l  e  i  n  t  h  e  a  *  e  a  *  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 *  *  o  *  *  *  e  *  *  e  c  t  *  t  o  t  h  e  v  e  t  e  

ET TU II RQ  
 *  a  n  *  

An examination of the key square suggests that the key word occupies the first two rows and that the last three rows are the remainder of the alphabet. Since we have already located ‘e’, this suggests that GU = d. We can also determine that GL =g and EU = k since only those letters can fit the gaps in the plain text.

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :* a * e t   
I :* o n * *   
G :c d f g h   
E :i k l * *   
R :* v w * *   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  h  e  i  *  d  e  c  i  *  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  *  i  n  *  t  e  *  o  n  e  *  i  *  *  o  i  n  t  *  e 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 v  e  n  t  w  o  f  i  v  e  n  o  *  t  h  a  n  d  f  i  v  e  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  w  o  d  o  t  f  i  v  e  f  i  v  e  f  o  *  *  w  e  *  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  h  e  i  *  a  t  t  h  e  *  a  *  e  o  f  a  l  a  *  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 g  e  t  *  e  e  w  h  i  l  e  i  n  t  h  e  a  *  e  a  *  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 *  *  o  *  *  *  e  *  *  e  c  t  *  t  o  t  h  e  v  e  t  e  

ET TU II RQ  
 *  a  n  *  

Now, the end of line 2 and the beginning of line 3 reads “*oint*even”. This suggests the phrase “point seven” and provides TQ = p and RQ = s.

Further the end of line 4 reads “fo**we*t”. This suggests the phrase “four west”. This provides IL = u and ET = r.

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :p a * e t   
I :* o n u *   
G :c d f g h   
E :i k l * r   
R :s v w * *   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  h  e  i  s  d  e  c  i  *  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  *  i  n  u  t  e  s  o  n  e  s  i  *  p  o  i  n  t  s  e 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 v  e  n  t  w  o  f  i  v  e  n  o  r  t  h  a  n  d  f  i  v  e  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  w  o  d  o  t  f  i  v  e  f  i  v  e  f  o  u  r  w  e  s  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  h  e  i  s  a  t  t  h  e  *  a  s  e  o  f  a  l  a  r  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 g  e  t  r  e  e  w  h  i  l  e  i  n  t  h  e  a  r  e  a  p  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 *  *  o  u  r  r  e  s  p  e  c  t  s  t  o  t  h  e  v  e  t  e  

ET TU II RQ  
 r  a  n  s  

Finally, line two suggests the phrase “minutes one six” providing IT = m and TI = x. Line 5 suggests the phrase “the base of” providing IQ = b. By the process of elimination this leaves EL = q, RL = y, and RT = z.

Using these identities and substituting we obtain:

   Q U I L T  
  ----------
T :p a x e t   
I :b o n u m   
G :c d f g h   
E :i k l q r   
R :s v w y z   
EI IU GQ TU TT EQ IU II IU GI GQ TU GQ GT TL EQ RQ GU TL GQ EQ IT
 l  o  c  a  t  i  o  n  o  f  c  a  c  h  e  i  s  d  e  c  i  m  

TU EI IT EQ II IL TT TL RQ IU II TL RQ EQ TI TQ IU EQ II TT RQ TL
 a  l  m  i  n  u  t  e  s  o  n  e  s  i  x  p  o  i  n  t  s  e 

RU TL II TT RI IU GI EQ RU TL II IU ET TT GT TU II GU GI EQ RU TL
 v  e  n  t  w  o  f  i  v  e  n  o  r  t  h  a  n  d  f  i  v  e  

TT RI IU GU IU TT GI EQ RU TL GI EQ RU TL GI IU IL ET RI TL RQ TT
 t  w  o  d  o  t  f  i  v  e  f  i  v  e  f  o  u  r  w  e  s  t   

GQ TU GQ GT TL EQ RQ TU TT TT GT TL IQ TU RQ TL IU GI TU EI TU ET
 c  a  c  h  e  i  s  a  t  t  h  e  b  a  s  e  o  f  a  l  a  r  

GL TL TT ET TL TL RI GT EQ EI TL EQ II TT GT TL TU ET TL TU TQ TU
 g  e  t  r  e  e  w  h  i  l  e  i  n  t  h  e  a  r  e  a  p  a

RL RL IU IL ET ET TL RQ TQ TL GQ TT RQ TT IU TT GT TL RU TL TT TL
 y  y  o  u  r  r  e  s  p  e  c  t  s  t  o  t  h  e  v  e  t  e  

ET TU II RQ  
 r  a  n  s  

The plain text is now completely decrypted. Note that the key square is a homage to the local geocachers TigQuilt and Pax Et Bonum. The cache coordinates are adjacent to Knox Lane. Hence, the title of the cache “Hard Knocks” is a pun.