Subsection3.1Enciphering the Transposition¶ permalink
A New Method How to Write Secretly by the Art of Combinations
To write by the method proposed, a certain number of letters are combined to lock and unlock the epistle. The differences of writing down the positions [of the letters] ... may be varied to a vast number; ...
The order of rows is agreed upon in parting.
The number of letters combined, which is the key, may be expressed in the epistle by some mathematical figure, as \(\scriptsize\triangle\) for three letters, \(\Box\) for 4, etc. or by some private mark.
They [the individuals communicating] frame a rectangular table of as many columns as there are letters combined.
The letters so combined are placed in their natural order along the top of the table.
Having determined of how many lines the table shall consist, the order of combinations agreed upon is set down in a row in the first column towards the left hand; as you may see in the following table.
The table being thus prepared for writing, they observe the order of the combinations, and write according to its direction.
When they have placed one letter of every column of all the lines, they begin a new, and so go on until the writing be finished.
And lastly, they take the letters out of the table according to their partitions, as so many barbarous words, upon paper apart and send it to the confidant. - John Falconer
Let's see if you can follow Falconer's directions. Below I set up Table 3.1 according to his description in steps (5) and (6) and used it to encipher the pangram “the quick brown fox jumps over the lazy sleeping dog.”
“\(\scriptsize\triangle\) EUS HJY TXZ UPE ISE QML COP BEN KVI WHO RRG OTD FL NEG OA”
Refection Questions:
Looking at Table 3.2 why is the from the beginning of the sentence written backwards? (Be sure to look carefully at the letters to the left of the row when answering.)
In the second row of the same table why is qui from quick written in the order it is written? (Again, be sure to look carefully at the letters to the left of the row when answering.)
In the last row we put down n from brown and the fo from fox, why are they in the order they are in and looking at the next table (Table 3.3) where do we put the x from fox and why?
How do we finish writing the rest of the message into the boxes in the table?
Refection Questions: Looking at the final message why is there a little triangle at the start of the message and why were the blocks of letters written in the order they were written?
In what ways is this different from other ciphers we have looked at? (Hint: in this cipher what does cipher text E represent, or cipher text F?)
Subsection3.2Decrypting the Transposition
After walking us through a careful exposition of how to find all the factors of a number and encouraging us to use this in order to find out the number of letters used for our key, Falconer lets us know that there is effectively an easier way to attack his own cipher.
“or rather for dispatch, take out the seeming words, and write them down in [columns] beginning at the first, and then proceed to the second, third, fourth, fifth, etc., until you have gone through them” - John Falconer
Let's follow Falconer's advice using the message we used when we introduced his cipher in Section 3:
“\(\scriptsize\triangle\) EUS HJY TXZ UPE ISE QML COP BEN KVI WHO RRG OTD FL NEG OA”
Which transforms to the following when we rewrite it following Falconer's directions:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
E
H
T
U
I
Q
C
B
K
W
R
O
F
N
O
2
U
J
X
P
S
M
O
E
V
H
R
T
L
E
A
3
S
Y
Z
E
E
L
P
N
I
O
G
D
\(\cdot\)
G
\(\cdot\)
Table3.4Transpose Table for a Falconer Cipher
“1. Search in the several lines for some of the particles [(words or n-grams)] of that language you shall suppose the epistle to have been writ in. If in English, make suppositions, e.g. for such little words as the, that, for, of, to, and, etc. and the like, without some of which no man can well express business of any moment.”
“2. Having supposed in any of the lines; for some one of those mentioned, or the like particles, you may prove the truth of your supposition, by taking out the opposite letters of all the lines: And if they do not make words, or syllables, or produce such letters as can probably follow one another in that order, your first supposition is false, and you must suppose anew.”
“3. Having by fresh suppositions found some useful word: And the letters of the other lines (in the same order) agreeing, the words or syllables arising from them, will direct you to some new [column] that goes before or after in the true order: And thus you may proceed till you have found out the whole writing, which by this time will be no great difficulty.” - John Falconer
Taking Falconer's advice from step one look at each line of Table 3.4 we created above as you consider the following.
Are there three letters in line one which we would expect to go together assuming this is written in English?
There is a Q in row one column six, what needs to come after that? What column is it in?
Looking now at line two what three letters do we again see that should go together?
In line three are there three letters you could put together to get a common English ending? Are there other letters you could arrange with them in order to get an entire word?
Finally, columns thirteen and fourteen only have two characters in them each, where do you think they belong if we were to rearrange the lines?
Use the observations you just made in order to rearrange the columns in the table, you should make a copy of this table to help you.
Col's
1
2
3
Table3.5Blank Transpose Table for a Falconer Cipher
Test your understanding by working on the following exercises of increasing size, be patient they are finicky and time consuming. As a hint, all of the quotes contain the word WINTER.
Transpose each block, you will end up with a table of four lines and fifteen columns.
Can you find common or even not so common words or combinations in any lines?
When you rearrange the columns do you see more words?
Stay organized, if you don't keep things lined up this will never work
A few closing observations:
Look carefully at how the blocks of characters (columns in the tables) are grouped after they are deciphered, i.e. did you ever get column one next to column thirteen or fourteen?
Why do you think this happened? How might it be related to the number of key letters?
Given your answers to the previous questions, how can we make use of Falconer's strategies for finding divisors and the number of key letters?
